{"_id": "0", "title": "CARD_EQ_CARD_IMP", "text": "|- (!s. (!t. (((FINITE t) /\\ (s =_c t)) ==> ((CARD s) = (CARD t)))))"}
{"_id": "1", "title": "MDIST_REFL", "text": "|- (!m. (!x. ((x IN (mspace m)) ==> (((mdist m) (x , x)) = (real_of_num (NUMERAL _0))))))"}
{"_id": "2", "title": "tendsto", "text": "|- (!f. (!l. (!net. ((((--> f) l) net) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\x. ((real_lt (distance ((f x) , l))) e))) net)))))))"}
{"_id": "3", "title": "ADD_AC", "text": "|- (((m + n) = (n + m)) /\\ ((((m + n) + p) = (m + (n + p))) /\\ ((m + (n + p)) = (n + (m + p)))))"}
{"_id": "4", "title": "Conforming.conforming_fan", "text": "|- (!x. (!V. (!E. ((conforming_fan (x , (V , E))) = ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((fan80 (x , (V , E))) /\\ ((conforming_bijection_fan (x , (V , E))) /\\ ((conforming_half_space_fan (x , (V , E))) /\\ ((conforming_solid_angle_fan (x , (V , E))) /\\ (conforming_diagonal_fan (x , (V , E))))))))))))"}
{"_id": "5", "title": "IM_DEF", "text": "|- (!z. ((Im z) = (z $ (NUMERAL (BIT0 (BIT1 _0))))))"}
{"_id": "6", "title": "CLOSURE_HALFSPACE_LT", "text": "|- (!a. (!b. ((~ (a = (vec (NUMERAL _0)))) ==> ((closure (GSPEC (\\GEN%PVAR%806. (?x. (((SETSPEC GEN%PVAR%806) ((real_lt ((dot a) x)) b)) x))))) = (GSPEC (\\GEN%PVAR%807. (?x. (((SETSPEC GEN%PVAR%807) ((real_le ((dot a) x)) b)) x))))))))"}
{"_id": "7", "title": "INT_EQ_IMP_LE", "text": "|- (!x. (!y. ((x = y) ==> ((int_le x) y))))"}
{"_id": "8", "title": "LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_BOREL", "text": "|- (!f. (!s. (!t. ((((measurable_on f) s) /\\ ((lebesgue_measurable s) /\\ (borel t))) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3660. (?x. (((SETSPEC GEN%PVAR%3660) ((x IN s) /\\ ((f x) IN t))) x)))))))))"}
{"_id": "9", "title": "IN_UNION", "text": "|- (!s. (!t. (!x. ((x IN (s UNION t)) = ((x IN s) \\/ (x IN t))))))"}
{"_id": "10", "title": "REALLIM_ATREAL", "text": "|- (!f. (!l. (!a. ((((---> f) l) (atreal a)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\\ ((real_lt (real_abs ((real_sub x) a))) d)) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))))"}
{"_id": "11", "title": "INF", "text": "|- (!s. (((~ (s = EMPTY)) /\\ (?b. (!x. ((x IN s) ==> ((real_le b) x))))) ==> ((!x. ((x IN s) ==> ((real_le (inf s)) x))) /\\ (!b. ((!x. ((x IN s) ==> ((real_le b) x))) ==> ((real_le b) (inf s)))))))"}
{"_id": "12", "title": "Collect_geom2.VBVYGGT", "text": "|- (!v1. (!v2. (!v3. (!v4. ((((CARD (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\\ (~ (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))))) ==> (((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) IN ((hull affine) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) /\\ ((?r. (!v. ((v IN (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) ==> (r = (distance ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) , v)))))) /\\ ((!p. (((p IN ((hull affine) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) /\\ (?r. (!v. ((v IN (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) ==> (r = (distance (p , v))))))) ==> (p = (circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))))) /\\ ((LET (\\x12. (LET_END ((LET (\\x13. (LET_END ((LET (\\x14. (LET_END ((LET (\\x23. (LET_END ((LET (\\x24. (LET_END ((LET (\\x34. (LET_END ((LET (\\chi11. (LET_END ((LET (\\chi22. (LET_END ((LET (\\chi33. (LET_END ((LET (\\chi44. (LET_END ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((delta x12) x13) x14) x23) x24) x34)))) ((vector_add ((% chi11) v1)) ((vector_add ((% chi22) v2)) ((vector_add ((% chi33) v3)) ((% chi44) v4))))))))) ((((((chi x34) x24) x14) x23) x13) x12))))) ((((((chi x34) x13) x23) x14) x24) x12))))) ((((((chi x12) x24) x23) x14) x13) x34))))) ((((((chi x12) x13) x14) x23) x24) x34))))) ((real_pow (distance (v3 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v2 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0)))))))))))))"}
{"_id": "13", "title": "Hypermap.lemma_in_face", "text": "|- (!H. (!x. (!n. ((((POWER (face_map H)) n) x) IN ((face H) x)))))"}
{"_id": "14", "title": "VSUM_SING_NUMSEG", "text": "|- (((vsum (n .. n)) f) = (f n))"}
{"_id": "15", "title": "MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED", "text": "|- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!t. ((closed t) ==> (measurable (GSPEC (\\GEN%PVAR%3646. (?x. (((SETSPEC GEN%PVAR%3646) ((x IN s) /\\ ((f x) IN t))) x)))))))))))"}
{"_id": "16", "title": "SUBSET_ANTISYM", "text": "|- (!s. (!t. (((s SUBSET t) /\\ (t SUBSET s)) ==> (s = t))))"}
{"_id": "17", "title": "Ssrnat.mul0n", "text": "|- (!n. (((NUMERAL _0) * n) = (NUMERAL _0)))"}
{"_id": "18", "title": "CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE", "text": "|- (!f. (!s. ((!x. (((cauchy x) /\\ (!n. ((x n) IN s))) ==> (cauchy (f o x)))) ==> (?g. (((continuous_on g) (closure s)) /\\ (!x. ((x IN s) ==> ((g x) = (f x)))))))))"}
{"_id": "19", "title": "Appendix.BBprime2_v39", "text": "|- (!vv. (!s. (((BBprime2_v39 s) vv) = (((BBprime_v39 s) vv) /\\ (((BBindex_v39 s) vv) = (BBindex_min_v39 s))))))"}
{"_id": "20", "title": "SERIES_DIRICHLET_COMPLEX_VERY_EXPLICIT", "text": "|- (!f. (!g. (!B. (!p. ((((real_lt (real_of_num (NUMERAL _0))) B) /\\ (((NUMERAL (BIT1 _0)) <= p) /\\ ((!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) f))) B)))) /\\ ((!n. ((p <= n) ==> ((real (g n)) /\\ ((real_le (real_of_num (NUMERAL _0))) (Re (g n)))))) /\\ (!n. ((p <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n))))))))) ==> (!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) (\\k. ((complex_mul (f k)) (g k)))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul B) (vector_norm (g m)))))))))))))"}
{"_id": "21", "title": "REAL_MAX_ACI", "text": "|- ((((real_max x) y) = ((real_max y) x)) /\\ ((((real_max ((real_max x) y)) z) = ((real_max x) ((real_max y) z))) /\\ ((((real_max x) ((real_max y) z)) = ((real_max y) ((real_max x) z))) /\\ ((((real_max x) x) = x) /\\ (((real_max x) ((real_max x) y)) = ((real_max x) y))))))"}
{"_id": "22", "title": "MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ", "text": "|- (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> ((measurable s) = (measurable t)))))"}
{"_id": "23", "title": "Nonlin_def.dih_x_div_sqrtdelta_posbranch", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((LET (\\d_x4. (LET_END ((LET (\\d. (LET_END ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1))) d_x4)) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x1) d))) ((real_pow d_x4) (NUMERAL (BIT0 (BIT1 _0)))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) ((((((delta_x4 x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "24", "title": "IM_COMPLEX_INV_LE_0", "text": "|- (!z. (((real_le (Im (complex_inv z))) (real_of_num (NUMERAL _0))) = ((real_le (real_of_num (NUMERAL _0))) (Im z))))"}
{"_id": "25", "title": "REAL_SUB_ADD2", "text": "|- (!x. (!y. (((real_add y) ((real_sub x) y)) = x)))"}
{"_id": "26", "title": "REAL_INTEGRAL", "text": "|- (((real_integrable_on f) s) ==> (((real_integral s) f) = (drop ((integral ((IMAGE lift) s)) (lift o (f o drop))))))"}
{"_id": "27", "title": "REAL_ADD_AC", "text": "|- ((((real_add m) n) = ((real_add n) m)) /\\ ((((real_add ((real_add m) n)) p) = ((real_add m) ((real_add n) p))) /\\ (((real_add m) ((real_add n) p)) = ((real_add n) ((real_add m) p)))))"}
{"_id": "28", "title": "absolutely_real_integrable_on", "text": "|- (!f. (!s. (((absolutely_real_integrable_on f) s) = (((real_integrable_on f) s) /\\ ((real_integrable_on (\\x. (real_abs (f x)))) s)))))"}
{"_id": "29", "title": "real_convex_on", "text": "|- (!s. (!f. (((real_convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\\ ((y IN s) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))"}
{"_id": "30", "title": "CONTINUOUS_LIFT_PRODUCT", "text": "|- (!net. (!f. (!t. (((FINITE t) /\\ (!i. ((i IN t) ==> ((continuous (\\x. (lift ((f x) i)))) net)))) ==> ((continuous (\\x. (lift ((product t) (f x))))) net)))))"}
{"_id": "31", "title": "MIN", "text": "|- (!m. (!n. (((MIN m) n) = (((COND (m <= n)) m) n))))"}
{"_id": "32", "title": "CONTINUOUS_AT_CEXP", "text": "|- (!z. ((continuous cexp) (at z)))"}
{"_id": "33", "title": "Collect_geom.DELTA_POS_4POINTS", "text": "|- (!x1. (!x2. (!x3. (!x4. ((real_le (real_of_num (NUMERAL _0))) ((((((delta ((real_pow (distance (x1 , x2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x1 , x3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x1 , x4))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x2 , x3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x2 , x4))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x3 , x4))) (NUMERAL (BIT0 (BIT1 _0))))))))))"}
{"_id": "34", "title": "CSQRT_PRINCIPAL", "text": "|- (!z. (((real_lt (real_of_num (NUMERAL _0))) (Re (csqrt z))) \\/ (((Re (csqrt z)) = (real_of_num (NUMERAL _0))) /\\ ((real_le (real_of_num (NUMERAL _0))) (Im (csqrt z))))))"}
{"_id": "35", "title": "Hypermap_iso.hyp_iso_card_components", "text": "|- (!H. (!G. (!f. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> (((CARD ((face H) x)) = (CARD ((face G) (f x)))) /\\ (((CARD ((node H) x)) = (CARD ((node G) (f x)))) /\\ ((CARD ((edge H) x)) = (CARD ((edge G) (f x))))))))))))"}
{"_id": "36", "title": "DIM_EQ_SPAN", "text": "|- (!s. (!t. (((s SUBSET t) /\\ ((dim t) <= (dim s))) ==> ((span s) = (span t)))))"}
{"_id": "37", "title": "CLOSED_TRANSLATION", "text": "|- (!s. (!a. ((closed s) ==> (closed ((IMAGE (\\x. ((vector_add a) x))) s)))))"}
{"_id": "38", "title": "Lp_ineqs_def.y7_fan", "text": "|- (!V. (!E. (!x. (((y7_fan (V , E)) x) = (vector_norm (FST (((POWER (inverse (f_fan_pair_ext (V , E)))) (NUMERAL (BIT0 (BIT1 _0)))) x)))))))"}
{"_id": "39", "title": "IN_ELIM_PAIR_THM", "text": "|- (!P. (!a. (!b. (((a , b) IN (GSPEC (\\GEN%PVAR%18. (?x. (?y. (((SETSPEC GEN%PVAR%18) ((P x) y)) (x , y))))))) = ((P a) b)))))"}
{"_id": "40", "title": "Hypermap_and_fan.NODE_HYPERMAP_OF_FAN_ALT", "text": "|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((v , w) IN (dart1_of_fan (V , E)))) ==> (((node (hypermap_of_fan (V , E))) (v , w)) = (GSPEC (\\GEN%PVAR%5635. (?u. (((SETSPEC GEN%PVAR%5635) (u IN (((set_of_edge v) V) E))) (v , u)))))))))))"}
{"_id": "41", "title": "REAL_INTEGRABLE_EQ", "text": "|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((real_integrable_on f) s)) ==> ((real_integrable_on g) s)))))"}
{"_id": "42", "title": "PAIR", "text": "|- (!x. (((FST x) , (SND x)) = x))"}
{"_id": "43", "title": "COMPLEX_NORM_NZ", "text": "|- (!z. (((real_lt (real_of_num (NUMERAL _0))) (vector_norm z)) = (~ (z = (Cx (real_of_num (NUMERAL _0)))))))"}
{"_id": "44", "title": "SYMMETRIC_MATRIX", "text": "|- (!A. (((transp A) = A) = ((adjoint (\\x. ((matrix_vector_mul A) x))) = (\\x. ((matrix_vector_mul A) x)))))"}
{"_id": "45", "title": "AFFINE_BOUNDED_EQ_LOWDIM", "text": "|- (!s. ((affine s) ==> ((bounded s) = ((int_le (aff_dim s)) (int_of_num (NUMERAL _0))))))"}
{"_id": "46", "title": "Float.FLOAT_ADD_PP", "text": "|- (!a. (!b. (!m. (!n. ((n <= m) ==> (((real_add ((float a) (int_of_num m))) ((float b) (int_of_num n))) = ((float ((int_add ((int_mul (int_of_num ((NUMERAL (BIT0 (BIT1 _0))) EXP (m - n)))) a)) b)) (int_of_num n))))))))"}
{"_id": "47", "title": "DIVISION_OF_NONTRIVIAL", "text": "|- (!s. (!a. (!b. ((((division_of s) (closed_interval ((CONS (a , b)) NIL))) /\\ (~ ((content (closed_interval ((CONS (a , b)) NIL))) = (real_of_num (NUMERAL _0))))) ==> ((division_of (GSPEC (\\GEN%PVAR%2756. (?k. (((SETSPEC GEN%PVAR%2756) ((k IN s) /\\ (~ ((content k) = (real_of_num (NUMERAL _0)))))) k))))) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "48", "title": "TRIVIAL_LIMIT_SEQUENTIALLY", "text": "|- (~ (trivial_limit sequentially))"}
{"_id": "49", "title": "TC_TRANS_R", "text": "|- (!R. (!x. (!y. (!z. ((((R x) y) /\\ (((TC R) y) z)) ==> (((TC R) x) z))))))"}
{"_id": "50", "title": "COUNTABLE", "text": "|- (!t. ((COUNTABLE t) = (UNIV >=_c t)))"}
{"_id": "51", "title": "COMPLEX_POW_INV", "text": "|- (!x. (!n. (((complex_pow (complex_inv x)) n) = (complex_inv ((complex_pow x) n)))))"}
{"_id": "52", "title": "DOT_CAUCHY_SCHWARZ_EQUAL", "text": "|- (!x. (!y. ((((real_pow ((dot x) y)) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_mul ((dot x) x)) ((dot y) y))) = (collinear ((vec (NUMERAL _0)) INSERT (x INSERT (y INSERT EMPTY)))))))"}
{"_id": "53", "title": "CONTINUOUS_ATREAL", "text": "|- (!f. (!x. (((continuous f) (atreal x)) = (((--> f) (f x)) (atreal x)))))"}
{"_id": "54", "title": "HAS_REAL_INTEGRAL_RESTRICT_INTER", "text": "|- (!f. (!s. (!t. ((((has_real_integral (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) i) t) = (((has_real_integral f) i) (s INTER t))))))"}
{"_id": "55", "title": "HAS_REAL_DERIVATIVE_ADD", "text": "|- (!f. (!f'. (!g. (!g'. (!net. (((((has_real_derivative f) f') net) /\\ (((has_real_derivative g) g') net)) ==> (((has_real_derivative (\\x. ((real_add (f x)) (g x)))) ((real_add f') g')) net)))))))"}
{"_id": "56", "title": "ROW_TRANSP", "text": "|- (!A. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (((row i) (transp A)) = ((column i) A)))))"}
{"_id": "57", "title": "Trigonometry2.LEMMA_SUM_ALL_OVER_CYCLIC_SET", "text": "|- (!W. (!xicm. (((FINITE W) /\\ (((CARD W) = n) /\\ ((!x. ((W x) ==> (~ (x = (vec (NUMERAL _0)))))) /\\ (((polar_cycle_on xicm) W) /\\ (W p))))) ==> ((?p. (?q. ((W p) /\\ ((W q) /\\ (~ (p = q)))))) ==> (((sum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\\i. ((arg_diff (((ITER i) xicm) p)) (((ITER (i + (NUMERAL (BIT1 _0)))) xicm) p)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))))"}
{"_id": "58", "title": "VECTOR_NEG_COMPONENT", "text": "|- (!x. (!i. (((vector_neg x) $ i) = (real_neg (x $ i)))))"}
{"_id": "59", "title": "Hypermap_iso.hyp_iso_inverse_comm", "text": "|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((((inverse (edge_map G)) (f x)) = (f ((inverse (edge_map H)) x))) /\\ ((((inverse (node_map G)) (f x)) = (f ((inverse (node_map H)) x))) /\\ (((inverse (face_map G)) (f x)) = (f ((inverse (face_map H)) x)))))))))))"}
{"_id": "60", "title": "AR_TRANSLATION", "text": "|- (!a. (!s. ((AR ((IMAGE (\\x. ((vector_add a) x))) s)) = (AR s))))"}
{"_id": "61", "title": "AFFINE_INTERS", "text": "|- ((!s. ((s IN f) ==> (affine s))) ==> (affine (INTERS f)))"}
{"_id": "62", "title": "Pack2.KIUMVTC", "text": "|- (!p. (!r. (!S. ((packing S) ==> (FINITE (S INTER (ball (p , r))))))))"}
{"_id": "63", "title": "LIM_COMPLEX_RMUL", "text": "|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\\x. ((complex_mul (f x)) c))) ((complex_mul l) c)) net)))))"}
{"_id": "64", "title": "REAL_LE_MUL", "text": "|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)))))"}
{"_id": "65", "title": "CLOSED_INTER", "text": "|- (!s. (!t. (((closed s) /\\ (closed t)) ==> (closed (s INTER t)))))"}
{"_id": "66", "title": "Seq.subseq_uniq", "text": "|- (!s1. (!s2. (((subseq s1) s2) ==> ((uniq s2) ==> (uniq s1)))))"}
{"_id": "67", "title": "Seq.filter_subseq", "text": "|- (!a. (!s. ((subseq ((filter a) s)) s)))"}
{"_id": "68", "title": "SIMPLEX_TRANSLATION_EQ", "text": "|- (!a. (!s. (!n. (((simplex n) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((simplex n) s)))))"}
{"_id": "69", "title": "DIFFERENTIABLE_NORM_AT", "text": "|- (!a. ((~ (a = (vec (NUMERAL _0)))) ==> ((differentiable (\\x. (lift (vector_norm x)))) (at a))))"}
{"_id": "70", "title": "CARD_EXP_CANTOR", "text": "|- (!s. (s <_c ((^_c UNIV) s)))"}
{"_id": "71", "title": "NOT_IMP", "text": "|- (!t1. (!t2. ((~ (t1 ==> t2)) = (t1 /\\ (~ t2)))))"}
{"_id": "72", "title": "Ssrnat.muln0", "text": "|- (!n. ((n * (NUMERAL _0)) = (NUMERAL _0)))"}
{"_id": "73", "title": "Sphere.acs_sqrt_x1_d4", "text": "|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((acs_sqrt_x1_d4 x1) x2) x3) x4) x5) x6) = (acs ((real_div (sqrt x1)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))"}
{"_id": "74", "title": "LE_ADDR", "text": "|- (!m. (!n. (n <= (m + n))))"}
{"_id": "75", "title": "Audit_formal_proof.chk_ball_annulus_def", "text": "|- (!v. ((v IN ball_annulus) = (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v)) /\\ ((real_le (vector_norm v)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))"}
{"_id": "76", "title": "COMPACT_INTER", "text": "|- (!s. (!t. (((compact s) /\\ (compact t)) ==> (compact (s INTER t)))))"}
{"_id": "77", "title": "Add_triangle.f_split_fan_face", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face (hypermap_of_fan (V , E))) (v , w))))) ==> ((((LET (\\w1. (\\d2. (\\f. (LET_END (((f (v , w)) = (w , w1)) /\\ (((f (w , w1)) = (w1 , v)) /\\ (((f (w1 , w)) = d2) /\\ (((f ((((((sigma_fan (vec (NUMERAL _0))) V) E) w1) v) , w1)) = (w1 , w)) /\\ (!d. (((d IN (dart_of_fan ((split_fan_face (V , E)) (v , w)))) /\\ (~ ((d = (v , w)) \\/ ((d = (w , w1)) \\/ ((d = (w1 , w)) \\/ (d = ((((((sigma_fan (vec (NUMERAL _0))) V) E) w1) v) , w1))))))) ==> ((f d) = ((f_fan_pair (V , E)) d))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) ((f_fan_pair (V , E)) (v , w))) (f_fan_pair ((split_fan_face (V , E)) (v , w)))))))))))"}
{"_id": "78", "title": "Lp_ineqs_proofs.f_fan_pair_ext_power_in_dart", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN (dart_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) IN (dart_of_fan (V , E)))))))))"}
{"_id": "79", "title": "ORTHOGONAL_TRANSFORMATION_INJECTIVE", "text": "|- (!f. ((orthogonal_transformation f) ==> (!x. (!y. (((f x) = (f y)) ==> (x = y))))))"}
{"_id": "80", "title": "CLOSURE_EQ_EMPTY", "text": "|- (!s. (((closure s) = EMPTY) = (s = EMPTY)))"}
{"_id": "81", "title": "Fan.MONO_SIGMA_FAN", "text": "|- (!x. (!V. (!E. (!v. (!u. (!w. (((FAN (x , (V , E))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((v INSERT (w INSERT EMPTY)) IN E) /\\ ((((((sigma_fan x) V) E) v) u) = (((((sigma_fan x) V) E) v) w))))) ==> (u = w))))))))"}
{"_id": "82", "title": "Hypermap.node", "text": "|- (!H. (!x. (((node H) x) = ((orbit_map (node_map H)) x))))"}
{"_id": "83", "title": "Trigonometry2.ups_x", "text": "|- (!x1. (!x2. (!x6. ((((ups_x x1) x2) x6) = ((real_add ((real_sub ((real_sub ((real_mul (real_neg x1)) x1)) ((real_mul x2) x2))) ((real_mul x6) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x1) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x1) x2))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x2) x6)))))))))"}
{"_id": "84", "title": "Sphere.hplus", "text": "|- (hplus = ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))"}
{"_id": "85", "title": "real_measurable", "text": "|- (!s. ((real_measurable s) = (?m. ((has_real_measure s) m))))"}
{"_id": "86", "title": "SPAN_0", "text": "|- ((vec (NUMERAL _0)) IN (span s))"}
{"_id": "87", "title": "DIMINDEX_4", "text": "|- ((dimindex UNIV) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))"}
{"_id": "88", "title": "FORALL_DROP", "text": "|- ((!x. (P x)) = (!x. (P (drop x))))"}
{"_id": "89", "title": "MIDPOINT_IN_SEGMENT", "text": "|- ((!a. (!b. ((midpoint (a , b)) IN (closed_segment ((CONS (a , b)) NIL))))) /\\ (!a. (!b. (((midpoint (a , b)) IN (open_segment (a , b))) = (~ (a = b))))))"}
{"_id": "90", "title": "RELATIVE_ORIENTATION", "text": "|- (!A. (!f. (!s. (((rows A) = (GSPEC (\\GEN%PVAR%2550. (?v. (((SETSPEC GEN%PVAR%2550) ((extreme_point_of v) s)) v))))) ==> (((relative_orientation f) s) = (real_sgn ((real_div (det ((maprows f) A))) (det A))))))))"}
{"_id": "91", "title": "REAL_POLY_NEG_CLAUSES", "text": "|- ((!x. ((real_neg x) = ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x))) /\\ (!x. (!y. (((real_sub x) y) = ((real_add x) ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) y))))))"}
{"_id": "92", "title": "has_bounded_variation_on", "text": "|- (!f. (!s. (((has_bounded_variation_on f) s) = ((has_bounded_setvariation_on (\\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))) s))))"}
{"_id": "93", "title": "REAL_SUB_RDISTRIB", "text": "|- (!x. (!y. (!z. (((real_mul ((real_sub x) y)) z) = ((real_sub ((real_mul x) z)) ((real_mul y) z))))))"}
{"_id": "94", "title": "Seq.nseq", "text": "|- (!n. (!x. (((nseq n) x) = (((ncons n) x) NIL))))"}
{"_id": "95", "title": "Hypermap_and_fan.FULLY_SURROUNDED", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) = (!v. ((v IN V) ==> ((surrounded_node (V , E)) v)))))))"}
{"_id": "96", "title": "Ssrnat.mul1n", "text": "|- (!n. (((NUMERAL (BIT1 _0)) * n) = n))"}
{"_id": "97", "title": "ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_PAIRWISE", "text": "|- (!A. ((orthogonal_matrix A) = (((CARD (rows A)) = (dimindex UNIV)) /\\ ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((vector_norm ((row i) A)) = (real_of_num (NUMERAL (BIT1 _0)))))) /\\ ((pairwise orthogonal) (rows A))))))"}
{"_id": "98", "title": "Appendix.s_init_list_v39", "text": "|- (s_init_list_v39 = ((LET (\\upperbd. (LET_END ((LET (\\a_pro. (LET_END ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) (d_tame (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (d_tame (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (d_tame (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) (d_tame (NUMERAL (BIT1 (BIT1 _0))))) (((cs_adj (NUMERAL (BIT1 (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) (((cs_adj (NUMERAL (BIT1 (BIT1 _0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) sqrt8)) (((cs_adj (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))) (((cs_adj (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((((a_pro (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((((a_pro (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) sqrt8) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) ((CONS ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((((a_pro (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) sqrt8)) ((((a_pro (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) sqrt8) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) upperbd))) NIL))))))))))) (\\k. (\\p. (\\a1. (\\a2. (\\i. (\\j. (((COND ((i MOD k) = (j MOD k))) (real_of_num (NUMERAL _0))) (((COND (((i MOD k) INSERT ((j MOD k) INSERT EMPTY)) = ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT EMPTY)))) p) (((COND (((j MOD k) = ((SUC i) MOD k)) \\/ (((SUC j) MOD k) = (i MOD k)))) a1) a2))))))))))))) (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))))"}
{"_id": "99", "title": "NUMSEG_OFFSET_IMAGE", "text": "|- (!m. (!n. (!p. (((m + p) .. (n + p)) = ((IMAGE (\\i. (i + p))) (m .. n))))))"}
{"_id": "100", "title": "CONVERGENT_IMP_BOUNDED", "text": "|- (!s. (!l. ((((--> s) l) sequentially) ==> (bounded ((IMAGE s) UNIV)))))"}
{"_id": "101", "title": "HAS_DERIVATIVE_CONST", "text": "|- (!c. (!net. (((has_derivative (\\x. c)) (\\h. (vec (NUMERAL _0)))) net)))"}
{"_id": "102", "title": "Collect_geom.DELTA_COEFS", "text": "|- (!x12. (!x13. (!x14. (!x23. (!x24. (!x34. (((((((delta x12) x13) x14) x23) x24) x34) = ((real_add ((real_mul (real_neg x12)) ((real_pow x34) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul (((((b_coef x12) x13) x14) x23) x24)) x34)) (((((c_coef x12) x13) x14) x23) x24))))))))))"}
{"_id": "103", "title": "LIM_NULL_COMPLEX_SUB", "text": "|- (!net. (!f. (!g. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> (\\x. ((vector_sub (f x)) (g x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))"}
{"_id": "104", "title": "CLOSED_IN_SUBSET_TRANS", "text": "|- (!s. (!t. (!u. ((((closed_in ((subtopology euclidean) u)) s) /\\ ((s SUBSET t) /\\ (t SUBSET u))) ==> ((closed_in ((subtopology euclidean) t)) s)))))"}
{"_id": "105", "title": "REAL_CONTINUOUS_AT_ID", "text": "|- (!x. ((real_continuous (\\x. x)) (atreal x)))"}
{"_id": "106", "title": "AFFINE_HULL_LINEAR_IMAGE", "text": "|- (!f. (!s. ((linear f) ==> (((hull affine) ((IMAGE f) s)) = ((IMAGE f) ((hull affine) s))))))"}
{"_id": "107", "title": "int_sgn_th", "text": "|- (!x. ((real_of_int (int_sgn x)) = (real_sgn (real_of_int x))))"}
{"_id": "108", "title": "ABS_DROP", "text": "|- (!x. ((vector_norm x) = (real_abs (drop x))))"}
{"_id": "109", "title": "ARC_LINEAR_IMAGE_EQ", "text": "|- (!f. (!g. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((arc (f o g)) = (arc g)))))"}
{"_id": "110", "title": "higher_complex_derivative", "text": "|- ((((higher_complex_derivative (NUMERAL _0)) f) = f) /\\ (!n. (((higher_complex_derivative (SUC n)) f) = (complex_derivative ((higher_complex_derivative n) f)))))"}
{"_id": "111", "title": "ANR_FROM_UNION_AND_INTER_LOCAL", "text": "|- (!s. (!t. ((((closed_in ((subtopology euclidean) (s UNION t))) s) /\\ (((closed_in ((subtopology euclidean) (s UNION t))) t) /\\ ((ANR (s UNION t)) /\\ (ANR (s INTER t))))) ==> ((ANR s) /\\ (ANR t)))))"}
{"_id": "112", "title": "CONTINUOUS_ON_CLOSEST_POINT", "text": "|- (!s. (!t. (((convex s) /\\ ((closed s) /\\ (~ (s = EMPTY)))) ==> ((continuous_on (closest_point s)) t))))"}
{"_id": "113", "title": "REAL_MIN_LE", "text": "|- (!x. (!y. (!z. (((real_le ((real_min x) y)) z) = (((real_le x) z) \\/ ((real_le y) z))))))"}
{"_id": "114", "title": "Functional_equation.functional_arclength_x_123'", "text": "|- (arclength_x_123' = (uni (acs , ((div6 ((add6 proj_x1) ((sub6 proj_x2) proj_x3))) (uni (sqrt , ((scalar6 ((mul6 proj_x1) proj_x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))"}
{"_id": "115", "title": "NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE", "text": "|- (!s. ((negligible s) ==> (lebesgue_measurable s)))"}
{"_id": "116", "title": "CONVEX_HULL_LINEAR_IMAGE", "text": "|- (!f. (!s. ((linear f) ==> (((hull convex) ((IMAGE f) s)) = ((IMAGE f) ((hull convex) s))))))"}
{"_id": "117", "title": "Xwitccn.PERIODIC_PROPERTY", "text": "|- (((~ (k = (NUMERAL _0))) /\\ ((periodic vv) k)) ==> (!i. ((vv (i MOD k)) = (vv i))))"}
{"_id": "118", "title": "DOT_POS_LT", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) ((dot x) x)) = (~ (x = (vec (NUMERAL _0))))))"}
{"_id": "119", "title": "LE_0", "text": "|- (!n. ((NUMERAL _0) <= n))"}
{"_id": "120", "title": "HAS_VECTOR_DERIVATIVE_CONST", "text": "|- (!c. (!net. (((has_vector_derivative (\\x. c)) (vec (NUMERAL _0))) net)))"}
{"_id": "121", "title": "REAL_SGN_POW", "text": "|- (!x. (!n. ((real_sgn ((real_pow x) n)) = ((real_pow (real_sgn x)) n))))"}
{"_id": "122", "title": "Misc_defs_and_lemmas.min_least", "text": "|- (!X. (!c. ((X c) ==> ((X (min_num X)) /\\ ((min_num X) <= c)))))"}
{"_id": "123", "title": "SUM_CASES", "text": "|- (!s. (!P. (!f. (!g. ((FINITE s) ==> (((sum s) (\\x. (((COND (P x)) (f x)) (g x)))) = ((real_add ((sum (GSPEC (\\GEN%PVAR%213. (?x. (((SETSPEC GEN%PVAR%213) ((x IN s) /\\ (P x))) x))))) f)) ((sum (GSPEC (\\GEN%PVAR%214. (?x. (((SETSPEC GEN%PVAR%214) ((x IN s) /\\ (~ (P x)))) x))))) g))))))))"}
{"_id": "124", "title": "independent", "text": "|- (!s. ((independent s) = (~ (dependent s))))"}
{"_id": "125", "title": "Seq.foldr", "text": "|- (((((foldr f) z0) ((CONS x) s)) = ((f x) (((foldr f) z0) s))) /\\ ((((foldr f) z0) NIL) = z0))"}
{"_id": "126", "title": "REAL_SUMMABLE", "text": "|- (((real_summable s) f) = ((summable s) (lift o f)))"}
{"_id": "127", "title": "DOT_RADD", "text": "|- (!x. (!y. (!z. (((dot x) ((vector_add y) z)) = ((real_add ((dot x) y)) ((dot x) z))))))"}
{"_id": "128", "title": "Real_ext.REAL_LE_LMUL_IMP", "text": "|- (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_le y) z)) ==> ((real_le ((real_mul x) y)) ((real_mul x) z))))))"}
{"_id": "129", "title": "CNJ_MUL", "text": "|- (!w. (!z. ((cnj ((complex_mul w) z)) = ((complex_mul (cnj w)) (cnj z)))))"}
{"_id": "130", "title": "Fatugpd.norm_mul_normalize", "text": "|- (!v. (((% (vector_norm v)) (normalize v)) = v))"}
{"_id": "131", "title": "Hypermap.lemma_transitive_permutation", "text": "|- (!L. (!x. (((in_dart_of_loop x) L) ==> ((dart_of_loop L) = ((orbit_map (loop_map L)) x)))))"}
{"_id": "132", "title": "COMPLEX_ADD_LID", "text": "|- (!x. (((vector_add (Cx (real_of_num (NUMERAL _0)))) x) = x))"}
{"_id": "133", "title": "Hypermap_and_fan.FINITE_ORBIT_MAP", "text": "|- (!s. (!f. (!x. (!n. (((FINITE s) /\\ (((permutes f) s) /\\ ((CARD ((orbit_map f) x)) = n))) ==> (((orbit_map f) x) = (GSPEC (\\GEN%PVAR%5631. (?k. (((SETSPEC GEN%PVAR%5631) (k < n)) (((POWER f) k) x)))))))))))"}
{"_id": "134", "title": "LEBESGUE_MEASURABLE_CLOSED_IN", "text": "|- (!s. (!t. ((((closed_in ((subtopology euclidean) s)) t) /\\ (lebesgue_measurable s)) ==> (lebesgue_measurable t))))"}
{"_id": "135", "title": "COMPLEX_NORM_POW", "text": "|- (!z. (!n. ((vector_norm ((complex_pow z) n)) = ((real_pow (vector_norm z)) n))))"}
{"_id": "136", "title": "Conforming.SIGMA_FAN_OF_FANADD_AT_POINT3", "text": "|- (!x. (!V. (!E. (!E1. (!v. (!u. (!w. (((FAN (x , (V , E))) /\\ ((FAN (x , (V , E1))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((fan80 (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))) ==> ((((((sigma_fan x) V) E1) w) v) = u)))))))))"}
{"_id": "137", "title": "SUBSET_UNION", "text": "|- ((!s. (!t. (s SUBSET (s UNION t)))) /\\ (!s. (!t. (s SUBSET (t UNION s)))))"}
{"_id": "138", "title": "REVERSEPATH_LINEAR_IMAGE", "text": "|- (!f. (!g. ((linear f) ==> ((reversepath (f o g)) = (f o (reversepath g))))))"}
{"_id": "139", "title": "TRANSITIVE_STEPWISE_LE", "text": "|- (!R. (((!x. ((R x) x)) /\\ ((!x. (!y. (!z. ((((R x) y) /\\ ((R y) z)) ==> ((R x) z))))) /\\ (!n. ((R n) (SUC n))))) ==> (!m. (!n. ((m <= n) ==> ((R m) n))))))"}
{"_id": "140", "title": "VECTOR_SUB_LZERO", "text": "|- (((vector_sub (vec (NUMERAL _0))) x) = (vector_neg x))"}
{"_id": "141", "title": "UNIONS_MONO_IMAGE", "text": "|- ((!x. ((x IN s) ==> ((f x) SUBSET (g x)))) ==> ((UNIONS ((IMAGE f) s)) SUBSET (UNIONS ((IMAGE g) s))))"}
{"_id": "142", "title": "OPEN_INTER", "text": "|- (!s. (!t. (((open s) /\\ (open t)) ==> (open (s INTER t)))))"}
{"_id": "143", "title": "Functional_equation.functional_gchi4_x'", "text": "|- (gchi4_x' = ((mul6 (uni (gchi , proj_y4))) dih4_x'))"}
{"_id": "144", "title": "SETDIST_SYM", "text": "|- (!s. (!t. ((setdist (s , t)) = (setdist (t , s)))))"}
{"_id": "145", "title": "PI_NZ", "text": "|- (~ (pi = (real_of_num (NUMERAL _0))))"}
{"_id": "146", "title": "NSUM_SING_NUMSEG", "text": "|- (!f. (!n. (((nsum (n .. n)) f) = (f n))))"}
{"_id": "147", "title": "AR_IMP_ABSOLUTE_RETRACT", "text": "|- (!s. (!u. (!s'. (((AR s) /\\ (((homeomorphic s) s') /\\ ((closed_in ((subtopology euclidean) u)) s'))) ==> ((retract_of s') u)))))"}
{"_id": "148", "title": "HAS_BOUNDED_VARIATION_DARBOUX_STRICT", "text": "|- (!f. (!a. (!b. (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = (?g. (?h. ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (g x))) (drop (g y)))))) /\\ ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_lt (drop x)) (drop y)))) ==> ((real_lt (drop (h x))) (drop (h y)))))) /\\ (!x. ((f x) = ((vector_sub (g x)) (h x))))))))))))"}
{"_id": "149", "title": "PATH_CONNECTED_PUNCTURED_CONVEX", "text": "|- (!s. (!a. (((convex s) /\\ (~ ((aff_dim s) = (int_of_num (NUMERAL (BIT1 _0)))))) ==> (path_connected (s DELETE a)))))"}
{"_id": "150", "title": "CONTINUOUS_AT_CSQRT", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((continuous csqrt) (at z))))"}
{"_id": "151", "title": "Bump.MCELL2", "text": "|- (!V. (!ul. (((mcell2 V) ul) = (((mcell (NUMERAL (BIT0 (BIT1 _0)))) V) ul))))"}
{"_id": "152", "title": "INT_OF_NUM_MUL", "text": "|- (!m. (!n. (((int_mul (int_of_num m)) (int_of_num n)) = (int_of_num (m * n)))))"}
{"_id": "153", "title": "REAL_LEBESGUE_MEASURABLE_COMPL", "text": "|- (!s. ((real_lebesgue_measurable (UNIV DIFF s)) = (real_lebesgue_measurable s)))"}
{"_id": "154", "title": "PATH_CONNECTED_PCROSS", "text": "|- (!s. (!t. (((path_connected s) /\\ (path_connected t)) ==> (path_connected (s PCROSS t)))))"}
{"_id": "155", "title": "DIFFERENTIAL_COMPONENT_ZERO_AT_MAXMIN", "text": "|- (!f. (!f'. (!x. (!s. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ ((k <= (dimindex UNIV)) /\\ ((x IN s) /\\ ((open s) /\\ ((((has_derivative f) f') (at x)) /\\ ((!w. ((w IN s) ==> ((real_le ((f w) $ k)) ((f x) $ k)))) \\/ (!w. ((w IN s) ==> ((real_le ((f x) $ k)) ((f w) $ k)))))))))) ==> (!h. (((f' h) $ k) = (real_of_num (NUMERAL _0))))))))))"}
{"_id": "156", "title": "CONTINUOUS_ON_LIFT_COMPONENT", "text": "|- (!i. (!s. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((continuous_on (\\x. (lift (x $ i)))) s))))"}
{"_id": "157", "title": "AR_STRONG_DEFORMATION_RETRACT_OF_AR", "text": "|- (!s. (!t. (((AR s) /\\ ((AR t) /\\ ((closed_in ((subtopology euclidean) s)) t))) ==> (?r. (((((homotopic_with (\\h. (!x. ((x IN t) ==> ((h x) = x))))) (s , s)) (\\x. x)) r) /\\ ((retraction (s , t)) r))))))"}
{"_id": "158", "title": "MEASURABLE_CBALL", "text": "|- (!a. (!r. (measurable (cball (a , r)))))"}
{"_id": "159", "title": "Lp_ineqs_proofs.split_list_list", "text": "|- (!L. (!ds. (((split_list_list L) ds) = (((foldr (\\d. (\\L. ((split_list_hyp L) d)))) L) ds))))"}
{"_id": "160", "title": "CARD_EQ_TRANS", "text": "|- (!s. (!t. (!u. (((s =_c t) /\\ (t =_c u)) ==> (s =_c u)))))"}
{"_id": "161", "title": "Nonlinear_lemma.lmfun_lfun", "text": "|- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((lmfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (lfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "162", "title": "ITERATE_DELTA", "text": "|- (!op. ((monoidal op) ==> (!f. (!a. (!s. ((((iterate op) s) (\\x. (((COND (x = a)) (f x)) (neutral op)))) = (((COND (a IN s)) (f a)) (neutral op))))))))"}
{"_id": "163", "title": "HAS_SIZE_FUNSPACE_UNIV", "text": "|- (!m. (!n. (((UNIV HAS_SIZE m) /\\ (UNIV HAS_SIZE n)) ==> (UNIV HAS_SIZE (n EXP m)))))"}
{"_id": "164", "title": "Tame_list.POWER_TO_ITER", "text": "|- (!n. (((POWER f) n) = ((ITER n) f)))"}
{"_id": "165", "title": "Ssrnat.addnn", "text": "|- (!n. ((n + n) = (double n)))"}
{"_id": "166", "title": "INT_LE_SUB_LADD", "text": "|- (!x. (!y. (!z. (((int_le x) ((int_sub y) z)) = ((int_le ((int_add x) z)) y)))))"}
{"_id": "167", "title": "INT_OF_NUM_EQ", "text": "|- (!m. (!n. (((int_of_num m) = (int_of_num n)) = (m = n))))"}
{"_id": "168", "title": "Tame_defs2.core", "text": "|- (!L. (!r. (((core L) r) = ((MAP (FST o HD)) ((parts L) r)))))"}
{"_id": "169", "title": "MEASURABLE_IMP_LEBESGUE_MEASURABLE", "text": "|- (!s. ((measurable s) ==> (lebesgue_measurable s)))"}
{"_id": "170", "title": "Tame_defs.scriptL", "text": "|- (!V. ((scriptL V) = ((sum V) (\\v. (lmfun ((real_div (vector_norm v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))"}
{"_id": "171", "title": "EXISTS_THM", "text": "|- (? = (\\P. (P (@ P))))"}
{"_id": "172", "title": "POLYTOPE_SEGMENT", "text": "|- (!a. (!b. (polytope (closed_segment ((CONS (a , b)) NIL)))))"}
{"_id": "173", "title": "COMPLEX_MUL_SYM", "text": "|- (!x. (!y. (((complex_mul x) y) = ((complex_mul y) x))))"}
{"_id": "174", "title": "Tame_defs.tame_1", "text": "|- (!H. ((tame_1 H) = ((plain_hypermap H) /\\ (planar_hypermap H))))"}
{"_id": "175", "title": "CONVEX_HALFSPACE_LT", "text": "|- (!a. (!b. (convex (GSPEC (\\GEN%PVAR%1439. (?x. (((SETSPEC GEN%PVAR%1439) ((real_lt ((dot a) x)) b)) x)))))))"}
{"_id": "176", "title": "BOUNDED_SUBSET", "text": "|- (!s. (!t. (((bounded t) /\\ (s SUBSET t)) ==> (bounded s))))"}
{"_id": "177", "title": "LET_END_DEF", "text": "|- (!t. ((LET_END t) = t))"}
{"_id": "178", "title": "LIM_NULL_COMPARISON", "text": "|- (!net. (!f. (!g. ((((eventually (\\x. ((real_le (vector_norm (f x))) (g x)))) net) /\\ (((--> (\\x. (lift (g x)))) (vec (NUMERAL _0))) net)) ==> (((--> f) (vec (NUMERAL _0))) net)))))"}
{"_id": "179", "title": "ABSOLUTELY_REAL_INTEGRABLE_0", "text": "|- (!s. ((absolutely_real_integrable_on (\\x. (real_of_num (NUMERAL _0)))) s))"}
{"_id": "180", "title": "ANR_TRANSLATION", "text": "|- (!a. (!s. ((ANR ((IMAGE (\\x. ((vector_add a) x))) s)) = (ANR s))))"}
{"_id": "181", "title": "GABS_DEF", "text": "|- (!P. ((GABS P) = (@ P)))"}
{"_id": "182", "title": "extreme_point_of", "text": "|- (!s. (!x. (((extreme_point_of x) s) = ((x IN s) /\\ (!a. (!b. (((a IN s) /\\ (b IN s)) ==> (~ (x IN (open_segment (a , b)))))))))))"}
{"_id": "183", "title": "RC_SC", "text": "|- (!R. ((RC (SC R)) = (SC (RC R))))"}
{"_id": "184", "title": "FINITE_EMPTY", "text": "|- (FINITE EMPTY)"}
{"_id": "185", "title": "HAS_INTEGRAL_COMPLEX_LMUL", "text": "|- (!f. (!y. (!i. (!c. ((((has_integral f) y) i) ==> (((has_integral (\\x. ((complex_mul c) (f x)))) ((complex_mul c) y)) i))))))"}
{"_id": "186", "title": "INTEGRABLE_DECREASING_PRODUCT", "text": "|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g y)) (g x)))))) ==> ((integrable_on (\\x. ((% (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "187", "title": "List_hypermap_iso.fan_list_iso", "text": "|- (!v. (!f. (!V. (!E. (!w. (((fan_list_iso (f , (V , E))) (v , w)) = (((fan_list_nodes_iso (f , (V , E))) v) , ((fan_list_nodes_iso (f , (V , E))) w))))))))"}
{"_id": "188", "title": "HAS_BOUNDED_SETVARIATION_WORKS_ON_INTERVAL", "text": "|- (!f. (!a. (!b. (((has_bounded_setvariation_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le ((sum d) (\\k. (vector_norm (f k))))) ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)))) /\\ (!B. ((!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le ((sum d) (\\k. (vector_norm (f k))))) B))) ==> ((real_le ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)) B))))))))"}
{"_id": "189", "title": "LIM_INV_N_OFFSET", "text": "|- (!z. (((--> (\\n. (complex_inv ((vector_add (Cx (real_of_num n))) z)))) (Cx (real_of_num (NUMERAL _0)))) sequentially))"}
{"_id": "190", "title": "LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF", "text": "|- (!s. (!t. (((lebesgue_measurable s) /\\ (negligible ((s DIFF t) UNION (t DIFF s)))) ==> (lebesgue_measurable t))))"}
{"_id": "191", "title": "COMPLEX_SUB_RZERO", "text": "|- (!x. (((vector_sub x) (Cx (real_of_num (NUMERAL _0)))) = x))"}
{"_id": "192", "title": "OPEN_IN_OPEN", "text": "|- (!s. (!u. (((open_in ((subtopology euclidean) u)) s) = (?t. ((open t) /\\ (s = (u INTER t)))))))"}
{"_id": "193", "title": "Calc_derivative.invert_den_eq", "text": "|- (!a. (!b. ((((real_div a) b) = (real_of_num (NUMERAL _0))) = (((real_mul a) b) = (real_of_num (NUMERAL _0))))))"}
{"_id": "194", "title": "CONVEX_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((convex ((IMAGE f) s)) = (convex s)))))"}
{"_id": "195", "title": "HAS_SIZE_0", "text": "|- (!s. ((s HAS_SIZE (NUMERAL _0)) = (s = EMPTY)))"}
{"_id": "196", "title": "Fan.sigma_fan_in_set_of_edge", "text": "|- (!x. (!V. (!E. (!v. (!u. (((FAN (x , (V , E))) /\\ (u IN (((set_of_edge v) V) E))) ==> ((((((sigma_fan x) V) E) v) u) IN (((set_of_edge v) V) E))))))))"}
{"_id": "197", "title": "Localization.has_orders", "text": "|- (!k. (!f. (((has_orders f) k) = ((!i. ((((NUMERAL _0) < i) /\\ (i < k)) ==> (~ (((ITER i) f) = I)))) /\\ (((ITER k) f) = I)))))"}
{"_id": "198", "title": "SETDIST_SUBSETS_EQ", "text": "|- (!s. (!t. (!s'. (!t'. (((s' SUBSET s) /\\ ((t' SUBSET t) /\\ (!x. (!y. (((x IN s) /\\ (y IN t)) ==> (?x'. (?y'. ((x' IN s') /\\ ((y' IN t') /\\ ((real_le (distance (x' , y'))) (distance (x , y)))))))))))) ==> ((setdist (s' , t')) = (setdist (s , t))))))))"}
{"_id": "199", "title": "SUBSET_INTERIOR_EQ", "text": "|- (!s. ((s SUBSET (interior s)) = (open s)))"}
{"_id": "200", "title": "REAL_LE_INV2", "text": "|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_le x) y)) ==> ((real_le (real_inv y)) (real_inv x)))))"}
{"_id": "201", "title": "Add_triangle.azim_dart1_eq_azim", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((((v , w) IN (dart1_of_fan (V , E))) /\\ ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0)))) ==> (((azim_dart (V , E)) (v , w)) = ((((azim (vec (NUMERAL _0))) v) w) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))"}
{"_id": "202", "title": "FSTCART_SUB", "text": "|- (!x. (!y. ((fstcart ((vector_sub x) y)) = ((vector_sub (fstcart x)) (fstcart y)))))"}
{"_id": "203", "title": "OPEN_SEGMENT_LINEAR_IMAGE", "text": "|- (!f. (!a. (!b. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((open_segment ((f a) , (f b))) = ((IMAGE f) (open_segment (a , b))))))))"}
{"_id": "204", "title": "AFF_DIM_OPEN", "text": "|- (!s. (((open s) /\\ (~ (s = EMPTY))) ==> ((aff_dim s) = (int_of_num (dimindex UNIV)))))"}
{"_id": "205", "title": "SUM_DEGENERATE", "text": "|- (!f. (!s. ((~ (FINITE (GSPEC (\\GEN%PVAR%194. (?x. (((SETSPEC GEN%PVAR%194) ((x IN s) /\\ (~ ((f x) = (real_of_num (NUMERAL _0)))))) x)))))) ==> (((sum s) f) = (real_of_num (NUMERAL _0))))))"}
{"_id": "206", "title": "UNIONS_IMAGE", "text": "|- (!f. (!s. ((UNIONS ((IMAGE f) s)) = (GSPEC (\\GEN%PVAR%29. (?y. (((SETSPEC GEN%PVAR%29) (?x. ((x IN s) /\\ (y IN (f x))))) y)))))))"}
{"_id": "207", "title": "PAIRWISE_DISJOINT_COMPONENTS", "text": "|- (!u. ((pairwise DISJOINT) (components u)))"}
{"_id": "208", "title": "Localization.v_prime", "text": "|- (!V. (!FF. (((v_prime V) FF) = (GSPEC (\\GEN%PVAR%7534. (?v. (((SETSPEC GEN%PVAR%7534) ((v IN V) /\\ (?w. ((v , w) IN FF)))) v)))))))"}
{"_id": "209", "title": "matrix_neg", "text": "|- (!A. ((matrix_neg A) = (lambdai. (lambdaj. (real_neg ((A $ i) $ j))))))"}
{"_id": "210", "title": "DIMINDEX_1", "text": "|- ((dimindex UNIV) = (NUMERAL (BIT1 _0)))"}
{"_id": "211", "title": "DISJOINT_RELATIVE_INTERIOR_CONVEX_HULL", "text": "|- (!s. (((~ (affine_dependent s)) /\\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (((relative_interior ((hull convex) s)) INTER s) = EMPTY)))"}
{"_id": "212", "title": "Collect_geom2.NOT_COL_EQ_UPS_X_POS", "text": "|- (!v1. (!v2. (!v3. ((~ (collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))) = ((real_lt (real_of_num (NUMERAL _0))) (((ups_x ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0))))))))))"}
{"_id": "213", "title": "COUNTABLE_DELETE", "text": "|- (!x. (!s. ((COUNTABLE (s DELETE x)) = (COUNTABLE s))))"}
{"_id": "214", "title": "PATH_SUBPATH", "text": "|- (!u. (!v. (!g. (((path g) /\\ ((u IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ (v IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))))) ==> (path (((subpath u) v) g))))))"}
{"_id": "215", "title": "RC_CLOSED", "text": "|- (!R. (((RC R) = R) = (!x. ((R x) x))))"}
{"_id": "216", "title": "CROSS_RADD", "text": "|- (!x. (!y. (!z. (((cross x) ((vector_add y) z)) = ((vector_add ((cross x) y)) ((cross x) z))))))"}
{"_id": "217", "title": "Tame_defs2.transform_assumption", "text": "|- (!L. ((transform_assumption_v2 L) = (!N. (!r. (!x. (((good_list L) /\\ ((good_list_nodes L) /\\ ((((marked_list L) N) r) x))) ==> ((((marked_list L) (((((ntrans L) N) r) x) (NUMERAL (BIT1 _0)))) (((((rtrans L) N) r) x) (NUMERAL (BIT1 _0)))) x)))))))"}
{"_id": "218", "title": "EXTREME_POINTS_OF_CONVEX_HULL_AFFINE_INDEPENDENT", "text": "|- (!s. ((~ (affine_dependent s)) ==> ((GSPEC (\\GEN%PVAR%2247. (?x. (((SETSPEC GEN%PVAR%2247) ((extreme_point_of x) ((hull convex) s))) x)))) = s)))"}
{"_id": "219", "title": "INSEG_SUBSET_FL", "text": "|- (!l. (!m. (((inseg m) l) ==> (!x. (((fl m) x) ==> ((fl l) x))))))"}
{"_id": "220", "title": "CARD_ADD_C", "text": "|- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> ((CARD ((+_c s) t)) = ((CARD s) + (CARD t))))))"}
{"_id": "221", "title": "LINEAR_1", "text": "|- (!f. ((linear f) = (?c. (f = (\\x. ((% c) x))))))"}
{"_id": "222", "title": "EXCLUDED_MIDDLE", "text": "|- (!t. (t \\/ (~ t)))"}
{"_id": "223", "title": "REAL_DIFFSQ", "text": "|- (!x. (!y. (((real_mul ((real_add x) y)) ((real_sub x) y)) = ((real_sub ((real_mul x) x)) ((real_mul y) y)))))"}
{"_id": "224", "title": "CSIN_ADD", "text": "|- (!w. (!z. ((csin ((vector_add w) z)) = ((vector_add ((complex_mul (csin w)) (ccos z))) ((complex_mul (ccos w)) (csin z))))))"}
{"_id": "225", "title": "CARD_PSUBSET", "text": "|- (!a. (!b. (((a PSUBSET b) /\\ (FINITE b)) ==> ((CARD a) < (CARD b)))))"}
{"_id": "226", "title": "OPEN_IN_TRANS", "text": "|- (!s. (!t. (!u. ((((open_in ((subtopology euclidean) t)) s) /\\ ((open_in ((subtopology euclidean) u)) t)) ==> ((open_in ((subtopology euclidean) u)) s)))))"}
{"_id": "227", "title": "Add_triangle.dart_switch", "text": "|- (!V. (!E. (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) = ((w , v) IN (dart_of_fan (V , E))))))))"}
{"_id": "228", "title": "DET_DEPENDENT_ROWS", "text": "|- (!A. ((dependent (rows A)) ==> ((det A) = (real_of_num (NUMERAL _0)))))"}
{"_id": "229", "title": "ADD_SUB", "text": "|- (!m. (!n. (((m + n) - n) = m)))"}
{"_id": "230", "title": "poset", "text": "|- (!l. ((poset l) = ((!x. (((fl l) x) ==> (l (x , x)))) /\\ ((!x. (!y. (!z. (((l (x , y)) /\\ (l (y , z))) ==> (l (x , z)))))) /\\ (!x. (!y. (((l (x , y)) /\\ (l (y , x))) ==> (x = y))))))))"}
{"_id": "231", "title": "HAS_REAL_MEASURE_EMPTY", "text": "|- ((has_real_measure EMPTY) (real_of_num (NUMERAL _0)))"}
{"_id": "232", "title": "MEM_LIST_OF_SET", "text": "|- (!s. ((FINITE s) ==> (!x. (((MEM x) (list_of_set s)) = (x IN s)))))"}
{"_id": "233", "title": "Functional_equation.functional_taum_x'", "text": "|- (taum_x' = ((add6 rhazim_x') ((add6 rhazim2_x) ((sub6 rhazim3_x) (constant6 ((real_mul ((real_add (real_of_num (NUMERAL (BIT1 _0)))) const1)) pi))))))"}
{"_id": "234", "title": "Fan.d_fan", "text": "|- (!x. (!V. (!E. ((d_fan (x , (V , E))) = ((d1_fan (x , (V , E))) UNION (d20_fan (x , (V , E))))))))"}
{"_id": "235", "title": "Nonlin_def.ldih_x_div_sqrtdelta_posbranch", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ldih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul (lfun ((real_div (sqrt x1)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "236", "title": "REAL_ADD_SYM", "text": "|- (!x. (!y. (((real_add x) y) = ((real_add y) x))))"}
{"_id": "237", "title": "Add_triangle.fan_map_ext", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((((f1_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((f_fan_pair (V , E)) d))) /\\ ((((((n_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((n_fan_pair (V , E)) d))) /\\ (((((e_fan (vec (NUMERAL _0))) V) E) ((ext_dart (V , E)) d)) = ((ext_dart (V , E)) ((e_fan_pair (V , E)) d))))))))))"}
{"_id": "238", "title": "RPOW_RPOW", "text": "|- (!x. (!y. (!z. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((rpow ((rpow x) y)) z) = ((rpow x) ((real_mul y) z)))))))"}
{"_id": "239", "title": "Seq2.list_sum_set_of_list", "text": "|- (!s. (!f. ((uniq s) ==> (((list_sum s) f) = ((sum (set_of_list s)) f)))))"}
{"_id": "240", "title": "AFFSIGN_SPECIAL_SCALE", "text": "|- (!sgn. (!s. (!t. (!a. (!v. (((FINITE s) /\\ ((FINITE t) /\\ ((~ ((vec (NUMERAL _0)) IN t)) /\\ ((~ (v IN t)) /\\ ((~ (((% a) v) IN t)) /\\ ((!x. ((sgn x) ==> (sgn ((real_div x) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) /\\ ((!x. (!y. (((sgn x) /\\ (sgn y)) ==> (sgn ((real_add x) y))))) /\\ ((real_lt (real_of_num (NUMERAL _0))) a)))))))) ==> ((((affsign sgn) ((vec (NUMERAL _0)) INSERT (((% a) v) INSERT s))) t) = (((affsign sgn) ((vec (NUMERAL _0)) INSERT (v INSERT s))) t))))))))"}
{"_id": "241", "title": "CLOSEST_POINT_IN_RELATIVE_FRONTIER", "text": "|- (!s. (!x. (((closed s) /\\ ((~ (s = EMPTY)) /\\ (x IN (((hull affine) s) DIFF (relative_interior s))))) ==> (((closest_point s) x) IN (relative_frontier s)))))"}
{"_id": "242", "title": "Betwn_corek_z_x.parts_rot", "text": "|- (((good_list L) /\\ (((all (\\x. ((MEM x) (list_of_darts L)))) r) /\\ ((is_edge_nondegenerate (hypermap_of_list L)) /\\ ((uniq r) /\\ ((final_list L) r))))) ==> (((parts L) ((rot n) r)) = ((rot n) ((parts L) r))))"}
{"_id": "243", "title": "Seq.uniq_leq_size", "text": "|- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) ==> ((MEM x) s2))) ==> ((sizel s1) <= (sizel s2))))))"}
{"_id": "244", "title": "LIM_NULL_CMUL", "text": "|- (!net. (!f. (!c. ((((--> f) (vec (NUMERAL _0))) net) ==> (((--> (\\x. ((% c) (f x)))) (vec (NUMERAL _0))) net)))))"}
{"_id": "245", "title": "INT_OF_NUM_GE", "text": "|- (!m. (!n. (((int_ge (int_of_num m)) (int_of_num n)) = (m >= n))))"}
{"_id": "246", "title": "REAL_CONTINUOUS_CONTINUOUS1", "text": "|- (((real_continuous f) net) = ((continuous (lift o f)) net))"}
{"_id": "247", "title": "REAL_CONTINUOUS_CONTINUOUS_ATREAL", "text": "|- (((real_continuous f) (atreal x)) = ((continuous (lift o (f o drop))) (at (lift x))))"}
{"_id": "248", "title": "SECOND_MEAN_VALUE_THEOREM_GEN_FULL", "text": "|- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((!x. ((x IN (open_interval (a , b))) ==> (((real_le u) (g x)) /\\ ((real_le (g x)) v)))) /\\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_interval ((CONS (a , b)) NIL))) /\\ (((has_integral (\\x. ((% (g x)) (f x)))) ((vector_add ((% u) ((integral (closed_interval ((CONS (a , c)) NIL))) f))) ((% v) ((integral (closed_interval ((CONS (c , b)) NIL))) f)))) (closed_interval ((CONS (a , b)) NIL))))))))))))"}
{"_id": "249", "title": "affine", "text": "|- (!s. ((affine s) = (!x. (!y. (!u. (!v. (((x IN s) /\\ ((y IN s) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vector_add ((% u) x)) ((% v) y)) IN s))))))))"}
{"_id": "250", "title": "UNIONS_0", "text": "|- ((UNIONS EMPTY) = EMPTY)"}
{"_id": "251", "title": "COMPLEX_SUB_RDISTRIB", "text": "|- (!x. (!y. (!z. (((complex_mul ((vector_sub x) y)) z) = ((vector_sub ((complex_mul x) z)) ((complex_mul y) z))))))"}
{"_id": "252", "title": "LIMPT_INJECTIVE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((limit_point_of (f x)) ((IMAGE f) s)) = ((limit_point_of x) s)))))"}
{"_id": "253", "title": "MOD_MOD_REFL", "text": "|- (!m. (!n. ((~ (n = (NUMERAL _0))) ==> (((m MOD n) MOD n) = (m MOD n)))))"}
{"_id": "254", "title": "REAL_AFFINITY_EQ", "text": "|- (!m. (!c. (!x. (!y. ((~ (m = (real_of_num (NUMERAL _0)))) ==> ((((real_add ((real_mul m) x)) c) = y) = (x = ((real_add ((real_mul (real_inv m)) y)) (real_neg ((real_div c) m))))))))))"}
{"_id": "255", "title": "NOT_CLAUSES", "text": "|- ((!t. ((~ (~ t)) = t)) /\\ (((~ T) = F) /\\ ((~ F) = T)))"}
{"_id": "256", "title": "CTAN_SUB", "text": "|- (!w. (!z. (((~ ((ccos w) = (Cx (real_of_num (NUMERAL _0))))) /\\ ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\\ (~ ((ccos ((vector_sub w) z)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((ctan ((vector_sub w) z)) = ((complex_div ((vector_sub (ctan w)) (ctan z))) ((vector_add (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_mul (ctan w)) (ctan z))))))))"}
{"_id": "257", "title": "ENDS_NOT_IN_SEGMENT", "text": "|- (!a. (!b. ((~ (a IN (open_segment (a , b)))) /\\ (~ (b IN (open_segment (a , b)))))))"}
{"_id": "258", "title": "TRIANGULATION_DISJOINT_RELATIVE_INTERIORS", "text": "|- (!t. (!c. (!d. (((triangulation t) /\\ ((c IN t) /\\ ((d IN t) /\\ (~ (c = d))))) ==> (((relative_interior c) INTER (relative_interior d)) = EMPTY)))))"}
{"_id": "259", "title": "eq_c", "text": "|- (!t. (!s. ((s =_c t) = (?f. ((!x. ((x IN s) ==> ((f x) IN t))) /\\ (!y. ((y IN t) ==> (?!x. ((x IN s) /\\ ((f x) = y))))))))))"}
{"_id": "260", "title": "MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_LE_DENSE", "text": "|- (!f. (!r. (((closure ((IMAGE lift) r)) = UNIV) ==> (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ ((k <= (dimindex UNIV)) /\\ (a IN r))) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3594. (?x. (((SETSPEC GEN%PVAR%3594) ((real_le ((f x) $ k)) a)) x))))))))))))"}
{"_id": "261", "title": "Planarity.properties1_inside_fan", "text": "|- (!x. (!u. (!w. ((((DISJOINT (x INSERT EMPTY)) (u INSERT (w INSERT EMPTY))) /\\ (((real_lt (real_of_num (NUMERAL _0))) a) /\\ ((real_lt a) (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) IN ((aff_ge (x INSERT EMPTY)) (u INSERT (w INSERT EMPTY))))))))"}
{"_id": "262", "title": "CONVEX_HULL_INSERT_SEGMENTS", "text": "|- (!s. (!a. (((hull convex) (a INSERT s)) = (((COND (s = EMPTY)) (a INSERT EMPTY)) (UNIONS (GSPEC (\\GEN%PVAR%1485. (?x. (((SETSPEC GEN%PVAR%1485) (x IN ((hull convex) s))) (closed_segment ((CONS (a , x)) NIL)))))))))))"}
{"_id": "263", "title": "REALLIM_SUB", "text": "|- (!net. (!f. (!g. (!l. (!m. (((((---> f) l) net) /\\ (((---> g) m) net)) ==> (((---> (\\x. ((real_sub (f x)) (g x)))) ((real_sub l) m)) net)))))))"}
{"_id": "264", "title": "POLYHEDRON_POSITIVE_ORTHANT", "text": "|- (polyhedron (GSPEC (\\GEN%PVAR%2275. (?x. (((SETSPEC GEN%PVAR%2275) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))"}
{"_id": "265", "title": "NSUM_PERMUTE", "text": "|- (!f. (!p. (!s. (((permutes p) s) ==> (((nsum s) f) = ((nsum s) (f o p)))))))"}
{"_id": "266", "title": "DIFFERENTIABLE_TRANSFORM_WITHIN", "text": "|- (!f. (!g. (!x. (!s. (!d. ((((real_lt (real_of_num (NUMERAL _0))) d) /\\ ((x IN s) /\\ ((!x'. (((x' IN s) /\\ ((real_lt (distance (x' , x))) d)) ==> ((f x') = (g x')))) /\\ ((differentiable f) ((within (at x)) s))))) ==> ((differentiable g) ((within (at x)) s))))))))"}
{"_id": "267", "title": "real_negligible", "text": "|- (!s. ((real_negligible s) = (negligible ((IMAGE lift) s))))"}
{"_id": "268", "title": "REAL_LE_LDIV_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le ((real_div x) z)) y) = ((real_le x) ((real_mul y) z)))))))"}
{"_id": "269", "title": "LOCALLY_PATH_CONNECTED_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((locally path_connected) ((IMAGE f) s)) = ((locally path_connected) s)))))"}
{"_id": "270", "title": "HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_AT", "text": "|- (!f. (!s. (!x. ((((holomorphic_on f) s) /\\ ((open s) /\\ (x IN s))) ==> ((complex_differentiable f) (at x))))))"}
{"_id": "271", "title": "CLOSED_SUBSET", "text": "|- (!s. (!t. (((s SUBSET t) /\\ (closed s)) ==> ((closed_in ((subtopology euclidean) t)) s))))"}
{"_id": "272", "title": "CLOSED_IN_IMP_SUBSET", "text": "|- (!top. (!s. (!t. (((closed_in ((subtopology top) s)) t) ==> (t SUBSET s)))))"}
{"_id": "273", "title": "Functional_equation.proj_y4", "text": "|- (!x1. (!x2. (!x3. (!x5. (!x6. (!x4. (((((((proj_y4 x1) x2) x3) x4) x5) x6) = (sqrt x4))))))))"}
{"_id": "274", "title": "INTER_SUBSET", "text": "|- ((!s. (!t. ((s INTER t) SUBSET s))) /\\ (!s. (!t. ((t INTER s) SUBSET s))))"}
{"_id": "275", "title": "ROOT_POS_LT", "text": "|- (!n. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) ((root n) x)))))"}
{"_id": "276", "title": "VSUM_RESTRICT_SET", "text": "|- (!P. (!s. (!f. (((vsum (GSPEC (\\GEN%PVAR%111. (?x. (((SETSPEC GEN%PVAR%111) ((x IN s) /\\ (P x))) x))))) f) = ((vsum s) (\\x. (((COND (P x)) (f x)) (vec (NUMERAL _0)))))))))"}
{"_id": "277", "title": "REAL_POLY_CLAUSES", "text": "|- ((!x. (!y. (!z. (((real_add x) ((real_add y) z)) = ((real_add ((real_add x) y)) z))))) /\\ ((!x. (!y. (((real_add x) y) = ((real_add y) x)))) /\\ ((!x. (((real_add (real_of_num (NUMERAL _0))) x) = x)) /\\ ((!x. (!y. (!z. (((real_mul x) ((real_mul y) z)) = ((real_mul ((real_mul x) y)) z))))) /\\ ((!x. (!y. (((real_mul x) y) = ((real_mul y) x)))) /\\ ((!x. (((real_mul (real_of_num (NUMERAL (BIT1 _0)))) x) = x)) /\\ ((!x. (((real_mul (real_of_num (NUMERAL _0))) x) = (real_of_num (NUMERAL _0)))) /\\ ((!x. (!y. (!z. (((real_mul x) ((real_add y) z)) = ((real_add ((real_mul x) y)) ((real_mul x) z)))))) /\\ ((!x. (((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0))))) /\\ (!x. (!n. (((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n))))))))))))))"}
{"_id": "278", "title": "CARD_NUMSEG_1", "text": "|- (!n. ((CARD ((NUMERAL (BIT1 _0)) .. n)) = n))"}
{"_id": "279", "title": "MDIST_REVERSE_TRIANGLE", "text": "|- (!m. (!x. (!y. (!z. (((x IN (mspace m)) /\\ ((y IN (mspace m)) /\\ (z IN (mspace m)))) ==> ((real_le (real_abs ((real_sub ((mdist m) (x , y))) ((mdist m) (y , z))))) ((mdist m) (x , z))))))))"}
{"_id": "280", "title": "MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL", "text": "|- (!f. (!g. (!a. (!b. ((((measurable_on f) UNIV) /\\ ((!x. ((f x) IN (open_interval (a , b)))) /\\ ((continuous_on g) (open_interval (a , b))))) ==> ((measurable_on (g o f)) UNIV))))))"}
{"_id": "281", "title": "II_NZ", "text": "|- (~ (ii = (Cx (real_of_num (NUMERAL _0)))))"}
{"_id": "282", "title": "REAL_CONTINUOUS_AT_AZIM_SHARP", "text": "|- (!v. (!w. (!w1. (!w2. (((~ (collinear (v INSERT (w INSERT (w1 INSERT EMPTY))))) /\\ (~ (w2 IN ((aff_ge (v INSERT (w INSERT EMPTY))) (w1 INSERT EMPTY))))) ==> ((real_continuous (((azim v) w) w1)) (at w2)))))))"}
{"_id": "283", "title": "Misc_defs_and_lemmas.min_num", "text": "|- (!X. ((min_num X) = (@m. ((m IN X) /\\ (!n. ((n IN X) ==> (m <= n)))))))"}
{"_id": "284", "title": "real", "text": "|- (!z. ((real z) = ((Im z) = (real_of_num (NUMERAL _0)))))"}
{"_id": "285", "title": "CSQRT", "text": "|- (!z. (((complex_pow (csqrt z)) (NUMERAL (BIT0 (BIT1 _0)))) = z))"}
{"_id": "286", "title": "CONVEX_HAS_BOUNDED_VARIATION_EQ", "text": "|- (!f. (!s. ((((convex_on (drop o f)) s) /\\ (is_interval s)) ==> (((has_bounded_variation_on f) s) = (bounded ((IMAGE f) s))))))"}
{"_id": "287", "title": "DIFFERENTIABLE_CHAIN_WITHIN", "text": "|- (!f. (!g. (!x. (!s. ((((differentiable f) ((within (at x)) s)) /\\ ((differentiable g) ((within (at (f x))) ((IMAGE f) s)))) ==> ((differentiable (g o f)) ((within (at x)) s)))))))"}
{"_id": "288", "title": "LE_TRANS", "text": "|- (!m. (!n. (!p. (((m <= n) /\\ (n <= p)) ==> (m <= p)))))"}
{"_id": "289", "title": "REAL_LE_TOTAL", "text": "|- (!x. (!y. (((real_le x) y) \\/ ((real_le y) x))))"}
{"_id": "290", "title": "REAL_CLOSED_IN", "text": "|- (!s. ((real_closed s) = ((closed_in euclideanreal) s)))"}
{"_id": "291", "title": "Dih2k_hypermap.IMAGE_NN_OF_HYP_F_SY", "text": "|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\\ (((NUMERAL (BIT1 _0)) < (dimindex UNIV)) /\\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ ((j <= (dimindex UNIV)) /\\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (((IMAGE (nn_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (F_SY (vecmats l))) = (GSPEC (\\GEN%PVAR%7498. (?i. (((SETSPEC GEN%PVAR%7498) (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) (((row (SUC (i MOD (dimindex UNIV)))) (vecmats l)) , ((row i) (vecmats l)))))))))"}
{"_id": "292", "title": "FROM_INTER_NUMSEG", "text": "|- (!k. (!n. (((from k) INTER ((NUMERAL _0) .. n)) = (k .. n))))"}
{"_id": "293", "title": "REAL_INV_INV", "text": "|- (!x. ((real_inv (real_inv x)) = x))"}
{"_id": "294", "title": "COMPLEX_EQ_SUB_LADD", "text": "|- (!x. (!y. (!z. ((x = ((vector_sub y) z)) = (((vector_add x) z) = y)))))"}
{"_id": "295", "title": "POLYHEDRON_IMP_CONVEX", "text": "|- (!s. ((polyhedron s) ==> (convex s)))"}
{"_id": "296", "title": "Hypermap.lemma_loop_outside_node", "text": "|- (!H. (!NF. (!L. (!x. ((((is_normal H) NF) /\\ (L IN NF)) ==> (~ ((dart_of_loop L) SUBSET ((node H) x))))))))"}
{"_id": "297", "title": "orthogonal_matrix", "text": "|- (!Q. ((orthogonal_matrix Q) = ((((matrix_mul (transp Q)) Q) = (mat (NUMERAL (BIT1 _0)))) /\\ (((matrix_mul Q) (transp Q)) = (mat (NUMERAL (BIT1 _0)))))))"}
{"_id": "298", "title": "ITERATE_OP_GEN", "text": "|- (!op. ((monoidal op) ==> (!f. (!g. (!s. (((FINITE (((support op) f) s)) /\\ (FINITE (((support op) g) s))) ==> ((((iterate op) s) (\\x. ((op (f x)) (g x)))) = ((op (((iterate op) s) f)) (((iterate op) s) g)))))))))"}
{"_id": "299", "title": "Lp_ineqs_proofs.y7_hi", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y7_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))"}
{"_id": "300", "title": "Fan_defs.n_fan_pair_ext", "text": "|- (!V. (!E. (!x. (((n_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((n_fan_pair (V , E)) x)) x)))))"}
{"_id": "301", "title": "Seq2.list_sum_add", "text": "|- (!s. (!f1. (!f2. (((list_sum s) (\\x. ((real_add (f1 x)) (f2 x)))) = ((real_add ((list_sum s) f1)) ((list_sum s) f2))))))"}
{"_id": "302", "title": "Tame_lemmas.tauVEF_alt3", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\\f. (LET_END ((tauVEF (V , (E , f))) = ((real_add ((sum f) (\\x. ((real_sub ((real_sub ((real_mul ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul ((real_div sol0) pi)) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (lmfun (h_dart x)))))) ((azim_dart (V , E)) x))) pi)) sol0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_add pi) sol0))))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))"}
{"_id": "303", "title": "REAL_SUP_LE", "text": "|- (!b. (((~ (s = EMPTY)) /\\ (!x. ((x IN s) ==> ((real_le x) b)))) ==> ((real_le (sup s)) b)))"}
{"_id": "304", "title": "DOT_SYM", "text": "|- (!x. (!y. (((dot x) y) = ((dot y) x))))"}
{"_id": "305", "title": "REAL_ABS_ABS", "text": "|- (!x. ((real_abs (real_abs x)) = (real_abs x)))"}
{"_id": "306", "title": "CONNECTED_COMPONENT_EQ_EMPTY", "text": "|- (!s. (!x. ((((connected_component s) x) = EMPTY) = (~ (x IN s)))))"}
{"_id": "307", "title": "REAL_BOUNDED", "text": "|- ((real_bounded s) = (bounded ((IMAGE lift) s)))"}
{"_id": "308", "title": "SPAN_CARD_GE_DIM", "text": "|- (!v. (!b. (((v SUBSET (span b)) /\\ (FINITE b)) ==> ((dim v) <= (CARD b)))))"}
{"_id": "309", "title": "REAL_DIFFERENTIABLE_AT_EXP", "text": "|- (!x. ((real_differentiable exp) (atreal x)))"}
{"_id": "310", "title": "LIM_COMPLEX_LMUL", "text": "|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\\x. ((complex_mul c) (f x)))) ((complex_mul c) l)) net)))))"}
{"_id": "311", "title": "REAL_LT_INF_FINITE", "text": "|- (!s. (!a. (((FINITE s) /\\ (~ (s = EMPTY))) ==> (((real_lt a) (inf s)) = (!x. ((x IN s) ==> ((real_lt a) x)))))))"}
{"_id": "312", "title": "Fan.properties_of_set_of_edge_fan", "text": "|- (!x. (!V. (!E. (!v. (!u. ((FAN (x , (V , E))) ==> (((v INSERT (u INSERT EMPTY)) IN E) = (u IN (((set_of_edge v) V) E)))))))))"}
{"_id": "313", "title": "LIFT_CMUL", "text": "|- (!x. (!c. ((lift ((real_mul c) x)) = ((% c) (lift x)))))"}
{"_id": "314", "title": "Hypermap.NON_ZERO", "text": "|- (!n. (~ ((SUC n) = (NUMERAL _0))))"}
{"_id": "315", "title": "ARITH_LE", "text": "|- ((!m. (!n. (((NUMERAL m) <= (NUMERAL n)) = (m <= n)))) /\\ (((_0 <= _0) = T) /\\ ((!n. (((BIT0 n) <= _0) = (n <= _0))) /\\ ((!n. (((BIT1 n) <= _0) = F)) /\\ ((!n. ((_0 <= (BIT0 n)) = T)) /\\ ((!n. ((_0 <= (BIT1 n)) = T)) /\\ ((!m. (!n. (((BIT0 m) <= (BIT0 n)) = (m <= n)))) /\\ ((!m. (!n. (((BIT0 m) <= (BIT1 n)) = (m <= n)))) /\\ ((!m. (!n. (((BIT1 m) <= (BIT0 n)) = (m < n)))) /\\ (!m. (!n. (((BIT1 m) <= (BIT1 n)) = (m <= n)))))))))))))"}
{"_id": "316", "title": "HOMEOMORPHISM_INJECTIVE_CLOSED_MAP_EQ", "text": "|- (!f. (!s. (!t. ((((continuous_on f) s) /\\ ((((IMAGE f) s) = t) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))))) ==> ((?g. ((homeomorphism (s , t)) (f , g))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))"}
{"_id": "317", "title": "Appendix.scs_k_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_k_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = k))))))))))"}
{"_id": "318", "title": "ORTHOGONAL_LINEAR_IMAGE_EQ", "text": "|- (!f. (!x. (!y. (((linear f) /\\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((orthogonal (f x)) (f y)) = ((orthogonal x) y))))))"}
{"_id": "319", "title": "OUTSIDE_CONNECTED_COMPONENT_LT", "text": "|- (!s. ((outside s) = (GSPEC (\\GEN%PVAR%2011. (?x. (((SETSPEC GEN%PVAR%2011) (!B. (?y. (((real_lt B) (vector_norm y)) /\\ (((connected_component (UNIV DIFF s)) x) y))))) x))))))"}
{"_id": "320", "title": "TRACE_TRANSP", "text": "|- (!A. ((trace (transp A)) = (trace A)))"}
{"_id": "321", "title": "REALLIM_COMPLEX", "text": "|- (((---> s) l) = ((--> (Cx o s)) (Cx l)))"}
{"_id": "322", "title": "DIFFERENTIABLE_BOUND", "text": "|- (!f. (!f'. (!s. (!B. (((convex s) /\\ ((!x. ((x IN s) ==> (((has_derivative f) (f' x)) ((within (at x)) s)))) /\\ (!x. ((x IN s) ==> ((real_le (onorm (f' x))) B))))) ==> (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))))))"}
{"_id": "323", "title": "CLOSED_LIMPT", "text": "|- (!s. ((closed s) = (!x. (((limit_point_of x) s) ==> (x IN s)))))"}
{"_id": "324", "title": "Ssrbool.implybF", "text": "|- (!b. ((b ==> F) = (~ b)))"}
{"_id": "325", "title": "Functional_equation.domain6", "text": "|- (!h. (!f. (!g. ((((domain6 h) f) g) = (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((h x1) x2) x3) x4) x5) x6) ==> (((((((f x1) x2) x3) x4) x5) x6) = ((((((g x1) x2) x3) x4) x5) x6)))))))))))))"}
{"_id": "326", "title": "FINITE_INDEX_INRANGE", "text": "|- (!i. (?k. (((NUMERAL (BIT1 _0)) <= k) /\\ ((k <= (dimindex UNIV)) /\\ (!x. ((x $ i) = (x $ k)))))))"}
{"_id": "327", "title": "INTEGRABLE_CONST", "text": "|- (!a. (!b. (!c. ((integrable_on (\\x. c)) (closed_interval ((CONS (a , b)) NIL))))))"}
{"_id": "328", "title": "SIN_0", "text": "|- ((sin (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))"}
{"_id": "329", "title": "REAL_SGN_REAL_SGN", "text": "|- (!x. ((real_sgn (real_sgn x)) = (real_sgn x)))"}
{"_id": "330", "title": "CENTRE_IN_CBALL", "text": "|- (!x. (!e. ((x IN (cball (x , e))) = ((real_le (real_of_num (NUMERAL _0))) e))))"}
{"_id": "331", "title": "REAL_LEBESGUE_MEASURABLE_INTER", "text": "|- (!s. (!t. (((real_lebesgue_measurable s) /\\ (real_lebesgue_measurable t)) ==> (real_lebesgue_measurable (s INTER t)))))"}
{"_id": "332", "title": "DOT_LSUB", "text": "|- (!x. (!y. (!z. (((dot ((vector_sub x) y)) z) = ((real_sub ((dot x) z)) ((dot y) z))))))"}
{"_id": "333", "title": "EXP", "text": "|- ((!m. ((m EXP (NUMERAL _0)) = (NUMERAL (BIT1 _0)))) /\\ (!m. (!n. ((m EXP (SUC n)) = (m * (m EXP n))))))"}
{"_id": "334", "title": "TAGGED_PARTIAL_DIVISION_SUBSET", "text": "|- (!s. (!t. (!i. ((((tagged_partial_division_of s) i) /\\ (t SUBSET s)) ==> ((tagged_partial_division_of t) i)))))"}
{"_id": "335", "title": "CARD_EQ_IMP_LE", "text": "|- (!s. (!t. ((s =_c t) ==> (s <=_c t))))"}
{"_id": "336", "title": "MOD_MOD", "text": "|- (!m. (!n. (!p. ((~ ((n * p) = (NUMERAL _0))) ==> (((m MOD (n * p)) MOD n) = (m MOD n))))))"}
{"_id": "337", "title": "IN_INTERIOR", "text": "|- (!x. (!s. ((x IN (interior s)) = (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ ((ball (x , e)) SUBSET s))))))"}
{"_id": "338", "title": "Lp_ineqs_def.sol_fan", "text": "|- (!f. (!V. (!E. (((sol_fan (V , E)) f) = (real_abs ((real_add ((sum f) (\\x. ((real_sub ((azim_dart (V , E)) x)) pi)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))))))"}
{"_id": "339", "title": "INTERIOR_INTER", "text": "|- (!s. (!t. ((interior (s INTER t)) = ((interior s) INTER (interior t)))))"}
{"_id": "340", "title": "CONTINUOUS_AT_TRANSLATION", "text": "|- (!a. (!z. (!f. (((continuous f) (at ((vector_add a) z))) = ((continuous (\\x. (f ((vector_add a) x)))) (at z))))))"}
{"_id": "341", "title": "Seq.cat", "text": "|- (!x. (!t. (!s2. ((((cat NIL) s2) = s2) /\\ (((cat ((CONS x) t)) s2) = ((CONS x) ((cat t) s2)))))))"}
{"_id": "342", "title": "LE_SUC_LT", "text": "|- (!m. (!n. (((SUC m) <= n) = (m < n))))"}
{"_id": "343", "title": "Hypermap_iso.hyp_iso", "text": "|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) = ((((BIJ f) (dart H)) (dart G)) /\\ (!x. ((x IN (dart H)) ==> ((((edge_map G) (f x)) = (f ((edge_map H) x))) /\\ ((((node_map G) (f x)) = (f ((node_map H) x))) /\\ (((face_map G) (f x)) = (f ((face_map H) x))))))))))))"}
{"_id": "344", "title": "sign", "text": "|- (!p. ((sign p) = (((COND (evenperm p)) (real_of_num (NUMERAL (BIT1 _0)))) (real_neg (real_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "345", "title": "CLOSED_IN_UNION_COMPLEMENT_COMPONENTS", "text": "|- (!u. (!s. (!c. ((((locally connected) u) /\\ (((closed_in ((subtopology euclidean) u)) s) /\\ (c SUBSET (components (u DIFF s))))) ==> ((closed_in ((subtopology euclidean) u)) (s UNION (UNIONS c)))))))"}
{"_id": "346", "title": "Pack1.ineq_lm5_3_step3", "text": "|- (!S. (!p. (!r. (!A. ((((real_le (real_of_num (NUMERAL _0))) r) /\\ ((packing S) /\\ ((saturated S) /\\ ((fcc_compatible A) S)))) ==> ((real_le ((real_mul (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (CARD (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0))))))))))) ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\\v. ((real_add (A v)) (measure ((voronoi_open S) v)))))))))))"}
{"_id": "347", "title": "HAS_REAL_INTEGRAL_LMUL", "text": "|- (!f. (!k. (!s. (!c. ((((has_real_integral f) k) s) ==> (((has_real_integral (\\x. ((real_mul c) (f x)))) ((real_mul c) k)) s))))))"}
{"_id": "348", "title": "HAS_COMPLEX_DERIVATIVE_POW_AT", "text": "|- (!f. (!f'. (!x. (!n. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative (\\x. ((complex_pow (f x)) n))) ((complex_mul (Cx (real_of_num n))) ((complex_mul ((complex_pow (f x)) (n - (NUMERAL (BIT1 _0))))) f'))) (at x)))))))"}
{"_id": "349", "title": "CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN", "text": "|- (!f. (!s. (((continuous_on f) s) = (!x. ((x IN s) ==> ((continuous f) ((within (at x)) s)))))))"}
{"_id": "350", "title": "SUM_GP", "text": "|- (!x. (!m. (!n. (((sum (m .. n)) (\\i. ((real_pow x) i))) = (((COND (n < m)) (real_of_num (NUMERAL _0))) (((COND (x = (real_of_num (NUMERAL (BIT1 _0))))) (real_of_num ((n + (NUMERAL (BIT1 _0))) - m))) ((real_div ((real_sub ((real_pow x) m)) ((real_pow x) (SUC n)))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) x))))))))"}
{"_id": "351", "title": "REAL_LT_IMP_LE", "text": "|- (!x. (!y. (((real_lt x) y) ==> ((real_le x) y))))"}
{"_id": "352", "title": "CONTENT_1", "text": "|- (!a. (!b. (((real_le (drop a)) (drop b)) ==> ((content (closed_interval ((CONS (a , b)) NIL))) = ((real_sub (drop b)) (drop a))))))"}
{"_id": "353", "title": "TAN_NEG", "text": "|- (!x. ((tan (real_neg x)) = (real_neg (tan x))))"}
{"_id": "354", "title": "REAL_LTE_TRANS", "text": "|- (!x. (!y. (!z. ((((real_lt x) y) /\\ ((real_le y) z)) ==> ((real_lt x) z)))))"}
{"_id": "355", "title": "SIMPLE_IMAGE", "text": "|- (!f. (!s. ((GSPEC (\\GEN%PVAR%11. (?x. (((SETSPEC GEN%PVAR%11) (x IN s)) (f x))))) = ((IMAGE f) s))))"}
{"_id": "356", "title": "BOUNDED_POS", "text": "|- (!s. ((bounded s) = (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\\ (!x. ((x IN s) ==> ((real_le (vector_norm x)) b)))))))"}
{"_id": "357", "title": "int_abs_th", "text": "|- (!x. ((real_of_int (int_abs x)) = (real_abs (real_of_int x))))"}
{"_id": "358", "title": "Hypermap_iso.orbit_comm", "text": "|- (!s. (!g2. (!f. (!g1. ((!x. ((x IN s) ==> ((g1 x) IN s))) ==> ((!x. ((x IN s) ==> ((f (g1 x)) = (g2 (f x))))) ==> (!x. ((x IN s) ==> (((orbit_map g2) (f x)) = ((IMAGE f) ((orbit_map g1) x)))))))))))"}
{"_id": "359", "title": "ABSOLUTELY_REAL_INTEGRABLE_MIN", "text": "|- (!f. (!g. (!s. ((((absolutely_real_integrable_on f) s) /\\ ((absolutely_real_integrable_on g) s)) ==> ((absolutely_real_integrable_on (\\x. ((real_min (f x)) (g x)))) s)))))"}
{"_id": "360", "title": "HD", "text": "|- ((HD ((CONS h) t)) = h)"}
{"_id": "361", "title": "Lp_ineqs_def.y6_fan", "text": "|- (!x. ((y6_fan x) = (distance x)))"}
{"_id": "362", "title": "REAL_DIV_RMUL", "text": "|- (!x. (!y. ((~ (y = (real_of_num (NUMERAL _0)))) ==> (((real_mul ((real_div x) y)) y) = x))))"}
{"_id": "363", "title": "int_le", "text": "|- (!x. (!y. (((int_le x) y) = ((real_le (real_of_int x)) (real_of_int y)))))"}
{"_id": "364", "title": "Hypermap.FACE_NOT_EMPTY", "text": "|- (!H. (!x. ((NUMERAL (BIT1 _0)) <= (CARD ((face H) x)))))"}
{"_id": "365", "title": "CONVEX_IMP_CONTRACTIBLE", "text": "|- (!s. ((convex s) ==> (contractible s)))"}
{"_id": "366", "title": "MIDPOINT_EQ_ENDPOINT", "text": "|- (!a. (!b. ((((midpoint (a , b)) = a) = (a = b)) /\\ ((((midpoint (a , b)) = b) = (a = b)) /\\ (((a = (midpoint (a , b))) = (a = b)) /\\ ((b = (midpoint (a , b))) = (a = b)))))))"}
{"_id": "367", "title": "MEASURABLE_MEASURABLE_PREIMAGE_CLOSED", "text": "|- (!f. (!s. (!t. ((((measurable_on f) s) /\\ ((measurable s) /\\ (closed t))) ==> (measurable (GSPEC (\\GEN%PVAR%3655. (?x. (((SETSPEC GEN%PVAR%3655) ((x IN s) /\\ ((f x) IN t))) x)))))))))"}
{"_id": "368", "title": "PATH_IMAGE_JOIN", "text": "|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((path_image ((++ g1) g2)) = ((path_image g1) UNION (path_image g2))))))"}
{"_id": "369", "title": "POWER_SERIES_CONV_IMP_ABSCONV_WEAK", "text": "|- (!a. (!k. (!w. (!z. ((((summable k) (\\n. ((complex_mul (a n)) ((complex_pow z) n)))) /\\ ((real_lt (vector_norm w)) (vector_norm z))) ==> ((summable k) (\\n. ((complex_mul (Cx (vector_norm (a n)))) ((complex_pow w) n)))))))))"}
{"_id": "370", "title": "MATRIX_VECTOR_MUL_LINEAR", "text": "|- (!A. (linear (\\x. ((matrix_vector_mul A) x))))"}
{"_id": "371", "title": "HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homotopy_equivalent s) ((IMAGE f) t)) = ((homotopy_equivalent s) t))))))"}
{"_id": "372", "title": "CONVEX_HULL_TRANSLATION", "text": "|- (!a. (!s. (((hull convex) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) ((hull convex) s)))))"}
{"_id": "373", "title": "REAL_LEBESGUE_MEASURABLE", "text": "|- (!s. ((real_lebesgue_measurable s) = (lebesgue_measurable ((IMAGE lift) s))))"}
{"_id": "374", "title": "Yxionxl.YXIONXL3", "text": "|- (!s. (!i. ((is_scs_v39 s) ==> ((scs_arrow_v39 (s INSERT EMPTY)) (((scs_prop_equ_v39 s) i) INSERT EMPTY)))))"}
{"_id": "375", "title": "RIGHT_POLAR_DECOMPOSITION_UNIQUE", "text": "|- (!A. (?!P. (?U. ((orthogonal_matrix U) /\\ ((positive_semidefinite P) /\\ (((matrix_mul U) P) = A))))))"}
{"_id": "376", "title": "span", "text": "|- (!s. ((span s) = ((hull subspace) s)))"}
{"_id": "377", "title": "INTEGRABLE_CMUL", "text": "|- (!f. (!c. (!s. (((integrable_on f) s) ==> ((integrable_on (\\x. ((% c) (f x)))) s)))))"}
{"_id": "378", "title": "ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV_GEN", "text": "|- (!f. (!s. (!t. (((is_interval s) /\\ ((bounded s) /\\ (((continuous_on f) s) /\\ (((has_bounded_variation_on f) s) /\\ ((COUNTABLE t) /\\ (!x. ((x IN (s DIFF t)) ==> ((differentiable f) ((within (at x)) s))))))))) ==> ((absolutely_continuous_on f) s)))))"}
{"_id": "379", "title": "MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT", "text": "|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV))) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3635. (?x. (((SETSPEC GEN%PVAR%3635) ((x IN s) /\\ ((real_gt ((f x) $ k)) a))) x))))))))))))"}
{"_id": "380", "title": "List_hypermap_iso.fan_imp_finite_set", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (FINITE V))))"}
{"_id": "381", "title": "VECTOR_MUL_ASSOC", "text": "|- (!a. (!b. (!x. (((% a) ((% b) x)) = ((% ((real_mul a) b)) x)))))"}
{"_id": "382", "title": "HAS_BOUNDED_VARIATION_ON_ID", "text": "|- (!s. ((bounded s) ==> ((has_bounded_variation_on (\\x. x)) s)))"}
{"_id": "383", "title": "Ssrbool.andb_idr", "text": "|- (!a. (!b. ((a ==> b) ==> ((a /\\ b) = a))))"}
{"_id": "384", "title": "PATH_COMPONENT_OF_SUBSET", "text": "|- (!s. (!t. (!x. (((s SUBSET t) /\\ (((path_component s) x) y)) ==> (((path_component t) x) y)))))"}
{"_id": "385", "title": "Lp_ineqs_proofs.dart4_yf_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((f_fan_pair_ext (V , E)) d)) = (y2_fan d)) /\\ (((y2_fan ((f_fan_pair_ext (V , E)) d)) = ((y7_fan (V , E)) d)) /\\ ((((y3_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y1_fan d)) /\\ ((((y5_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y6_fan d)) /\\ (((y6_fan ((f_fan_pair_ext (V , E)) d)) = ((y9_fan (V , E)) d)) /\\ ((((y7_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y3_fan (V , E)) d)) /\\ ((((y8_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y5_fan (V , E)) d)) /\\ (((y9_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y8_fan (V , E)) d)))))))))))))))))"}
{"_id": "386", "title": "REAL_LE_ANTISYM", "text": "|- (!x. (!y. ((((real_le x) y) /\\ ((real_le y) x)) = (x = y))))"}
{"_id": "387", "title": "Lp_ineqs_def.yn_fan", "text": "|- (!v. ((yn_fan v) = (vector_norm v)))"}
{"_id": "388", "title": "POLYTOPE_IMP_CONVEX", "text": "|- (!s. ((polytope s) ==> (convex s)))"}
{"_id": "389", "title": "COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS", "text": "|- (!s. (!t. ((((locally connected) s) \\/ ((compact s) /\\ (ANR t))) ==> ((!f. (!g. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((continuous_on g) s) /\\ (((IMAGE g) s) SUBSET t)))) ==> ((((homotopic_with (\\x. T)) (s , t)) f) g)))) = (!c. ((c IN (components s)) ==> (!f. (!g. ((((continuous_on f) c) /\\ ((((IMAGE f) c) SUBSET t) /\\ (((continuous_on g) c) /\\ (((IMAGE g) c) SUBSET t)))) ==> ((((homotopic_with (\\x. T)) (c , t)) f) g))))))))))"}
{"_id": "390", "title": "BILINEAR_BOUNDED_POS", "text": "|- (!h. ((bilinear h) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\\ (!x. (!y. ((real_le (vector_norm ((h x) y))) ((real_mul B) ((real_mul (vector_norm x)) (vector_norm y))))))))))"}
{"_id": "391", "title": "Functional_equation.proj_x1", "text": "|- (((((((proj_x1 x1) x2) x3) x4) x5) x6) = x1)"}
{"_id": "392", "title": "DIAGONAL_POSITIVE_SEMIDEFINITE", "text": "|- (!A. (!i. (((positive_semidefinite A) /\\ (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) ==> ((real_le (real_of_num (NUMERAL _0))) ((A $ i) $ i)))))"}
{"_id": "393", "title": "azim_def", "text": "|- (!w1. (!w2. (!w. (!v. (((((azim v) w) w1) w2) = (((COND ((collinear (v INSERT (w INSERT (w1 INSERT EMPTY)))) \\/ (collinear (v INSERT (w INSERT (w2 INSERT EMPTY)))))) (real_of_num (NUMERAL _0))) (@theta. (((real_le (real_of_num (NUMERAL _0))) theta) /\\ (((real_lt theta) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\\ (?h1. (?h2. (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\\ ((((% (distance (w , v))) e3) = ((vector_sub w) v)) /\\ (~ (w = v)))) ==> (?psi. (?r1. (?r2. ((((vector_sub w1) v) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub w) v))))) /\\ ((((vector_sub w2) v) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) theta)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) theta)))) e2)) ((% h2) ((vector_sub w) v))))) /\\ (((real_lt (real_of_num (NUMERAL _0))) r1) /\\ ((real_lt (real_of_num (NUMERAL _0))) r2))))))))))))))))))))))"}
{"_id": "394", "title": "HOMEOMORPHIC_REFL", "text": "|- (!s. ((homeomorphic s) s))"}
{"_id": "395", "title": "Functional_equation.functional_dih4_x_div_sqrtdelta_posbranch", "text": "|- (dih4_x_div_sqrtdelta_posbranch = (rotate4 dih_x_div_sqrtdelta_posbranch))"}
{"_id": "396", "title": "Nonlinear_lemma.hminus_lt_h0", "text": "|- ((real_lt hminus) h0)"}
{"_id": "397", "title": "INFINITE", "text": "|- (!s. ((INFINITE s) = (~ (FINITE s))))"}
{"_id": "398", "title": "MBOUNDED_SUBSET", "text": "|- (!m. (!s. (!t. ((((mbounded m) t) /\\ (s SUBSET t)) ==> ((mbounded m) s)))))"}
{"_id": "399", "title": "ENR_TRANSLATION", "text": "|- (!a. (!s. ((ENR ((IMAGE (\\x. ((vector_add a) x))) s)) = (ENR s))))"}
{"_id": "400", "title": "I_THM", "text": "|- (!x. ((I x) = x))"}
{"_id": "401", "title": "Seq.leq_size_uniq", "text": "|- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) ==> ((MEM x) s2))) ==> (((sizel s2) <= (sizel s1)) ==> (uniq s2))))))"}
{"_id": "402", "title": "BOUNDED_BALL", "text": "|- (!x. (!e. (bounded (ball (x , e)))))"}
{"_id": "403", "title": "Seq.all_pred1P", "text": "|- (!x. (!s. ((s = ((nseq (sizel s)) x)) = ((all (pred1 x)) s))))"}
{"_id": "404", "title": "FORALL_1", "text": "|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (NUMERAL (BIT1 _0)))) ==> (P i))) = (P (NUMERAL (BIT1 _0))))"}
{"_id": "405", "title": "ALL_MEM", "text": "|- (!P. (!l. ((!x. (((MEM x) l) ==> (P x))) = ((ALL P) l))))"}
{"_id": "406", "title": "OPEN_CLOSED", "text": "|- (!s. ((open s) = (closed (UNIV DIFF s))))"}
{"_id": "407", "title": "PASTECART_IN_PCROSS", "text": "|- (!s. (!t. (!x. (!y. ((((pastecart x) y) IN (s PCROSS t)) = ((x IN s) /\\ (y IN t)))))))"}
{"_id": "408", "title": "CLOSURE_UNION", "text": "|- (!s. (!t. ((closure (s UNION t)) = ((closure s) UNION (closure t)))))"}
{"_id": "409", "title": "EULER", "text": "|- (!z. ((cexp z) = ((complex_mul (Cx (exp (Re z)))) ((vector_add (Cx (cos (Im z)))) ((complex_mul ii) (Cx (sin (Im z))))))))"}
{"_id": "410", "title": "COMPLEX_DIFFERENTIABLE_AT_CASN", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((complex_differentiable casn) (at z))))"}
{"_id": "411", "title": "HAS_REAL_INTEGRAL_STRADDLE_NULL", "text": "|- (!f. (!g. (!s. (((!x. ((x IN s) ==> (((real_le (real_of_num (NUMERAL _0))) (f x)) /\\ ((real_le (f x)) (g x))))) /\\ (((has_real_integral g) (real_of_num (NUMERAL _0))) s)) ==> (((has_real_integral f) (real_of_num (NUMERAL _0))) s)))))"}
{"_id": "412", "title": "Conforming.STEP3_REDUCE_FAN", "text": "|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((fan80 (x , (V , E))) /\\ ((ds IN (face_set (hypermap1_of_fanx (x , (V , E))))) /\\ (((CARD ds) > (NUMERAL (BIT1 (BIT1 _0)))) /\\ (((f1 INSERT (f2 INSERT (f3 INSERT EMPTY))) SUBSET ds) /\\ ((((((f1_fan x) V) E) f1) = f2) /\\ ((((((f1_fan x) V) E) f2) = f3) /\\ ((~ (((((f1_fan x) V) E) f3) = f1)) /\\ (((pr2 f1) = v) /\\ (((pr2 f2) = u) /\\ (((pr2 f3) = w) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))))))))) ==> (FAN (x , (V , E1)))))))))))))))"}
{"_id": "413", "title": "ALL", "text": "|- ((((ALL P) NIL) = T) /\\ (((ALL P) ((CONS h) t)) = ((P h) /\\ ((ALL P) t))))"}
{"_id": "414", "title": "LET_DEF", "text": "|- (!f. (!x. (((LET f) x) = (f x))))"}
{"_id": "415", "title": "HAUSDIST_EMPTY", "text": "|- ((!t. ((hausdist (EMPTY , t)) = (real_of_num (NUMERAL _0)))) /\\ (!s. ((hausdist (s , EMPTY)) = (real_of_num (NUMERAL _0)))))"}
{"_id": "416", "title": "CLOSED_SING", "text": "|- (!a. (closed (a INSERT EMPTY)))"}
{"_id": "417", "title": "int_ge", "text": "|- (!x. (!y. (((int_ge x) y) = ((real_ge (real_of_int x)) (real_of_int y)))))"}
{"_id": "418", "title": "Lp_ineqs_def.azim3_fan", "text": "|- (!V. (!E. (!x. (((azim3_fan (V , E)) x) = ((azim_dart (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))"}
{"_id": "419", "title": "Lp_main_estimate.in_e_prime", "text": "|- (!E. (!f. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN ((e_prime E) f)) = (((v INSERT (w INSERT EMPTY)) IN E) /\\ (((v , w) IN f) \\/ ((w , v) IN f))))))))"}
{"_id": "420", "title": "REAL_CLOSED", "text": "|- (!s. ((real_closed s) = (closed ((IMAGE lift) s))))"}
{"_id": "421", "title": "CONVEX_EMPTY", "text": "|- (convex EMPTY)"}
{"_id": "422", "title": "CONNECTED_WITH_OUTSIDE", "text": "|- (!s. (((closed s) /\\ (connected s)) ==> (connected (s UNION (outside s)))))"}
{"_id": "423", "title": "SUPERADMISSIBLE_COND", "text": "|- (!<<. (!p. (!P. (!s. (!h. (!k. ((((((admissible <<) p) s) P) /\\ (((((superadmissible <<) (\\f. (\\x. (((p f) x) /\\ ((P f) x))))) s) h) /\\ ((((superadmissible <<) (\\f. (\\x. (((p f) x) /\\ (~ ((P f) x)))))) s) k))) ==> ((((superadmissible <<) p) s) (\\f. (\\x. (((COND ((P f) x)) ((h f) x)) ((k f) x))))))))))))"}
{"_id": "424", "title": "PATH_CONNECTED_OPEN_DELETE", "text": "|- (!s. (!a. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ ((open s) /\\ (connected s))) ==> (path_connected (s DELETE a)))))"}
{"_id": "425", "title": "CLOSURE_TRANSLATION", "text": "|- (!a. (!s. ((closure ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) (closure s)))))"}
{"_id": "426", "title": "COMPACT_SPHERE", "text": "|- (!a. (!r. (compact (GSPEC (\\GEN%PVAR%555. (?y. (((SETSPEC GEN%PVAR%555) ((vector_norm ((vector_sub y) a)) = r)) y)))))))"}
{"_id": "427", "title": "NPRODUCT_CLAUSES_NUMSEG", "text": "|- ((!m. (((nproduct (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (NUMERAL (BIT1 _0))))) /\\ (!m. (!n. (((nproduct (m .. (SUC n))) f) = (((COND (m <= (SUC n))) (((nproduct (m .. n)) f) * (f (SUC n)))) ((nproduct (m .. n)) f))))))"}
{"_id": "428", "title": "REAL_MEASURE_UNIQUE", "text": "|- (!s. (!m. (((has_real_measure s) m) ==> ((real_measure s) = m))))"}
{"_id": "429", "title": "LINEAR_COMPOSE_ADD", "text": "|- (!f. (!g. (((linear f) /\\ (linear g)) ==> (linear (\\x. ((vector_add (f x)) (g x)))))))"}
{"_id": "430", "title": "orthogonal", "text": "|- (!x. (!y. (((orthogonal x) y) = (((dot x) y) = (real_of_num (NUMERAL _0))))))"}
{"_id": "431", "title": "REAL_SEGMENT_SEGMENT", "text": "|- ((!a. (!b. ((closed_real_segment ((CONS (a , b)) NIL)) = ((IMAGE drop) (closed_segment ((CONS ((lift a) , (lift b))) NIL)))))) /\\ (!a. (!b. ((open_real_segment (a , b)) = ((IMAGE drop) (open_segment ((lift a) , (lift b))))))))"}
{"_id": "432", "title": "COMPLEX_MUL_RINV", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((complex_mul z) (complex_inv z)) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "433", "title": "PATH_COMPONENT_TRANS", "text": "|- (!s. (!x. (!y. (((((path_component s) x) y) /\\ (((path_component s) y) z)) ==> (((path_component s) x) z)))))"}
{"_id": "434", "title": "FRONTIER_CLOSED", "text": "|- (!s. (closed (frontier s)))"}
{"_id": "435", "title": "Seq.undup_uniq", "text": "|- (!s. (uniq (undup s)))"}
{"_id": "436", "title": "DEPENDENT_IMP_AFFINE_DEPENDENT", "text": "|- (!a. (!s. (((dependent (GSPEC (\\GEN%PVAR%1511. (?x. (((SETSPEC GEN%PVAR%1511) (x IN s)) ((vector_sub x) a)))))) /\\ (~ (a IN s))) ==> (affine_dependent (a INSERT s)))))"}
{"_id": "437", "title": "BOUNDED_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((bounded ((IMAGE f) s)) = (bounded s)))))"}
{"_id": "438", "title": "GDELTA_EMPTY", "text": "|- (gdelta EMPTY)"}
{"_id": "439", "title": "EVENTUALLY_WITHIN", "text": "|- (!s. (!a. (!p. (((eventually p) ((within (at a)) s)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (((x IN s) /\\ (((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\\ ((real_lt (distance (x , a))) d))) ==> (p x)))))))))"}
{"_id": "440", "title": "HULL_ANTIMONO", "text": "|- (!P. (!Q. (!s. ((P SUBSET Q) ==> (((hull Q) s) SUBSET ((hull P) s))))))"}
{"_id": "441", "title": "Sphere.vol2r", "text": "|- (!r. (!y. (((vol2r y) r) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul pi) ((real_div ((real_sub ((real_mul r) r)) ((real_pow ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))))"}
{"_id": "442", "title": "Ssrnat.uphalfS", "text": "|- (!n. ((uphalf (SUC n)) = (SUC (half n))))"}
{"_id": "443", "title": "Functional_equation.functional_gchi4_x", "text": "|- (gchi4_x = ((mul6 (uni (gchi , proj_y4))) dih4_x))"}
{"_id": "444", "title": "Lp_ineqs_proofs.f_fan_pair_ext_power_inverse", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. (((d IN (dart_of_fan (V , E))) /\\ (k < (CARD ((face (hypermap_of_fan (V , E))) d)))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) = (((POWER (inverse (f_fan_pair_ext (V , E)))) ((CARD ((face (hypermap_of_fan (V , E))) d)) - k)) d))))))))"}
{"_id": "445", "title": "Seq.cats1", "text": "|- (!s. (!z. (((cat s) ((CONS z) NIL)) = ((rcons s) z))))"}
{"_id": "446", "title": "RELATIVE_INTERIOR_INTERIOR", "text": "|- (!s. ((((hull affine) s) = UNIV) ==> ((relative_interior s) = (interior s))))"}
{"_id": "447", "title": "SUB_ADD", "text": "|- (!m. (!n. ((n <= m) ==> (((m - n) + n) = m))))"}
{"_id": "448", "title": "HAS_BOUNDED_VARIATION_COMPOSE_HOMEOMORPHISM", "text": "|- (!f. (!f'. (!g. (!s. (!t. (((is_interval s) /\\ (((homeomorphism (s , t)) (f , f')) /\\ ((has_bounded_variation_on g) t))) ==> ((has_bounded_variation_on (g o f)) s)))))))"}
{"_id": "449", "title": "CONNECTED_IMP_PERFECT_AFF_DIM", "text": "|- (!s. (!x. (((connected s) /\\ ((~ ((aff_dim s) = (int_of_num (NUMERAL _0)))) /\\ (x IN s))) ==> ((limit_point_of x) s))))"}
{"_id": "450", "title": "GDELTA_UNIV", "text": "|- (gdelta UNIV)"}
{"_id": "451", "title": "LENGTH_MAP", "text": "|- (!l. (!f. ((LENGTH ((MAP f) l)) = (LENGTH l))))"}
{"_id": "452", "title": "Hypermap_and_fan.FACE_LAST_POINT", "text": "|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((v , w) IN (dart1_of_fan (V , E)))) ==> ((LET (\\w'. (LET_END ((w' , v) IN ((face (hypermap_of_fan (V , E))) (v , w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))"}
{"_id": "453", "title": "Hypermap.LT1_NZ", "text": "|- (!n. (((NUMERAL (BIT1 _0)) <= n) = ((NUMERAL _0) < n)))"}
{"_id": "454", "title": "SETDIST_ZERO", "text": "|- (!s. (!t. ((~ ((DISJOINT s) t)) ==> ((setdist (s , t)) = (real_of_num (NUMERAL _0))))))"}
{"_id": "455", "title": "CONVEX_DISTANCE", "text": "|- (!s. (!a. ((convex_on (\\x. (distance (a , x)))) s)))"}
{"_id": "456", "title": "POLYHEDRON_TRANSLATION_EQ", "text": "|- (!a. (!s. ((polyhedron ((IMAGE (\\x. ((vector_add a) x))) s)) = (polyhedron s))))"}
{"_id": "457", "title": "IS_HULL", "text": "|- (!P. (!s. ((!f. ((!s. ((s IN f) ==> (P s))) ==> (P (INTERS f)))) ==> ((P s) = (?t. (s = ((hull P) t)))))))"}
{"_id": "458", "title": "Miqmcsn.scs_3T4_prime2", "text": "|- (scs_3T4_prime2 = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , cstab)) NIL))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) NIL)) cstab) (NUMERAL (BIT1 (BIT1 _0))))))"}
{"_id": "459", "title": "AFFINE_DEPENDENT_EXPLICIT", "text": "|- (!p. ((affine_dependent p) = (?s. (?u. ((FINITE s) /\\ ((s SUBSET p) /\\ ((((sum s) u) = (real_of_num (NUMERAL _0))) /\\ ((?v. ((v IN s) /\\ (~ ((u v) = (real_of_num (NUMERAL _0)))))) /\\ (((vsum s) (\\v. ((% (u v)) v))) = (vec (NUMERAL _0)))))))))))"}
{"_id": "460", "title": "PATH_VECTOR_POLYNOMIAL_FUNCTION", "text": "|- (!g. ((vector_polynomial_function g) ==> (path g)))"}
{"_id": "461", "title": "NORM_CEXP", "text": "|- (!z. ((vector_norm (cexp z)) = (exp (Re z))))"}
{"_id": "462", "title": "ARC_IMP_PATH", "text": "|- (!g. ((arc g) ==> (path g)))"}
{"_id": "463", "title": "Seq.can_inj", "text": "|- (!f. (!g. ((!x. ((g (f x)) = x)) ==> (!x. (!y. (((f x) = (f y)) ==> (x = y)))))))"}
{"_id": "464", "title": "REAL_POW_NZ", "text": "|- (!x. (!n. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (~ (((real_pow x) n) = (real_of_num (NUMERAL _0)))))))"}
{"_id": "465", "title": "Hypermap_iso.bij_ext", "text": "|- (!f. (!g. (!s. (!t. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> ((((BIJ f) s) t) ==> (((BIJ g) s) t)))))))"}
{"_id": "466", "title": "SUM_DIFF", "text": "|- (!f. (!s. (!t. (((FINITE s) /\\ (t SUBSET s)) ==> (((sum (s DIFF t)) f) = ((real_sub ((sum s) f)) ((sum t) f)))))))"}
{"_id": "467", "title": "REAL_SUB_RZERO", "text": "|- (!x. (((real_sub x) (real_of_num (NUMERAL _0))) = x))"}
{"_id": "468", "title": "TC_MONO", "text": "|- (!R. (!S. ((!x. (!y. (((R x) y) ==> ((S x) y)))) ==> (!x. (!y. ((((TC R) x) y) ==> (((TC S) x) y)))))))"}
{"_id": "469", "title": "Hypermap_iso.iso_number_of_nodes", "text": "|- (!f. (!G. (!H. (((hyp_iso f) (H , G)) ==> ((number_of_nodes G) = (number_of_nodes H))))))"}
{"_id": "470", "title": "HAS_SIZE", "text": "|- (!s. (!n. ((s HAS_SIZE n) = ((FINITE s) /\\ ((CARD s) = n)))))"}
{"_id": "471", "title": "SPHERE_UNION_BALL", "text": "|- (!a. (!r. (((sphere (a , r)) UNION (ball (a , r))) = (cball (a , r)))))"}
{"_id": "472", "title": "BOHL", "text": "|- (!f. (!s. (!a. ((((continuous_on f) s) /\\ ((convex s) /\\ ((compact s) /\\ (a IN (interior s))))) ==> ((?x. ((x IN s) /\\ ((f x) = x))) \\/ (?x. ((x IN (frontier s)) /\\ (x IN (open_segment (a , (f x)))))))))))"}
{"_id": "473", "title": "REFLECT_REAL_INTERVAL", "text": "|- ((!a. (!b. (((IMAGE real_neg) (closed_real_interval ((CONS (a , b)) NIL))) = (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))))) /\\ (!a. (!b. (((IMAGE real_neg) (open_real_interval (a , b))) = (open_real_interval ((real_neg b) , (real_neg a)))))))"}
{"_id": "474", "title": "Seq.count_predT", "text": "|- (!s. (((count predT) s) = (sizel s)))"}
{"_id": "475", "title": "LINEAR_CNJ", "text": "|- (linear cnj)"}
{"_id": "476", "title": "INTERIOR_SPHERE", "text": "|- (!a. (!r. ((interior (sphere (a , r))) = EMPTY)))"}
{"_id": "477", "title": "EXISTS_PAIR_THM", "text": "|- (!P. ((?p. (P p)) = (?p1. (?p2. (P (p1 , p2))))))"}
{"_id": "478", "title": "GEQ_DEF", "text": "|- (!a. (!b. (((GEQ a) b) = (a = b))))"}
{"_id": "479", "title": "CONIC_IMP_CONTRACTIBLE", "text": "|- (!s. ((conic s) ==> (contractible s)))"}
{"_id": "480", "title": "RE_CLOG_POS_LE", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((real_le (real_abs (Im (clog z)))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_le (real_of_num (NUMERAL _0))) (Re z)))))"}
{"_id": "481", "title": "ABSOLUTELY_SETCONTINUOUS_ON_NEG", "text": "|- (!f. (!s. (((absolutely_setcontinuous_on (\\x. (vector_neg (f x)))) s) = ((absolutely_setcontinuous_on f) s))))"}
{"_id": "482", "title": "EXP_2", "text": "|- (!n. ((n EXP (NUMERAL (BIT0 (BIT1 _0)))) = (n * n)))"}
{"_id": "483", "title": "EXP_MONO_LE", "text": "|- (!x. (!y. (!n. (((x EXP n) <= (y EXP n)) = ((x <= y) \\/ (n = (NUMERAL _0)))))))"}
{"_id": "484", "title": "FORALL_DOT_EQ_0", "text": "|- ((!y. ((!x. (((dot x) y) = (real_of_num (NUMERAL _0)))) = (y = (vec (NUMERAL _0))))) /\\ (!x. ((!y. (((dot x) y) = (real_of_num (NUMERAL _0)))) = (x = (vec (NUMERAL _0))))))"}
{"_id": "485", "title": "PASTECART_SUB", "text": "|- (!x1. (!y1. (!x2. (!y2. (((vector_sub ((pastecart x1) y1)) ((pastecart x2) y2)) = ((pastecart ((vector_sub x1) x2)) ((vector_sub y1) y2)))))))"}
{"_id": "486", "title": "REAL_MUL_RID", "text": "|- (!x. (((real_mul x) (real_of_num (NUMERAL (BIT1 _0)))) = x))"}
{"_id": "487", "title": "epigraph", "text": "|- (!s. (!f. (((epigraph s) f) = (GSPEC (\\GEN%PVAR%1703. (?xy. (((SETSPEC GEN%PVAR%1703) (((fstcart xy) IN s) /\\ ((real_le (f (fstcart xy))) (drop (sndcart xy))))) xy)))))))"}
{"_id": "488", "title": "real_differentiable", "text": "|- (!f. (!net. (((real_differentiable f) net) = (?f'. (((has_real_derivative f) f') net)))))"}
{"_id": "489", "title": "INT_NOT_LE", "text": "|- (!x. (!y. ((~ ((int_le x) y)) = ((int_lt y) x))))"}
{"_id": "490", "title": "HAS_FRECHET_DERIVATIVE_UNIQUE_AT", "text": "|- (!f. (!f'. (!x. ((((has_derivative f) f') (at x)) ==> (((frechet_derivative f) (at x)) = f')))))"}
{"_id": "491", "title": "LT_SUC", "text": "|- (!m. (!n. (((SUC m) < (SUC n)) = (m < n))))"}
{"_id": "492", "title": "ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE", "text": "|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((integrable_on f) s))))"}
{"_id": "493", "title": "PATHFINISH_LINEAR_IMAGE", "text": "|- (!f. (!g. ((linear f) ==> ((pathfinish (f o g)) = (f (pathfinish g))))))"}
{"_id": "494", "title": "Hypermap_iso.image_ext_eq", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((IMAGE f) s) = ((IMAGE g) s))))))"}
{"_id": "495", "title": "HAS_COMPLEX_DERIVATIVE_CATN", "text": "|- (!z. ((((Re z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Im z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative catn) (complex_inv ((vector_add (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow z) (NUMERAL (BIT0 (BIT1 _0))))))) (at z))))"}
{"_id": "496", "title": "NEGLIGIBLE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((negligible ((IMAGE f) s)) = (negligible s)))))"}
{"_id": "497", "title": "SIN_CIRCLE", "text": "|- (!x. (((real_add ((real_pow (sin x)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (cos x)) (NUMERAL (BIT0 (BIT1 _0))))) = (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "498", "title": "BIT1", "text": "|- (!n. ((BIT1 n) = (SUC (n + n))))"}
{"_id": "499", "title": "Lp_ineqs_proofs.face4_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((LET (\\w'. (LET_END ((LET (\\u. (LET_END (((v , w) IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((((f_fan_pair_ext (V , E)) (v , w)) = (w , u)) /\\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) (v , w)) = (u , w')) /\\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) (v , w)) = (w' , v)) /\\ ((((f_fan_pair_ext (V , E)) (w , u)) = (u , w')) /\\ ((((f_fan_pair_ext (V , E)) (u , w')) = (w' , v)) /\\ (((f_fan_pair_ext (V , E)) (w' , v)) = (v , w))))))) /\\ ((((v , w) IN (dart1_of_fan (V , E))) /\\ (((w , u) IN (dart1_of_fan (V , E))) /\\ (((u , w') IN (dart1_of_fan (V , E))) /\\ ((w' , v) IN (dart1_of_fan (V , E)))))) /\\ (((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , u) INSERT ((u , w') INSERT ((w' , v) INSERT EMPTY))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))"}
{"_id": "500", "title": "rational", "text": "|- (!x. ((rational x) = (?m. (?n. ((integer m) /\\ ((integer n) /\\ ((~ (n = (real_of_num (NUMERAL _0)))) /\\ (x = ((real_div m) n)))))))))"}
{"_id": "501", "title": "complex_add", "text": "|- (((vector_add w) z) = (complex (((real_add (Re w)) (Re z)) , ((real_add (Im w)) (Im z)))))"}
{"_id": "502", "title": "CONNECTED_LIMIT_POINTS_EQ_CLOSURE", "text": "|- (!s. (((connected s) /\\ (~ (?a. (s = (a INSERT EMPTY))))) ==> ((GSPEC (\\GEN%PVAR%584. (?x. (((SETSPEC GEN%PVAR%584) ((limit_point_of x) s)) x)))) = (closure s))))"}
{"_id": "503", "title": "SERIES_ADD", "text": "|- (!x. (!x0. (!y. (!y0. (!s. (((((sums x) x0) s) /\\ (((sums y) y0) s)) ==> (((sums (\\n. ((vector_add (x n)) (y n)))) ((vector_add x0) y0)) s)))))))"}
{"_id": "504", "title": "Fan.power_map_points", "text": "|- (((((((((power_map_points f) x) V) E) v) w) (NUMERAL _0)) = w) /\\ ((((((((power_map_points f) x) V) E) v) w) (SUC n)) = (((((f x) V) E) v) (((((((power_map_points f) x) V) E) v) w) n))))"}
{"_id": "505", "title": "TOPSPACE_EUCLIDEAN_SUBTOPOLOGY", "text": "|- (!s. ((topspace ((subtopology euclidean) s)) = s))"}
{"_id": "506", "title": "Nonlinear_lemma.sqrtxx", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((sqrt ((real_mul x) x)) = x)))"}
{"_id": "507", "title": "FINITE_IMP_COMPACT", "text": "|- (!s. ((FINITE s) ==> (compact s)))"}
{"_id": "508", "title": "REAL_POW_NEG", "text": "|- (!x. (!n. (((real_pow (real_neg x)) n) = (((COND (EVEN n)) ((real_pow x) n)) (real_neg ((real_pow x) n))))))"}
{"_id": "509", "title": "EQ_MULT_RCANCEL", "text": "|- (!m. (!n. (!p. (((m * p) = (n * p)) = ((m = n) \\/ (p = (NUMERAL _0)))))))"}
{"_id": "510", "title": "FINITE_DIFF", "text": "|- (!s. (!t. ((FINITE s) ==> (FINITE (s DIFF t)))))"}
{"_id": "511", "title": "Fnjlbxs.vector_of_list", "text": "|- (!list. (!d. (((vector_of_list d) list) = (lambdai. (((COND (((i - (NUMERAL (BIT1 _0))) DIV d) < (LENGTH list))) (((EL ((i - (NUMERAL (BIT1 _0))) DIV d)) list) $ (((i - (NUMERAL (BIT1 _0))) MOD d) + (NUMERAL (BIT1 _0))))) (real_of_num (NUMERAL _0)))))))"}
{"_id": "512", "title": "Hypermap_and_fan.NOT_IN_DART1_OF_FAN", "text": "|- (!V. (!E. (!v. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (~ ((v , v) IN (dart1_of_fan (V , E))))))))"}
{"_id": "513", "title": "IN_REAL_INTERVAL", "text": "|- (!a. (!b. (!x. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) = (((real_le a) x) /\\ ((real_le x) b))) /\\ ((x IN (open_real_interval (a , b))) = (((real_lt a) x) /\\ ((real_lt x) b)))))))"}
{"_id": "514", "title": "SPAN_OPEN", "text": "|- (!s. (((open s) /\\ (~ (s = EMPTY))) ==> ((span s) = UNIV)))"}
{"_id": "515", "title": "DIVISION", "text": "|- (!m. (!n. ((~ (n = (NUMERAL _0))) ==> ((m = (((m DIV n) * n) + (m MOD n))) /\\ ((m MOD n) < n)))))"}
{"_id": "516", "title": "Seq.mask", "text": "|- ((((mask NIL) s') = NIL) /\\ ((((mask m') NIL) = NIL) /\\ (((mask ((CONS b) m')) ((CONS x) s')) = (((COND b) ((CONS x) ((mask m') s'))) ((mask m') s')))))"}
{"_id": "517", "title": "RELATIVE_FRONTIER_TRANSLATION", "text": "|- (!a. (!s. ((relative_frontier ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) (relative_frontier s)))))"}
{"_id": "518", "title": "CONVEX_IMP_LOCALLY_CONNECTED", "text": "|- (!s. ((convex s) ==> ((locally connected) s)))"}
{"_id": "519", "title": "vector_sub", "text": "|- (!x. (!y. (((vector_sub x) y) = (lambdai. ((real_sub (x $ i)) (y $ i))))))"}
{"_id": "520", "title": "PAIRWISE_ORTHOGONAL_INDEPENDENT", "text": "|- (!s. ((((pairwise orthogonal) s) /\\ (~ ((vec (NUMERAL _0)) IN s))) ==> (independent s)))"}
{"_id": "521", "title": "INTERIOR_CLOSURE_INTER_OPEN", "text": "|- (!s. (!t. (((open s) /\\ (open t)) ==> ((interior (closure (s INTER t))) = ((interior (closure s)) INTER (interior (closure t)))))))"}
{"_id": "522", "title": "Lvducxu.FIRST_AAUHTVE2", "text": "|- ((FAN (x , (V , E))) ==> ((FINITE ((darts_of_hyp E) V)) /\\ (((permutes (ee_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\\ (((permutes (nn_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\\ (((permutes (ff_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\\ ((((ee_of_hyp (x , (V , E))) o ((nn_of_hyp (x , (V , E))) o (ff_of_hyp (x , (V , E))))) = I) /\\ (((ee_of_hyp (x , (V , E))) o (ee_of_hyp (x , (V , E)))) = I)))))))"}
{"_id": "523", "title": "CONTINUOUS_COMPLEX_DIV", "text": "|- (!net. (!f. (!g. ((((continuous f) net) /\\ (((continuous g) net) /\\ (~ ((g (netlimit net)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((continuous (\\x. ((complex_div (f x)) (g x)))) net)))))"}
{"_id": "524", "title": "SEPARATION_NORMAL", "text": "|- (!s. (!t. (((closed s) /\\ ((closed t) /\\ ((s INTER t) = EMPTY))) ==> (?u. (?v. ((open u) /\\ ((open v) /\\ ((s SUBSET u) /\\ ((t SUBSET v) /\\ ((u INTER v) = EMPTY))))))))))"}
{"_id": "525", "title": "FORALL_IN_GSPEC", "text": "|- ((!P. (!f. ((!z. ((z IN (GSPEC (\\GEN%PVAR%21. (?x. (((SETSPEC GEN%PVAR%21) (P x)) (f x)))))) ==> (Q z))) = (!x. ((P x) ==> (Q (f x))))))) /\\ ((!P. (!f. ((!z. ((z IN (GSPEC (\\GEN%PVAR%22. (?x. (?y. (((SETSPEC GEN%PVAR%22) ((P x) y)) ((f x) y))))))) ==> (Q z))) = (!x. (!y. (((P x) y) ==> (Q ((f x) y)))))))) /\\ ((!P. (!f. ((!z. ((z IN (GSPEC (\\GEN%PVAR%23. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%23) (((P w) x) y)) (((f w) x) y)))))))) ==> (Q z))) = (!w. (!x. (!y. ((((P w) x) y) ==> (Q (((f w) x) y))))))))) /\\ (!P. (!f. ((!z. ((z IN (GSPEC (\\GEN%PVAR%24. (?v. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%24) ((((P v) w) x) y)) ((((f v) w) x) y))))))))) ==> (Q z))) = (!v. (!w. (!x. (!y. (((((P v) w) x) y) ==> (Q ((((f v) w) x) y)))))))))))))"}
{"_id": "526", "title": "CX_INV", "text": "|- (!x. ((Cx (real_inv x)) = (complex_inv (Cx x))))"}
{"_id": "527", "title": "SIMPLY_CONNECTED_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((simply_connected ((IMAGE f) s)) = (simply_connected s)))))"}
{"_id": "528", "title": "DIAMETER_LE", "text": "|- (!s. ((((~ (s = EMPTY)) \\/ ((real_le (real_of_num (NUMERAL _0))) d)) /\\ (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub x) y))) d))))) ==> ((real_le (diameter s)) d)))"}
{"_id": "529", "title": "SUM_LT", "text": "|- (!f. (!g. (!s. (((FINITE s) /\\ ((!x. ((x IN s) ==> ((real_le (f x)) (g x)))) /\\ (?x. ((x IN s) /\\ ((real_lt (f x)) (g x)))))) ==> ((real_lt ((sum s) f)) ((sum s) g))))))"}
{"_id": "530", "title": "HELLY_CLOSED", "text": "|- (!f. ((((FINITE f) ==> ((CARD f) >= ((dimindex UNIV) + (NUMERAL (BIT1 _0))))) /\\ ((!s. ((s IN f) ==> ((convex s) /\\ (closed s)))) /\\ ((?s. ((s IN f) /\\ (bounded s))) /\\ (!t. (((t SUBSET f) /\\ ((FINITE t) /\\ ((CARD t) = ((dimindex UNIV) + (NUMERAL (BIT1 _0)))))) ==> (~ ((INTERS t) = EMPTY))))))) ==> (~ ((INTERS f) = EMPTY))))"}
{"_id": "531", "title": "HAS_DERIVATIVE_ZERO_UNIQUE_STRONG_INTERVAL", "text": "|- (!f. (!a. (!b. (!k. (!y. (((COUNTABLE k) /\\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (((f a) = y) /\\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF k)) ==> (((has_derivative f) (\\h. (vec (NUMERAL _0)))) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = y)))))))))"}
{"_id": "532", "title": "List_hypermap.darts_of_list", "text": "|- (!L. ((darts_of_list L) = (set_of_list (list_of_darts L))))"}
{"_id": "533", "title": "Seq.size_nil", "text": "|- ((sizel NIL) = (NUMERAL _0))"}
{"_id": "534", "title": "INTERS_IMAGE", "text": "|- (!f. (!s. ((INTERS ((IMAGE f) s)) = (GSPEC (\\GEN%PVAR%30. (?y. (((SETSPEC GEN%PVAR%30) (!x. ((x IN s) ==> (y IN (f x))))) y)))))))"}
{"_id": "535", "title": "mbounded", "text": "|- (!s. (!m. (((mbounded m) s) = (?c. (?b. (s SUBSET ((mcball m) (c , b))))))))"}
{"_id": "536", "title": "SEGMENT_TRANSLATION", "text": "|- ((!c. (!a. (!b. ((closed_segment ((CONS (((vector_add c) a) , ((vector_add c) b))) NIL)) = ((IMAGE (\\x. ((vector_add c) x))) (closed_segment ((CONS (a , b)) NIL))))))) /\\ (!c. (!a. (!b. ((open_segment (((vector_add c) a) , ((vector_add c) b))) = ((IMAGE (\\x. ((vector_add c) x))) (open_segment (a , b))))))))"}
{"_id": "537", "title": "HAS_REAL_DERIVATIVE_FROM_COMPLEX_AT", "text": "|- (!f. (!f'. (!x. (((((has_complex_derivative f) f') (at (Cx x))) /\\ (!z. ((real z) ==> (real (f z))))) ==> (((has_real_derivative (Re o (f o Cx))) (Re f')) (atreal x))))))"}
{"_id": "538", "title": "CLOSURE_EQ", "text": "|- (!s. (((closure s) = s) = (closed s)))"}
{"_id": "539", "title": "I_DEF", "text": "|- (I = (\\x. x))"}
{"_id": "540", "title": "TAGGED_DIVISION_UNIONS", "text": "|- (!iset. (!pfn. (((FINITE iset) /\\ ((!i. ((i IN iset) ==> ((tagged_division_of (pfn i)) i))) /\\ (!i1. (!i2. (((i1 IN iset) /\\ ((i2 IN iset) /\\ (~ (i1 = i2)))) ==> (((interior i1) INTER (interior i2)) = EMPTY)))))) ==> ((tagged_division_of (UNIONS ((IMAGE pfn) iset))) (UNIONS iset)))))"}
{"_id": "541", "title": "RTC", "text": "|- (!R. ((RTC R) = (RC (TC R))))"}
{"_id": "542", "title": "NSUM_UNION_NONZERO", "text": "|- (!f. (!s. (!t. (((FINITE s) /\\ ((FINITE t) /\\ (!x. ((x IN (s INTER t)) ==> ((f x) = (NUMERAL _0)))))) ==> (((nsum (s UNION t)) f) = (((nsum s) f) + ((nsum t) f)))))))"}
{"_id": "543", "title": "IMAGE_SNDCART_PCROSS", "text": "|- (!s. (!t. (((IMAGE sndcart) (s PCROSS t)) = (((COND (s = EMPTY)) EMPTY) t))))"}
{"_id": "544", "title": "retraction", "text": "|- (!s. (!t. (!r. (((retraction (s , t)) r) = ((t SUBSET s) /\\ (((continuous_on r) s) /\\ ((((IMAGE r) s) SUBSET t) /\\ (!x. ((x IN t) ==> ((r x) = x))))))))))"}
{"_id": "545", "title": "ABSOLUTELY_INTEGRABLE_NORM", "text": "|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((absolutely_integrable_on (\\x. (lift (vector_norm (f x))))) s))))"}
{"_id": "546", "title": "HAS_MEASURE_LINEAR_IMAGE_SAME", "text": "|- (!f. (!s. (((linear f) /\\ ((measurable s) /\\ ((real_abs (det (matrix f))) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((has_measure ((IMAGE f) s)) (measure s)))))"}
{"_id": "547", "title": "ITERATE_DIFF", "text": "|- (!op. ((monoidal op) ==> (!f. (!s. (!t. (((FINITE s) /\\ (t SUBSET s)) ==> (((op (((iterate op) (s DIFF t)) f)) (((iterate op) t) f)) = (((iterate op) s) f))))))))"}
{"_id": "548", "title": "REAL_DIV_1", "text": "|- (!x. (((real_div x) (real_of_num (NUMERAL (BIT1 _0)))) = x))"}
{"_id": "549", "title": "EVENTUALLY_AT", "text": "|- (!a. (!p. (((eventually p) (at a)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\\ ((real_lt (distance (x , a))) d)) ==> (p x))))))))"}
{"_id": "550", "title": "REAL_LET_TRANS", "text": "|- (!x. (!y. (!z. ((((real_le x) y) /\\ ((real_lt y) z)) ==> ((real_lt x) z)))))"}
{"_id": "551", "title": "CARD_LE_ANTISYM", "text": "|- (!s. (!t. (((s <=_c t) /\\ (t <=_c s)) = (s =_c t))))"}
{"_id": "552", "title": "VECTOR_SUB_COMPONENT", "text": "|- (!x. (!y. (!i. ((((vector_sub x) y) $ i) = ((real_sub (x $ i)) (y $ i))))))"}
{"_id": "553", "title": "Hypermap.EQ_SUC", "text": "|- (!m. (!n. (((SUC m) = (SUC n)) = (m = n))))"}
{"_id": "554", "title": "Packing3.VORONOI_LIST_SING", "text": "|- (!V. (!u. (((voronoi_list V) ((CONS u) NIL)) = ((voronoi_closed V) u))))"}
{"_id": "555", "title": "LIM_COMPONENTWISE", "text": "|- (!net. (!f. ((((--> f) l) net) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (((---> (\\x. ((f x) $ i))) (l $ i)) net))))))"}
{"_id": "556", "title": "CONVEX_SUMS", "text": "|- (!s. (!t. (((convex s) /\\ (convex t)) ==> (convex (GSPEC (\\GEN%PVAR%1471. (?x. (?y. (((SETSPEC GEN%PVAR%1471) ((x IN s) /\\ (y IN t))) ((vector_add x) y))))))))))"}
{"_id": "557", "title": "CONTENT_PASTECART", "text": "|- (!a. (!b. (!c. (!d. ((content (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) = ((real_mul (content (closed_interval ((CONS (a , b)) NIL)))) (content (closed_interval ((CONS (c , d)) NIL)))))))))"}
{"_id": "558", "title": "NSUM_CLAUSES", "text": "|- ((!f. (((nsum EMPTY) f) = (NUMERAL _0))) /\\ (!x. (!f. (!s. ((FINITE s) ==> (((nsum (x INSERT s)) f) = (((COND (x IN s)) ((nsum s) f)) ((f x) + ((nsum s) f)))))))))"}
{"_id": "559", "title": "REAL_CONTINUOUS_AT_SQRT", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_continuous sqrt) (atreal x))))"}
{"_id": "560", "title": "CROSS_LADD", "text": "|- (!x. (!y. (!z. (((cross ((vector_add x) y)) z) = ((vector_add ((cross x) z)) ((cross y) z))))))"}
{"_id": "561", "title": "CEXP_NEG", "text": "|- (!z. ((cexp (vector_neg z)) = (complex_inv (cexp z))))"}
{"_id": "562", "title": "Lp_ineqs_proofs.dart3_yf_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((f_fan_pair_ext (V , E)) d)) = (y2_fan d)) /\\ (((y2_fan ((f_fan_pair_ext (V , E)) d)) = ((y3_fan (V , E)) d)) /\\ ((((y3_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y1_fan d)) /\\ ((((y4_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y5_fan (V , E)) d)) /\\ ((((y5_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y6_fan d)) /\\ ((y6_fan ((f_fan_pair_ext (V , E)) d)) = ((y4_fan (V , E)) d)))))))))))))))"}
{"_id": "563", "title": "LIM_TRANSFORM_WITHINREAL_SET_IMP", "text": "|- (!f. (!l. (!a. (!s. (!t. ((((eventually (\\x. ((x IN t) ==> (x IN s)))) (atreal a)) /\\ (((--> f) l) ((within (atreal a)) s))) ==> (((--> f) l) ((within (atreal a)) t))))))))"}
{"_id": "564", "title": "INTERS_0", "text": "|- ((INTERS EMPTY) = UNIV)"}
{"_id": "565", "title": "INTEGER_CLOSED", "text": "|- ((!n. (integer (real_of_num n))) /\\ ((!x. (!y. (((integer x) /\\ (integer y)) ==> (integer ((real_add x) y))))) /\\ ((!x. (!y. (((integer x) /\\ (integer y)) ==> (integer ((real_sub x) y))))) /\\ ((!x. (!y. (((integer x) /\\ (integer y)) ==> (integer ((real_mul x) y))))) /\\ ((!x. (!r. ((integer x) ==> (integer ((real_pow x) r))))) /\\ ((!x. ((integer x) ==> (integer (real_neg x)))) /\\ ((!x. ((integer x) ==> (integer (real_abs x)))) /\\ ((!x. (!y. (((integer x) /\\ (integer y)) ==> (integer ((real_max x) y))))) /\\ (!x. (!y. (((integer x) /\\ (integer y)) ==> (integer ((real_min x) y)))))))))))))"}
{"_id": "566", "title": "Conforming.add_edge_imp_card_set_edge_ge1_fan", "text": "|- (!x. (!V. (!E. (!E1. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ (E1 = (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))))) ==> (!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E1)) > (NUMERAL (BIT1 _0))))))))))"}
{"_id": "567", "title": "OPEN_IN_DIFF", "text": "|- (!top. (!s. (!t. ((((open_in top) s) /\\ ((closed_in top) t)) ==> ((open_in top) (s DIFF t))))))"}
{"_id": "568", "title": "Sphere.bump", "text": "|- (!h. ((bump h) = ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) ((real_div ((real_pow ((real_sub h) h0)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub hplus) h0)) (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "569", "title": "FACE_OF_CONIC", "text": "|- (!s. (!f. (((conic s) /\\ ((face_of f) s)) ==> (conic f))))"}
{"_id": "570", "title": "KIRSZBRAUN", "text": "|- (!f. (!s. (!B. ((((real_le (real_of_num (NUMERAL _0))) B) /\\ (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))))) ==> (?g. ((!x. (!y. ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))) /\\ (!x. ((x IN s) ==> ((g x) = (f x))))))))))"}
{"_id": "571", "title": "Hypermap.res", "text": "|- (!f. (!s. (!x. ((((res f) s) x) = (((COND (x IN s)) (f x)) x)))))"}
{"_id": "572", "title": "Lp_ineqs_def.y9_fan", "text": "|- (!V. (!E. (!x. (((y9_fan (V , E)) x) = (distance ((f_fan_pair_ext (V , E)) x))))))"}
{"_id": "573", "title": "REAL_EXP_NEG", "text": "|- (!x. ((exp (real_neg x)) = (real_inv (exp x))))"}
{"_id": "574", "title": "Sphere.rotate2", "text": "|- (!f. (!x2. (!x3. (!x1. (!x5. (!x6. (!x4. ((((((((rotate2 f) x1) x2) x3) x4) x5) x6) = ((((((f x2) x3) x1) x5) x6) x4)))))))))"}
{"_id": "575", "title": "Trigonometry2.AFF2", "text": "|- (!u. (!v. ((aff (u INSERT (v INSERT EMPTY))) = (GSPEC (\\GEN%PVAR%5314. (?x. (((SETSPEC GEN%PVAR%5314) (?t. (x = ((vector_add ((% t) u)) ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) v))))) x)))))))"}
{"_id": "576", "title": "AFFINE", "text": "|- (!V. ((affine V) = (!s. (!u. (((FINITE s) /\\ ((~ (s = EMPTY)) /\\ ((s SUBSET V) /\\ (((sum s) u) = (real_of_num (NUMERAL (BIT1 _0))))))) ==> (((vsum s) (\\x. ((% (u x)) x))) IN V))))))"}
{"_id": "577", "title": "PATHSTART_IN_PATH_IMAGE", "text": "|- (!g. ((pathstart g) IN (path_image g)))"}
{"_id": "578", "title": "MEM_EL", "text": "|- (!l. (!n. ((n < (LENGTH l)) ==> ((MEM ((EL n) l)) l))))"}
{"_id": "579", "title": "LAMBDA_UNIQUE", "text": "|- (!f. (!g. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((f $ i) = (g i)))) = ((lambda g) = f))))"}
{"_id": "580", "title": "INJ", "text": "|- (!t. (!s. (!f. ((((INJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))))))))"}
{"_id": "581", "title": "Appendix.MMs_v39", "text": "|- (!vv. (!s. (((MMs_v39 s) vv) = (((BBprime2_v39 s) vv) /\\ ((!i. (((scs_str_v39 s) i) ==> (((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + ((scs_k_v39 s) - (NUMERAL (BIT1 _0)))))) = pi))) /\\ ((!i. (((scs_lo_v39 s) i) ==> ((vector_norm (vv i)) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) /\\ ((!i. (((scs_hi_v39 s) i) ==> ((vector_norm (vv i)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) /\\ ((!i. (!j. ((real_le (((scs_am_v39 s) i) j)) (distance ((vv i) , (vv j)))))) /\\ (!i. (!j. ((real_le (distance ((vv i) , (vv j)))) (((scs_bm_v39 s) i) j))))))))))))"}
{"_id": "582", "title": "COLLINEAR_HYPERPLANE_2", "text": "|- (!a. (!b. ((((dimindex UNIV) <= (NUMERAL (BIT0 (BIT1 _0)))) /\\ (~ (a = (vec (NUMERAL _0))))) ==> (collinear (GSPEC (\\GEN%PVAR%1598. (?x. (((SETSPEC GEN%PVAR%1598) (((dot a) x) = b)) x))))))))"}
{"_id": "583", "title": "VECTOR_MUL_RZERO", "text": "|- (((% c) (vec (NUMERAL _0))) = (vec (NUMERAL _0)))"}
{"_id": "584", "title": "CONNECTED_COMPONENT_LINEAR_IMAGE", "text": "|- (!f. (!s. (!x. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> (((connected_component ((IMAGE f) s)) (f x)) = ((IMAGE f) ((connected_component s) x)))))))"}
{"_id": "585", "title": "REAL_INFSUM_UNIQUE", "text": "|- (!f. (!l. (!s. ((((real_sums f) l) s) ==> (((real_infsum s) f) = l)))))"}
{"_id": "586", "title": "sgn_lt", "text": "|- (sgn_lt = (\\t. ((real_lt t) (real_of_num (NUMERAL _0)))))"}
{"_id": "587", "title": "FORALL_4", "text": "|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\\ ((P (NUMERAL (BIT0 (BIT1 _0)))) /\\ ((P (NUMERAL (BIT1 (BIT1 _0)))) /\\ (P (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))"}
{"_id": "588", "title": "MBOUNDED_MCBALL", "text": "|- (!m. (!c. (!b. ((mbounded m) ((mcball m) (c , b))))))"}
{"_id": "589", "title": "ORTHOGONAL_SYM", "text": "|- (!x. (!y. (((orthogonal x) y) = ((orthogonal y) x))))"}
{"_id": "590", "title": "LIM_NULL_COMPLEX_RMUL_BOUNDED", "text": "|- (!net. (!f. (!g. (!B. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\\ ((eventually (\\a. (((f a) = (Cx (real_of_num (NUMERAL _0)))) \\/ ((real_le (vector_norm (g a))) B)))) net)) ==> (((--> (\\z. ((complex_mul (f z)) (g z)))) (Cx (real_of_num (NUMERAL _0)))) net))))))"}
{"_id": "591", "title": "CONTINUOUS_ON_SUBSET", "text": "|- (!f. (!s. (!t. ((((continuous_on f) s) /\\ (t SUBSET s)) ==> ((continuous_on f) t)))))"}
{"_id": "592", "title": "Functional_equation.proj_y1", "text": "|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((proj_y1 x1) x2) x3) x4) x5) x6) = (sqrt x1))))))))"}
{"_id": "593", "title": "inseg", "text": "|- (!m. (!l. (((inseg l) m) = (!x. (!y. ((l (x , y)) = ((m (x , y)) /\\ ((fl l) y))))))))"}
{"_id": "594", "title": "CSQRT_CEXP_CLOG", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((csqrt z) = (cexp ((complex_div (clog z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))"}
{"_id": "595", "title": "Lp_ineqs_def.rhazim2_fan", "text": "|- (!V. (!E. (!x. (((rhazim2_fan (V , E)) x) = ((rhazim_fan (V , E)) ((f_fan_pair_ext (V , E)) x))))))"}
{"_id": "596", "title": "CARD_EQ_CARD", "text": "|- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> ((s =_c t) = ((CARD s) = (CARD t))))))"}
{"_id": "597", "title": "Ssrnat.addn1", "text": "|- (!n. ((n + (NUMERAL (BIT1 _0))) = (SUC n)))"}
{"_id": "598", "title": "GDELTA_INTER", "text": "|- (!s. (!t. (((gdelta s) /\\ (gdelta t)) ==> (gdelta (s INTER t)))))"}
{"_id": "599", "title": "Miqmcsn.SCS_4M8_13_IS_SCS", "text": "|- (is_scs_v39 scs_4M8_13)"}
{"_id": "600", "title": "LT_NZ", "text": "|- (!n. (((NUMERAL _0) < n) = (~ (n = (NUMERAL _0)))))"}
{"_id": "601", "title": "CLOSURE_APPROACHABLE", "text": "|- (!x. (!s. ((x IN (closure s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?y. ((y IN s) /\\ ((real_lt (distance (y , x))) e))))))))"}
{"_id": "602", "title": "IN_MCBALL", "text": "|- (!m. (!x. (!r. (!y. ((y IN ((mcball m) (x , r))) = ((x IN (mspace m)) /\\ ((y IN (mspace m)) /\\ ((real_le ((mdist m) (x , y))) r))))))))"}
{"_id": "603", "title": "MATRIX_INV_0", "text": "|- ((matrix_inv (mat (NUMERAL _0))) = (mat (NUMERAL _0)))"}
{"_id": "604", "title": "POSITIVE_DEFINITE_2_DET", "text": "|- (!A. ((positive_definite A) = (((transp A) = A) /\\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((real_lt (real_of_num (NUMERAL _0))) (det A)))))))"}
{"_id": "605", "title": "COMPLEX_NORM_CX", "text": "|- (!x. ((vector_norm (Cx x)) = (real_abs x)))"}
{"_id": "606", "title": "ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES_LINEAR", "text": "|- (!f. (!g. (!s. ((linear g) ==> (((absolutely_integrable_on (\\x. ((% (real_abs (det (matrix g)))) (f (g x))))) s) = ((absolutely_integrable_on f) ((IMAGE g) s)))))))"}
{"_id": "607", "title": "LIFT_SUB", "text": "|- (!x. (!y. ((lift ((real_sub x) y)) = ((vector_sub (lift x)) (lift y)))))"}
{"_id": "608", "title": "SHIFTPATH_TRANSLATION", "text": "|- (!a. (!t. (!g. (((shiftpath t) ((\\x. ((vector_add a) x)) o g)) = ((\\x. ((vector_add a) x)) o ((shiftpath t) g))))))"}
{"_id": "609", "title": "NORM_GE_SQUARE", "text": "|- (!x. (((real_ge (vector_norm x)) a) = (((real_le a) (real_of_num (NUMERAL _0))) \\/ ((real_ge ((dot x) x)) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "610", "title": "INDEPENDENT_IMP_AFFINE_DEPENDENT_0", "text": "|- (!s. ((independent s) ==> (~ (affine_dependent ((vec (NUMERAL _0)) INSERT s)))))"}
{"_id": "611", "title": "COMPLEX_MUL_CCOS_CSIN", "text": "|- (!w. (!z. (((complex_mul (ccos w)) (csin z)) = ((complex_div ((vector_sub (csin ((vector_add w) z))) (csin ((vector_sub w) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "612", "title": "RECTIFIABLE_PATH_IMP_PATH", "text": "|- (!g. ((rectifiable_path g) ==> (path g)))"}
{"_id": "613", "title": "REAL_SGN_EQ", "text": "|- ((!x. (((real_sgn x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))) /\\ ((!x. (((real_sgn x) = (real_of_num (NUMERAL (BIT1 _0)))) = ((real_gt x) (real_of_num (NUMERAL _0))))) /\\ (!x. (((real_sgn x) = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) = ((real_lt x) (real_of_num (NUMERAL _0)))))))"}
{"_id": "614", "title": "ABSOLUTELY_SETCONTINUOUS_ON_DIVISION", "text": "|- (!f. (!a. (!b. (!d. ((((operative vector_add) f) /\\ ((division_of d) (closed_interval ((CONS (a , b)) NIL)))) ==> ((!k. ((k IN d) ==> ((absolutely_setcontinuous_on f) k))) = ((absolutely_setcontinuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "615", "title": "Seq.in_nil", "text": "|- (!x. (((MEM x) NIL) = F))"}
{"_id": "616", "title": "CBALL_MIN_INTER", "text": "|- (!x. (!d. (!e. ((cball (x , ((real_min d) e))) = ((cball (x , d)) INTER (cball (x , e)))))))"}
{"_id": "617", "title": "NEGLIGIBLE_TRANSLATION_EQ", "text": "|- (!a. (!s. ((negligible ((IMAGE (\\x. ((vector_add a) x))) s)) = (negligible s))))"}
{"_id": "618", "title": "SQRT_WORKS", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_le (real_of_num (NUMERAL _0))) (sqrt x)) /\\ (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = x))))"}
{"_id": "619", "title": "EXISTS_LIFT", "text": "|- ((?x. (P x)) = (?x. (P (lift x))))"}
{"_id": "620", "title": "CONTINUOUS_AT_CTAN", "text": "|- (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> ((continuous ctan) (at z))))"}
{"_id": "621", "title": "Ineq.dart_std4", "text": "|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. ((((((((((dart_std4 y1) y2) y3) y4) y5) y6) y7) y8) y9) = ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y7 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y8 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y9 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL)))))))))))))))))))"}
{"_id": "622", "title": "REAL_NOT_LE", "text": "|- (!x. (!y. ((~ ((real_le x) y)) = ((real_lt y) x))))"}
{"_id": "623", "title": "INT_LE_NEG", "text": "|- (!x. (!y. (((int_le (int_neg x)) (int_neg y)) = ((int_le y) x))))"}
{"_id": "624", "title": "Constants_approx.INTERVAL_INV", "text": "|- ((((interval_arith x) (a , b)) /\\ (((real_lt (real_of_num (NUMERAL _0))) a) \\/ ((real_lt b) (real_of_num (NUMERAL _0))))) ==> ((interval_arith (real_inv x)) ((real_inv b) , (real_inv a))))"}
{"_id": "625", "title": "RE_MUL_CX", "text": "|- (!x. (!z. (((Re ((complex_mul (Cx x)) z)) = ((real_mul x) (Re z))) /\\ ((Re ((complex_mul z) (Cx x))) = ((real_mul (Re z)) x)))))"}
{"_id": "626", "title": "Ssrnat.mul2n", "text": "|- (!m. (((NUMERAL (BIT0 (BIT1 _0))) * m) = (double m)))"}
{"_id": "627", "title": "Real_ext.REAL_MK_NZ_EXP", "text": "|- (!x. (~ ((exp x) = (real_of_num (NUMERAL _0)))))"}
{"_id": "628", "title": "HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_RIGHT", "text": "|- (!p. (!f. (!g. (!h. (!W. (!X. (!Y. ((((((homotopic_with (\\f. (p (f o h)))) (X , Y)) f) g) /\\ (((continuous_on h) W) /\\ (((IMAGE h) W) SUBSET X))) ==> ((((homotopic_with p) (W , Y)) (f o h)) (g o h))))))))))"}
{"_id": "629", "title": "FRECHET_DERIVATIVE_WORKS", "text": "|- (!f. (!net. (((differentiable f) net) = (((has_derivative f) ((frechet_derivative f) net)) net))))"}
{"_id": "630", "title": "SUM_LMUL", "text": "|- (!f. (!c. (!s. (((sum s) (\\x. ((real_mul c) (f x)))) = ((real_mul c) ((sum s) f))))))"}
{"_id": "631", "title": "Sphere.dih2_x", "text": "|- (!x2. (!x1. (!x3. (!x5. (!x4. (!x6. (((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih_x x2) x1) x3) x5) x4) x6))))))))"}
{"_id": "632", "title": "CX_EXP", "text": "|- (!x. ((Cx (exp x)) = (cexp (Cx x))))"}
{"_id": "633", "title": "Hypermap.node_map_and_darts", "text": "|- (!H. ((FINITE (dart H)) /\\ ((permutes (node_map H)) (dart H))))"}
{"_id": "634", "title": "BASIS_COMPONENT", "text": "|- (!k. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (((basis k) $ i) = (((COND (i = k)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))))"}
{"_id": "635", "title": "COMPLEX_NEG_0", "text": "|- ((vector_neg (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL _0))))"}
{"_id": "636", "title": "REAL_ABS_POW", "text": "|- (!x. (!n. ((real_abs ((real_pow x) n)) = ((real_pow (real_abs x)) n))))"}
{"_id": "637", "title": "VSUM_SUB", "text": "|- (!f. (!g. (!s. ((FINITE s) ==> (((vsum s) (\\x. ((vector_sub (f x)) (g x)))) = ((vector_sub ((vsum s) f)) ((vsum s) g)))))))"}
{"_id": "638", "title": "CLOSED_IMP_LOCALLY_COMPACT", "text": "|- (!s. ((closed s) ==> ((locally compact) s)))"}
{"_id": "639", "title": "LIM_UNION", "text": "|- (!f. (!x. (!l. (!s. (!t. (((((--> f) l) ((within (at x)) s)) /\\ (((--> f) l) ((within (at x)) t))) ==> (((--> f) l) ((within (at x)) (s UNION t)))))))))"}
{"_id": "640", "title": "TRANSP_TRANSP", "text": "|- (!A. ((transp (transp A)) = A))"}
{"_id": "641", "title": "MEASURABLE_TRANSLATION_EQ", "text": "|- (!a. (!s. ((measurable ((IMAGE (\\x. ((vector_add a) x))) s)) = (measurable s))))"}
{"_id": "642", "title": "Seq2.list_sum_EL", "text": "|- (!s. (!f. ((~ (s = NIL)) ==> (((list_sum s) f) = ((sum ((NUMERAL _0) .. ((sizel s) - (NUMERAL (BIT1 _0))))) (\\i. (f ((EL i) s))))))))"}
{"_id": "643", "title": "Hypermap_and_fan.E_N_F_IN_DART1_OF_FAN", "text": "|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (x IN (dart1_of_fan (V , E)))) ==> ((((e_fan_pair (V , E)) x) IN (dart1_of_fan (V , E))) /\\ ((((n_fan_pair (V , E)) x) IN (dart1_of_fan (V , E))) /\\ (((f_fan_pair (V , E)) x) IN (dart1_of_fan (V , E)))))))))"}
{"_id": "644", "title": "DIST_SYM", "text": "|- (!x. (!y. ((distance (x , y)) = (distance (y , x)))))"}
{"_id": "645", "title": "REAL_MUL_POS_LE", "text": "|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((x = (real_of_num (NUMERAL _0))) \\/ ((y = (real_of_num (NUMERAL _0))) \\/ ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) \\/ (((real_lt x) (real_of_num (NUMERAL _0))) /\\ ((real_lt y) (real_of_num (NUMERAL _0))))))))))"}
{"_id": "646", "title": "Seq.eqS0", "text": "|- (!n. (((SUC n) = (NUMERAL _0)) = F))"}
{"_id": "647", "title": "FORALL_IN_UNION", "text": "|- (!P. (!s. (!t. ((!x. ((x IN (s UNION t)) ==> (P x))) = ((!x. ((x IN s) ==> (P x))) /\\ (!x. ((x IN t) ==> (P x))))))))"}
{"_id": "648", "title": "REAL_INTEGRAL_EQ", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((real_integral s) f) = ((real_integral s) g))))))"}
{"_id": "649", "title": "CLOSED_IN_COMPACT", "text": "|- (!s. (!t. (((compact s) /\\ ((closed_in ((subtopology euclidean) s)) t)) ==> (compact t))))"}
{"_id": "650", "title": "matrix_add", "text": "|- (!A. (!B. (((matrix_add A) B) = (lambdai. (lambdaj. ((real_add ((A $ i) $ j)) ((B $ i) $ j)))))))"}
{"_id": "651", "title": "HAS_SIZE_1_EXISTS", "text": "|- (!s. ((s HAS_SIZE (NUMERAL (BIT1 _0))) = (?!x. (x IN s))))"}
{"_id": "652", "title": "Collect_geom.PER_SET2", "text": "|- ((a INSERT (b INSERT EMPTY)) = (b INSERT (a INSERT EMPTY)))"}
{"_id": "653", "title": "Float.INT_POW_NEG1", "text": "|- (!x. (!n. (((int_pow (int_neg (int_of_num x))) n) = ((int_mul ((int_pow (int_neg (int_of_num (NUMERAL (BIT1 _0))))) n)) ((int_pow (int_of_num x)) n)))))"}
{"_id": "654", "title": "Local_lemmas.CONDS_IN_HAFL_LINE", "text": "|- ((((real_le (real_of_num (NUMERAL _0))) t) /\\ (((vector_sub a) x) = ((% t) ((vector_sub b) x)))) ==> (a IN ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))"}
{"_id": "655", "title": "SUM_IMAGE", "text": "|- (!f. (!g. (!s. ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) ==> (((sum ((IMAGE f) s)) g) = ((sum s) (g o f)))))))"}
{"_id": "656", "title": "LE_1", "text": "|- ((!n. ((~ (n = (NUMERAL _0))) ==> ((NUMERAL _0) < n))) /\\ ((!n. ((~ (n = (NUMERAL _0))) ==> ((NUMERAL (BIT1 _0)) <= n))) /\\ ((!n. (((NUMERAL _0) < n) ==> (~ (n = (NUMERAL _0))))) /\\ ((!n. (((NUMERAL _0) < n) ==> ((NUMERAL (BIT1 _0)) <= n))) /\\ ((!n. (((NUMERAL (BIT1 _0)) <= n) ==> ((NUMERAL _0) < n))) /\\ (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (~ (n = (NUMERAL _0))))))))))"}
{"_id": "657", "title": "COMPACT_CONVEX_HULL", "text": "|- (!s. ((compact s) ==> (compact ((hull convex) s))))"}
{"_id": "658", "title": "Lp_ineqs_proofs.y1_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (y1_fan d))))))))))"}
{"_id": "659", "title": "LE_SUC", "text": "|- (!m. (!n. (((SUC m) <= (SUC n)) = (m <= n))))"}
{"_id": "660", "title": "frontier", "text": "|- (!s. ((frontier s) = ((closure s) DIFF (interior s))))"}
{"_id": "661", "title": "PATH_CONNECTED_CONVEX_DIFF_CARD_LT", "text": "|- (!u. (!s. (((convex u) /\\ ((~ (collinear u)) /\\ (s <_c UNIV))) ==> (path_connected (u DIFF s)))))"}
{"_id": "662", "title": "FINITE_SUBSET", "text": "|- (!s. (!t. (((FINITE t) /\\ (s SUBSET t)) ==> (FINITE s))))"}
{"_id": "663", "title": "o_ASSOC", "text": "|- (!f. (!g. (!h. ((f o (g o h)) = ((f o g) o h)))))"}
{"_id": "664", "title": "CONVEX_LINEAR_IMAGE", "text": "|- (!f. (!s. (((convex s) /\\ (linear f)) ==> (convex ((IMAGE f) s)))))"}
{"_id": "665", "title": "Fan.POWER_2", "text": "|- (!f. (((POWER f) (NUMERAL (BIT0 (BIT1 _0)))) = (f o f)))"}
{"_id": "666", "title": "Topology.FINITE_ORBITS_SIGMA_FAN", "text": "|- (!x. (!V. (!E. (!v. (!u. (((FAN (x , (V , E))) /\\ ((v INSERT (u INSERT EMPTY)) IN E)) ==> (FINITE (((((set_of_orbits_points_fan x) V) E) v) u))))))))"}
{"_id": "667", "title": "LE_REFL", "text": "|- (!n. (n <= n))"}
{"_id": "668", "title": "COMPLEX_DIFFERENTIABLE_AT_CLOG", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((complex_differentiable clog) (at z))))"}
{"_id": "669", "title": "MEASURE_NEGLIGIBLE_UNIONS_IMAGE", "text": "|- (!f. (!s. (((FINITE s) /\\ ((!x. ((x IN s) ==> (measurable (f x)))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ (~ (x = y)))) ==> (negligible ((f x) INTER (f y)))))))) ==> ((measure (UNIONS ((IMAGE f) s))) = ((sum s) (\\x. (measure (f x))))))))"}
{"_id": "670", "title": "Seq.gtS0", "text": "|- (!n. ((NUMERAL _0) < (SUC n)))"}
{"_id": "671", "title": "AR_SING", "text": "|- (!a. (AR (a INSERT EMPTY)))"}
{"_id": "672", "title": "Rnsyjxm.f_surr_localization_convex_local", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((LET (\\f. (LET_END (convex_local_fan (((v_prime V) f) , (((e_prime E) f) , f)))))) ((face (hypermap_of_fan (V , E))) d))))))))"}
{"_id": "673", "title": "IM_CLOG_POS_LT", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((((real_lt (real_of_num (NUMERAL _0))) (Im (clog z))) /\\ ((real_lt (Im (clog z))) pi)) = ((real_lt (real_of_num (NUMERAL _0))) (Im z)))))"}
{"_id": "674", "title": "CONNECTED_PCROSS", "text": "|- (!s. (!t. (((connected s) /\\ (connected t)) ==> (connected (s PCROSS t)))))"}
{"_id": "675", "title": "HOMEOMORPHISM_SYM", "text": "|- (!f. (!g. (!s. (!t. (((homeomorphism (s , t)) (f , g)) = ((homeomorphism (t , s)) (g , f)))))))"}
{"_id": "676", "title": "RE_CX", "text": "|- (!x. ((Re (Cx x)) = x))"}
{"_id": "677", "title": "CONVEX_HALFSPACE_GT", "text": "|- (!a. (!b. (convex (GSPEC (\\GEN%PVAR%1441. (?x. (((SETSPEC GEN%PVAR%1441) ((real_gt ((dot a) x)) b)) x)))))))"}
{"_id": "678", "title": "COMPLEX_DERIVATIVE_UNIQUE_AT", "text": "|- (!f. (!z. (!f'. (!f''. (((((has_complex_derivative f) f') (at z)) /\\ (((has_complex_derivative f) f'') (at z))) ==> (f' = f''))))))"}
{"_id": "679", "title": "VECTOR_POLYNOMIAL_FUNCTION_ADD", "text": "|- (!f. (!g. (((vector_polynomial_function f) /\\ (vector_polynomial_function g)) ==> (vector_polynomial_function (\\x. ((vector_add (f x)) (g x)))))))"}
{"_id": "680", "title": "Ssrnat.uphalf0", "text": "|- ((uphalf (NUMERAL _0)) = (NUMERAL _0))"}
{"_id": "681", "title": "POSITIVE_SEMIDEFINITE_2_DET", "text": "|- (!A. ((positive_semidefinite A) = (((transp A) = A) /\\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\\ (((real_le (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((real_le (real_of_num (NUMERAL _0))) (det A)))))))"}
{"_id": "682", "title": "HOMEOMORPHISM_OF_SUBSETS", "text": "|- (!f. (!g. (!s. (!t. (!s'. (!t'. ((((homeomorphism (s , t)) (f , g)) /\\ ((s' SUBSET s) /\\ ((t' SUBSET t) /\\ (((IMAGE f) s') = t')))) ==> ((homeomorphism (s' , t')) (f , g)))))))))"}
{"_id": "683", "title": "COS_DOUBLE", "text": "|- (!x. ((cos ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)) = ((real_sub ((real_pow (cos x)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (sin x)) (NUMERAL (BIT0 (BIT1 _0)))))))"}
{"_id": "684", "title": "CONIC_EMPTY", "text": "|- (conic EMPTY)"}
{"_id": "685", "title": "LINEPATH_TRANSLATION", "text": "|- (!a. (!b. (!c. ((linepath (((vector_add a) b) , ((vector_add a) c))) = ((\\x. ((vector_add a) x)) o (linepath (b , c)))))))"}
{"_id": "686", "title": "CONTRACTIBLE_IMP_SIMPLY_CONNECTED", "text": "|- (!s. ((contractible s) ==> (simply_connected s)))"}
{"_id": "687", "title": "REAL_NEG_SUB", "text": "|- (!x. (!y. ((real_neg ((real_sub x) y)) = ((real_sub y) x))))"}
{"_id": "688", "title": "INTEGRABLE_INCREASING_PRODUCT", "text": "|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g x)) (g y)))))) ==> ((integrable_on (\\x. ((% (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "689", "title": "Ssrnat.ltnNge", "text": "|- (!m. (!n. ((m < n) = (~ (n <= m)))))"}
{"_id": "690", "title": "CLOSED_CLOSED_PREIMAGES_IMP_CONTINUOUS_ON", "text": "|- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\\ ((compact t) /\\ ((!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%1941. (?x. (((SETSPEC GEN%PVAR%1941) ((x IN s) /\\ ((f x) = y))) x)))))) /\\ (!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c))))))) ==> ((continuous_on f) s)))))"}
{"_id": "691", "title": "FINITE_INSERT", "text": "|- (!s. (!x. ((FINITE (x INSERT s)) = (FINITE s))))"}
{"_id": "692", "title": "CONTINUOUS_COMPLEX_MUL", "text": "|- (!net. (!f. (!g. ((((continuous f) net) /\\ ((continuous g) net)) ==> ((continuous (\\x. ((complex_mul (f x)) (g x)))) net)))))"}
{"_id": "693", "title": "CARD_MUL_C", "text": "|- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> ((CARD ((*_c s) t)) = ((CARD s) * (CARD t))))))"}
{"_id": "694", "title": "ANALYTIC_ON_SUBSET", "text": "|- (!f. (!s. (!t. ((((analytic_on f) s) /\\ (t SUBSET s)) ==> ((analytic_on f) t)))))"}
{"_id": "695", "title": "POLYTOPE_TRANSLATION_EQ", "text": "|- (!a. (!s. ((polytope ((IMAGE (\\x. ((vector_add a) x))) s)) = (polytope s))))"}
{"_id": "696", "title": "MATRIX_VECTOR_MUL_RZERO", "text": "|- (!A. (((matrix_vector_mul A) (vec (NUMERAL _0))) = (vec (NUMERAL _0))))"}
{"_id": "697", "title": "Ardbzye.K_SCS_3T3", "text": "|- ((scs_k_v39 scs_3T3) = (NUMERAL (BIT1 (BIT1 _0))))"}
{"_id": "698", "title": "Fan.subset_d_fan", "text": "|- (!x. (!V. (!E. (((d1_fan (x , (V , E))) SUBSET (d_fan (x , (V , E)))) /\\ ((d20_fan (x , (V , E))) SUBSET (d_fan (x , (V , E))))))))"}
{"_id": "699", "title": "UNIFORMLY_CONTINUOUS_ON_COMPOSE", "text": "|- (!f. (!g. (!s. ((((uniformly_continuous_on f) s) /\\ ((uniformly_continuous_on g) ((IMAGE f) s))) ==> ((uniformly_continuous_on (g o f)) s)))))"}
{"_id": "700", "title": "HAS_MEASURE_UNION_NEGLIGIBLE_EQ", "text": "|- (!s. (!t. (!m. ((negligible t) ==> (((has_measure (s UNION t)) m) = ((has_measure s) m))))))"}
{"_id": "701", "title": "Hypermap_iso.res_inv_left", "text": "|- (!f. (!s. (!x. (((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) /\\ (x IN s)) ==> ((((res_inv f) s) (f x)) = x)))))"}
{"_id": "702", "title": "ARG_CNJ", "text": "|- (!z. ((Arg (cnj z)) = (((COND ((real z) /\\ ((real_le (real_of_num (NUMERAL _0))) (Re z)))) (Arg z)) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) (Arg z)))))"}
{"_id": "703", "title": "INTEGRAL_SPIKE_SET", "text": "|- (!f. (!g. (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> (((integral s) f) = ((integral t) f)))))))"}
{"_id": "704", "title": "HAS_MEASURE_DISJOINT_UNIONS", "text": "|- (!m. (!f. (((FINITE f) /\\ ((!s. ((s IN f) ==> ((has_measure s) (m s)))) /\\ (!s. (!t. (((s IN f) /\\ ((t IN f) /\\ (~ (s = t)))) ==> ((DISJOINT s) t)))))) ==> ((has_measure (UNIONS f)) ((sum f) m)))))"}
{"_id": "705", "title": "FORALL_LIFT", "text": "|- ((!x. (P x)) = (!x. (P (lift x))))"}
{"_id": "706", "title": "CONTINUOUS_MUL", "text": "|- (!net. (!f. (!c. ((((continuous (lift o c)) net) /\\ ((continuous f) net)) ==> ((continuous (\\x. ((% (c x)) (f x)))) net)))))"}
{"_id": "707", "title": "Real_ext.REAL_SV_EXP_0", "text": "|- ((exp (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "708", "title": "NSUM_LMUL", "text": "|- (!f. (!c. (!s. (((nsum s) (\\x. (c * (f x)))) = (c * ((nsum s) f))))))"}
{"_id": "709", "title": "HAS_BOUNDED_VARIATION_ON_VMUL", "text": "|- (!f. (!s. (!v. (((has_bounded_variation_on (\\x. (lift (f x)))) s) ==> ((has_bounded_variation_on (\\x. ((% (f x)) v))) s)))))"}
{"_id": "710", "title": "IM_NEG", "text": "|- (!x. ((Im (vector_neg x)) = (real_neg (Im x))))"}
{"_id": "711", "title": "CARD_LE_SUBSET", "text": "|- (!s. (!t. ((s SUBSET t) ==> (s <=_c t))))"}
{"_id": "712", "title": "SPHERE_LINEAR_IMAGE", "text": "|- (!f. (!x. (!r. (((linear f) /\\ ((!y. (?x. ((f x) = y))) /\\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((sphere ((f x) , r)) = ((IMAGE f) (sphere (x , r))))))))"}
{"_id": "713", "title": "RANK_TRANSP", "text": "|- (!A. ((rank (transp A)) = (rank A)))"}
{"_id": "714", "title": "HAS_REAL_DERIVATIVE_CONST", "text": "|- (!c. (!net. (((has_real_derivative (\\x. c)) (real_of_num (NUMERAL _0))) net)))"}
{"_id": "715", "title": "Fan.INVERSE1_SIGMA_FAN", "text": "|- (!x. (!V. (!E. (!v. ((FAN (x , (V , E))) ==> ((!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((v INSERT ((((((inverse1_sigma_fan x) V) E) v) w) INSERT EMPTY)) IN E))) /\\ ((!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((((((sigma_fan x) V) E) v) (((((inverse1_sigma_fan x) V) E) v) w)) = w))) /\\ (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> ((((((inverse1_sigma_fan x) V) E) v) (((((sigma_fan x) V) E) v) w)) = w))))))))))"}
{"_id": "716", "title": "SC_INC", "text": "|- (!R. (!x. (!y. (((R x) y) ==> (((SC R) x) y)))))"}
{"_id": "717", "title": "CPRODUCT_CLAUSES", "text": "|- ((!f. (((cproduct EMPTY) f) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))) /\\ (!x. (!f. (!s. ((FINITE s) ==> (((cproduct (x INSERT s)) f) = (((COND (x IN s)) ((cproduct s) f)) ((complex_mul (f x)) ((cproduct s) f)))))))))"}
{"_id": "718", "title": "DOT_LNEG", "text": "|- (!x. (!y. (((dot (vector_neg x)) y) = (real_neg ((dot x) y)))))"}
{"_id": "719", "title": "ARC_IMP_SIMPLE_PATH", "text": "|- (!g. ((arc g) ==> (simple_path g)))"}
{"_id": "720", "title": "Lp_ineqs_def.ye_fan", "text": "|- (!x. ((ye_fan x) = (distance x)))"}
{"_id": "721", "title": "HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNION", "text": "|- (!s1. (!s2. (!m1. (!m2. ((((has_real_measure s1) m1) /\\ (((has_real_measure s2) m2) /\\ (real_negligible (s1 INTER s2)))) ==> ((has_real_measure (s1 UNION s2)) ((real_add m1) m2)))))))"}
{"_id": "722", "title": "Pack1.saturated", "text": "|- (!S. ((saturated S) = (!x. (?y. ((y IN S) /\\ ((real_lt (distance (x , y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))"}
{"_id": "723", "title": "MATRIX_TRANSP_MULTIPLE_INV_LEFT", "text": "|- (!A. ((transp A) = ((matrix_mul (matrix_inv A)) ((matrix_mul A) (transp A)))))"}
{"_id": "724", "title": "SELF_ADJOINT_ORTHOGONAL_EIGENVECTORS", "text": "|- (!f. (!v. (!w. (!a. (!b. (((linear f) /\\ (((adjoint f) = f) /\\ (((f v) = ((% a) v)) /\\ (((f w) = ((% b) w)) /\\ (~ (a = b)))))) ==> ((orthogonal v) w)))))))"}
{"_id": "725", "title": "Seq.lastI", "text": "|- (!x. (!s. (((CONS x) s) = ((rcons ((belast x) s)) ((last x) s)))))"}
{"_id": "726", "title": "REAL_CONTINUOUS_WITHIN_ACS_STRONG", "text": "|- (!x. ((real_continuous acs) ((within (atreal x)) (GSPEC (\\GEN%PVAR%4911. (?t. (((SETSPEC GEN%PVAR%4911) ((real_le (real_abs t)) (real_of_num (NUMERAL (BIT1 _0))))) t)))))))"}
{"_id": "727", "title": "STEINHAUS", "text": "|- (!s. (((measurable s) /\\ ((real_lt (real_of_num (NUMERAL _0))) (measure s))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ ((ball ((vec (NUMERAL _0)) , d)) SUBSET (GSPEC (\\GEN%PVAR%3461. (?x. (?y. (((SETSPEC GEN%PVAR%3461) ((x IN s) /\\ (y IN s))) ((vector_sub x) y)))))))))))"}
{"_id": "728", "title": "INTERIOR_MAXIMAL", "text": "|- (!s. (!t. (((t SUBSET s) /\\ (open t)) ==> (t SUBSET (interior s)))))"}
{"_id": "729", "title": "RELATIVE_INTERIOR_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((relative_interior ((IMAGE f) s)) = ((IMAGE f) (relative_interior s))))))"}
{"_id": "730", "title": "ANALYTIC_ON_MUL", "text": "|- (!f. (!g. (!s. ((((analytic_on f) s) /\\ ((analytic_on g) s)) ==> ((analytic_on (\\z. ((complex_mul (f z)) (g z)))) s)))))"}
{"_id": "731", "title": "ORTHOGONAL_CLAUSES", "text": "|- ((!a. ((orthogonal a) (vec (NUMERAL _0)))) /\\ ((!a. (!x. (!c. (((orthogonal a) x) ==> ((orthogonal a) ((% c) x)))))) /\\ ((!a. (!x. (((orthogonal a) x) ==> ((orthogonal a) (vector_neg x))))) /\\ ((!a. (!x. (!y. ((((orthogonal a) x) /\\ ((orthogonal a) y)) ==> ((orthogonal a) ((vector_add x) y)))))) /\\ ((!a. (!x. (!y. ((((orthogonal a) x) /\\ ((orthogonal a) y)) ==> ((orthogonal a) ((vector_sub x) y)))))) /\\ ((!a. ((orthogonal (vec (NUMERAL _0))) a)) /\\ ((!a. (!x. (!c. (((orthogonal x) a) ==> ((orthogonal ((% c) x)) a))))) /\\ ((!a. (!x. (((orthogonal x) a) ==> ((orthogonal (vector_neg x)) a)))) /\\ ((!a. (!x. (!y. ((((orthogonal x) a) /\\ ((orthogonal y) a)) ==> ((orthogonal ((vector_add x) y)) a))))) /\\ (!a. (!x. (!y. ((((orthogonal x) a) /\\ ((orthogonal y) a)) ==> ((orthogonal ((vector_sub x) y)) a))))))))))))))"}
{"_id": "732", "title": "SQRT_WORKS_GEN", "text": "|- (!x. (((real_sgn (sqrt x)) = (real_sgn x)) /\\ (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = (real_abs x))))"}
{"_id": "733", "title": "REAL_EQ_LCANCEL_IMP", "text": "|- (!x. (!y. (!z. (((~ (z = (real_of_num (NUMERAL _0)))) /\\ (((real_mul z) x) = ((real_mul z) y))) ==> (x = y)))))"}
{"_id": "734", "title": "Nkezbfc_local.slicef", "text": "|- (!E. (!FF. (!w. (!v. (((((slicef E) FF) v) w) = ((GSPEC (\\GEN%PVAR%7012. (?f. (((SETSPEC GEN%PVAR%7012) (?u. ((u IN (((((slicev E) FF) v) w) DELETE w)) /\\ (f = (u , ((rho_node1 FF) u)))))) f)))) UNION ((w , v) INSERT EMPTY)))))))"}
{"_id": "735", "title": "SUP", "text": "|- (!s. (((~ (s = EMPTY)) /\\ (?b. (!x. ((x IN s) ==> ((real_le x) b))))) ==> ((!x. ((x IN s) ==> ((real_le x) (sup s)))) /\\ (!b. ((!x. ((x IN s) ==> ((real_le x) b))) ==> ((real_le (sup s)) b))))))"}
{"_id": "736", "title": "REALLIM_LMUL", "text": "|- (!f. (!l. (!c. ((((---> f) l) net) ==> (((---> (\\x. ((real_mul c) (f x)))) ((real_mul c) l)) net)))))"}
{"_id": "737", "title": "aff_le_def", "text": "|- (aff_le = (affsign sgn_le))"}
{"_id": "738", "title": "LIMPT_OF_UNIV", "text": "|- (!x. ((limit_point_of x) UNIV))"}
{"_id": "739", "title": "MATRIX_MUL_ASSOC", "text": "|- (!A. (!B. (!C. (((matrix_mul A) ((matrix_mul B) C)) = ((matrix_mul ((matrix_mul A) B)) C)))))"}
{"_id": "740", "title": "CCOS_EQ_MINUS1", "text": "|- (!z. (((ccos z) = (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) = (?n. ((integer n) /\\ (z = (Cx ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) (real_of_num (NUMERAL (BIT1 _0))))) pi)))))))"}
{"_id": "741", "title": "RAT_LEMMA4", "text": "|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> (((real_le ((real_div x1) y1)) ((real_div x2) y2)) = ((real_le ((real_mul x1) y2)) ((real_mul x2) y1))))"}
{"_id": "742", "title": "CONIC_IMP_PATH_CONNECTED", "text": "|- (!s. ((conic s) ==> (path_connected s)))"}
{"_id": "743", "title": "Ssrbool.if_same", "text": "|- (!b. (!vT. ((((COND b) vT) vT) = vT)))"}
{"_id": "744", "title": "Conforming.SIGMA_FAN_OF_FANADD_AT_POINT6", "text": "|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\\ ((FAN (x , (V , E1))) /\\ ((fan80 (x , (V , E))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ ((w1 = (((((inverse1_sigma_fan x) V) E) w) u)) /\\ (((w INSERT (w1 INSERT EMPTY)) IN E) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))) ==> ((((((sigma_fan x) V) E1) w) w1) = v)))))))))"}
{"_id": "745", "title": "ORTHOGONAL_SUBSPACE_DECOMP_EXISTS", "text": "|- (!s. (!x. (?y. (?z. ((y IN (span s)) /\\ ((!w. ((w IN (span s)) ==> ((orthogonal z) w))) /\\ (x = ((vector_add y) z))))))))"}
{"_id": "746", "title": "DROP_CMUL", "text": "|- (!x. (!c. ((drop ((% c) x)) = ((real_mul c) (drop x)))))"}
{"_id": "747", "title": "AFFSIGN_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!sgn. (!s. (!t. (!v. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((((affsign sgn) ((IMAGE f) s)) ((IMAGE f) t)) = ((IMAGE f) (((affsign sgn) s) t)))))))))"}
{"_id": "748", "title": "DOT_LZERO", "text": "|- (!x. (((dot (vec (NUMERAL _0))) x) = (real_of_num (NUMERAL _0))))"}
{"_id": "749", "title": "SUBSET_INTERIOR", "text": "|- (!s. (!t. ((s SUBSET t) ==> ((interior s) SUBSET (interior t)))))"}
{"_id": "750", "title": "LIM_NULL_NORM", "text": "|- (!net. (!f. ((((--> f) (vec (NUMERAL _0))) net) = (((--> (\\x. (lift (vector_norm (f x))))) (vec (NUMERAL _0))) net))))"}
{"_id": "751", "title": "Hypermap_iso.hypermap_of_fan_rep_alt", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((dart (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = (d_fan ((vec (NUMERAL _0)) , (V , E)))) /\\ (((edge_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((e_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E))))) /\\ (((node_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((n_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E))))) /\\ ((face_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) = ((res (((f1_fan (vec (NUMERAL _0))) V) E)) (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))))))"}
{"_id": "752", "title": "CLOPEN", "text": "|- (!s. (((closed s) /\\ (open s)) = ((s = EMPTY) \\/ (s = UNIV))))"}
{"_id": "753", "title": "ORTHOGONAL_TRANSFORMATION_INTO_SUBSPACE", "text": "|- (!s. (!t. (((subspace s) /\\ ((subspace t) /\\ ((dim s) <= (dim t)))) ==> (?f. ((orthogonal_transformation f) /\\ (((IMAGE f) s) SUBSET t))))))"}
{"_id": "754", "title": "HAS_COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN", "text": "|- (!f. (!g. (!f'. (!s. (!z. (((open s) /\\ ((z IN s) /\\ ((!w. ((w IN s) ==> ((f w) = (g w)))) /\\ (((has_complex_derivative f) f') (at z))))) ==> (((has_complex_derivative g) f') (at z))))))))"}
{"_id": "755", "title": "COMPLEX_MUL_CSIN_CSIN", "text": "|- (!w. (!z. (((complex_mul (csin w)) (csin z)) = ((complex_div ((vector_sub (ccos ((vector_sub w) z))) (ccos ((vector_add w) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "756", "title": "FORALL_IN_IMAGE", "text": "|- (!f. (!s. ((!y. ((y IN ((IMAGE f) s)) ==> (P y))) = (!x. ((x IN s) ==> (P (f x)))))))"}
{"_id": "757", "title": "column", "text": "|- (!A. (!j. (((column j) A) = (lambdai. ((A $ i) $ j)))))"}
{"_id": "758", "title": "VECTOR_ADD_LID", "text": "|- (!x. (((vector_add (vec (NUMERAL _0))) x) = x))"}
{"_id": "759", "title": "INTER_UNIV", "text": "|- ((!s. ((UNIV INTER s) = s)) /\\ (!s. ((s INTER UNIV) = s)))"}
{"_id": "760", "title": "cpow", "text": "|- (!z. (!w. (((cpow w) z) = (((COND (w = (Cx (real_of_num (NUMERAL _0))))) (Cx (real_of_num (NUMERAL _0)))) (cexp ((complex_mul z) (clog w)))))))"}
{"_id": "761", "title": "HOMOTOPIC_ON_EMPTY", "text": "|- (!t. (!f. (!g. ((((homotopic_with (\\x. T)) (EMPTY , t)) f) g))))"}
{"_id": "762", "title": "INCREASING_RIGHT_LIMIT_1", "text": "|- (!f. (!a. (!b. (!c. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f x))) (drop (f y)))))) /\\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (?l. (((--> f) l) ((within (at c)) (closed_interval ((CONS (c , b)) NIL))))))))))"}
{"_id": "763", "title": "ARITH_EXP", "text": "|- ((!m. (!n. (((NUMERAL m) EXP (NUMERAL n)) = (NUMERAL (m EXP n))))) /\\ (((_0 EXP _0) = (BIT1 _0)) /\\ ((!m. (((BIT0 m) EXP _0) = (BIT1 _0))) /\\ ((!m. (((BIT1 m) EXP _0) = (BIT1 _0))) /\\ ((!n. ((_0 EXP (BIT0 n)) = ((_0 EXP n) * (_0 EXP n)))) /\\ ((!m. (!n. (((BIT0 m) EXP (BIT0 n)) = (((BIT0 m) EXP n) * ((BIT0 m) EXP n))))) /\\ ((!m. (!n. (((BIT1 m) EXP (BIT0 n)) = (((BIT1 m) EXP n) * ((BIT1 m) EXP n))))) /\\ ((!n. ((_0 EXP (BIT1 n)) = _0)) /\\ ((!m. (!n. (((BIT0 m) EXP (BIT1 n)) = ((BIT0 m) * (((BIT0 m) EXP n) * ((BIT0 m) EXP n)))))) /\\ (!m. (!n. (((BIT1 m) EXP (BIT1 n)) = ((BIT1 m) * (((BIT1 m) EXP n) * ((BIT1 m) EXP n)))))))))))))))"}
{"_id": "764", "title": "ENR_IMP_ANR", "text": "|- (!s. ((ENR s) ==> (ANR s)))"}
{"_id": "765", "title": "Dih2k_hypermap.vecmat", "text": "|- (!f. (!j. (((vecmat j) f) = (lambdai. (f $ ((j * (dimindex UNIV)) + i))))))"}
{"_id": "766", "title": "Lp_ineqs_proofs.add_diag", "text": "|- (!V. (!v. (!w. (!E. (((add_diag (V , E)) (v , w)) = (((COND ((v , w) IN (dart1_of_fan (V , E)))) ((w INSERT ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) INSERT EMPTY)) INSERT E)) E))))))"}
{"_id": "767", "title": "Seq.perm_eq_mem", "text": "|- (!s1. (!s2. (((perm_eq s1) s2) ==> (!x. (((MEM x) s1) = ((MEM x) s2))))))"}
{"_id": "768", "title": "NEUTRAL_ADD", "text": "|- ((neutral +) = (NUMERAL _0))"}
{"_id": "769", "title": "Hypermap.number_of_nodes", "text": "|- (!H. ((number_of_nodes H) = (CARD (node_set H))))"}
{"_id": "770", "title": "Pack2.BOUNDED_VORONOI_CLOSED", "text": "|- (!s. (!v. ((saturated s) ==> (bounded ((voronoi_closed s) v)))))"}
{"_id": "771", "title": "Hypermap.FINITE_HYPERMAP_ORBITS", "text": "|- (!H. ((FINITE (edge_set H)) /\\ ((FINITE (node_set H)) /\\ (FINITE (face_set H)))))"}
{"_id": "772", "title": "Hypermap.is_node_nondegenerate", "text": "|- (!H. ((is_node_nondegenerate H) = (!x. ((x IN (dart H)) ==> (~ (((node_map H) x) = x))))))"}
{"_id": "773", "title": "FRECHET_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL", "text": "|- (!f. (!f'. (!f''. (!x. (!a. (!b. (((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i)))) /\\ ((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((((has_derivative f) f') ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) /\\ (((has_derivative f) f'') ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))) ==> (f' = f''))))))))"}
{"_id": "774", "title": "OPEN_COMPONENTS", "text": "|- (!u. (!s. (((open u) /\\ (s IN (components u))) ==> (open s))))"}
{"_id": "775", "title": "QUOTIENT_MAP_FROM_COMPOSITION", "text": "|- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((continuous_on g) t) /\\ ((((IMAGE g) t) SUBSET u) /\\ (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%667. (?x. (((SETSPEC GEN%PVAR%667) ((x IN s) /\\ (((g o f) x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))) ==> (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) t)) (GSPEC (\\GEN%PVAR%668. (?x. (((SETSPEC GEN%PVAR%668) ((x IN t) /\\ ((g x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v))))))))))"}
{"_id": "776", "title": "OUTSIDE_TRANSLATION", "text": "|- (!a. (!s. ((outside ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) (outside s)))))"}
{"_id": "777", "title": "OPEN_IMP_LOCALLY_COMPACT", "text": "|- (!s. ((open s) ==> ((locally compact) s)))"}
{"_id": "778", "title": "TC_CASES", "text": "|- (!R. (!a0. (!a1. ((((TC R) a0) a1) = (((R a0) a1) \\/ (?y. ((((TC R) a0) y) /\\ (((TC R) y) a1))))))))"}
{"_id": "779", "title": "LIM_ZERO_NEGINFINITY", "text": "|- (!f. (!l. ((((--> (\\x. (f ((real_div (real_of_num (NUMERAL (BIT1 _0)))) x)))) l) (atreal (real_of_num (NUMERAL _0)))) ==> (((--> f) l) at_neginfinity))))"}
{"_id": "780", "title": "NEGLIGIBLE_ON_INTERVALS", "text": "|- (!s. ((negligible s) = (!a. (!b. (negligible (s INTER (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "781", "title": "DIFFERENTIABLE_NEG", "text": "|- (!f. (!net. (((differentiable f) net) ==> ((differentiable (\\z. (vector_neg (f z)))) net))))"}
{"_id": "782", "title": "INTERIOR_SUBSET_RELATIVE_INTERIOR", "text": "|- (!s. ((interior s) SUBSET (relative_interior s)))"}
{"_id": "783", "title": "REAL_CONVEX_ON", "text": "|- (!f. (!s. (((real_convex_on f) s) = ((convex_on (f o drop)) ((IMAGE lift) s)))))"}
{"_id": "784", "title": "Fan.pr2", "text": "|- (pr2 = (GABS (\\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) v))))))))"}
{"_id": "785", "title": "SING_SUBSET", "text": "|- (!s. (!x. (((x INSERT EMPTY) SUBSET s) = (x IN s))))"}
{"_id": "786", "title": "REAL_MIN_MIN", "text": "|- (!x. (!y. (((real_le ((real_min x) y)) x) /\\ ((real_le ((real_min x) y)) y))))"}
{"_id": "787", "title": "Nonlin_def.sol_euler_x_div_sqrtdelta", "text": "|- (!x4. (!x5. (!x3. (!x1. (!x2. (!x6. (((((((sol_euler_x_div_sqrtdelta x1) x2) x3) x4) x5) x6) = ((LET (\\a. (LET_END ((real_div (matan ((real_div ((((((delta_x x1) x2) x3) x4) x5) x6)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0)))))))) a)))) ((real_add (sqrt ((real_mul x1) ((real_mul x2) x3)))) ((real_add ((real_mul (sqrt x1)) ((real_div ((real_add x2) ((real_sub x3) x4))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_add ((real_mul (sqrt x2)) ((real_div ((real_add x1) ((real_sub x3) x5))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul (sqrt x3)) ((real_div ((real_add x1) ((real_sub x2) x6))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))"}
{"_id": "788", "title": "INJECTIVE_ON_PREIMAGE", "text": "|- (!f. (!s. (!u. ((!t. (!t'. (((t SUBSET u) /\\ ((t' SUBSET u) /\\ ((GSPEC (\\GEN%PVAR%142. (?x. (((SETSPEC GEN%PVAR%142) ((x IN s) /\\ ((f x) IN t))) x)))) = (GSPEC (\\GEN%PVAR%143. (?x. (((SETSPEC GEN%PVAR%143) ((x IN s) /\\ ((f x) IN t'))) x))))))) ==> (t = t')))) = (u SUBSET ((IMAGE f) s))))))"}
{"_id": "789", "title": "Functional_equation.proj_y3", "text": "|- (!x1. (!x2. (!x4. (!x5. (!x6. (!x3. (((((((proj_y3 x1) x2) x3) x4) x5) x6) = (sqrt x3))))))))"}
{"_id": "790", "title": "MEASURABLE_ON_TRANSLATION", "text": "|- (!f. (!s. (!a. (((measurable_on f) ((IMAGE (\\x. ((vector_add a) x))) s)) ==> ((measurable_on (\\x. (f ((vector_add a) x)))) s)))))"}
{"_id": "791", "title": "RELATIVE_INTERIOR_EQ_EMPTY", "text": "|- (!s. ((convex s) ==> (((relative_interior s) = EMPTY) = (s = EMPTY))))"}
{"_id": "792", "title": "CLOSED_HALFSPACE_IM_GE", "text": "|- (!b. (closed (GSPEC (\\GEN%PVAR%4756. (?z. (((SETSPEC GEN%PVAR%4756) ((real_ge (Im z)) b)) z))))))"}
{"_id": "793", "title": "Add_triangle.f_fan_pair3", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((((v , w) IN (dart1_of_fan (V , E))) /\\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0))))) ==> ((((f_fan_pair (V , E)) (v , w)) = (w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))) /\\ ((((f_fan_pair (V , E)) (w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))) = ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v)) /\\ ((((f_fan_pair (V , E)) ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v)) = (v , w)) /\\ ((w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) IN (dart1_of_fan (V , E))))))))))))"}
{"_id": "794", "title": "ABSOLUTELY_INTEGRABLE_SPIKE", "text": "|- (!f. (!g. (!s. (!t. (((negligible s) /\\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((absolutely_integrable_on f) t) ==> ((absolutely_integrable_on g) t)))))))"}
{"_id": "795", "title": "PERMUTES_EMPTY", "text": "|- (!p. (((permutes p) EMPTY) = (p = I)))"}
{"_id": "796", "title": "ARITH_EVEN", "text": "|- ((!n. ((EVEN (NUMERAL n)) = (EVEN n))) /\\ (((EVEN _0) = T) /\\ ((!n. ((EVEN (BIT0 n)) = T)) /\\ (!n. ((EVEN (BIT1 n)) = F)))))"}
{"_id": "797", "title": "nsum", "text": "|- (nsum = (iterate +))"}
{"_id": "798", "title": "List_hypermap.mem_list_of_darts", "text": "|- (!d. (!L. (((MEM d) (list_of_darts L)) = (?l. (((MEM l) L) /\\ ((MEM d) (list_pairs l)))))))"}
{"_id": "799", "title": "Planarity.PROPERTIES_TRIANGLE_FAN", "text": "|- (!x. (!V. (!E. (!v. (!u. (!w. (((FAN (x , (V , E))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ (((w INSERT (v INSERT EMPTY)) IN E) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ (fan80 (x , (V , E))))))))) ==> (((((((sigma_fan x) V) E) v) u) = w) /\\ ((((((sigma_fan x) V) E) w) v) = u)))))))))"}
{"_id": "800", "title": "ARC_TRANSLATION_EQ", "text": "|- (!a. (!g. ((arc ((\\x. ((vector_add a) x)) o g)) = (arc g))))"}
{"_id": "801", "title": "Ssrnat.sqrn_add", "text": "|- (!m. (!n. (((m + n) EXP (NUMERAL (BIT0 (BIT1 _0)))) = (((m EXP (NUMERAL (BIT0 (BIT1 _0)))) + (n EXP (NUMERAL (BIT0 (BIT1 _0))))) + ((NUMERAL (BIT0 (BIT1 _0))) * (m * n))))))"}
{"_id": "802", "title": "STARLIKE_IMP_PATH_CONNECTED", "text": "|- (!s. ((starlike s) ==> (path_connected s)))"}
{"_id": "803", "title": "Hypermap.COM_POWER", "text": "|- (!n. (!f. (((POWER f) (SUC n)) = (f o ((POWER f) n)))))"}
{"_id": "804", "title": "REAL_SUB_REFL", "text": "|- (!x. (((real_sub x) x) = (real_of_num (NUMERAL _0))))"}
{"_id": "805", "title": "Seq.size_drop", "text": "|- (!n0. (!s. ((sizel ((dropl n0) s)) = ((sizel s) - n0))))"}
{"_id": "806", "title": "IMAGE_ID", "text": "|- (!s. (((IMAGE (\\x. x)) s) = s))"}
{"_id": "807", "title": "REAL_ROOT_POW", "text": "|- (!n. (!x. (((ODD n) \\/ ((~ (n = (NUMERAL _0))) /\\ ((real_le (real_of_num (NUMERAL _0))) x))) ==> (((root n) ((real_pow x) n)) = x))))"}
{"_id": "808", "title": "HOMOTOPIC_WITH_COMPOSE", "text": "|- (!P. (!Q. (!R. (!f. (!f'. (!g. (!g'. (!s. (!t. (!u. (((!f. (!g. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ ((P f) /\\ (((continuous_on g) t) /\\ ((((IMAGE g) t) SUBSET u) /\\ (Q g)))))) ==> (R (g o f))))) /\\ (((((homotopic_with P) (s , t)) f) f') /\\ ((((homotopic_with Q) (t , u)) g) g'))) ==> ((((homotopic_with R) (s , u)) (g o f)) (g' o f')))))))))))))"}
{"_id": "809", "title": "COMPLEX_ADD_SYM", "text": "|- (!x. (!y. (((vector_add x) y) = ((vector_add y) x))))"}
{"_id": "810", "title": "ETA_AX", "text": "|- (!t. ((\\x. (t x)) = t))"}
{"_id": "811", "title": "HOMEOMORPHIC_COMPACTNESS", "text": "|- (!s. (!t. (((homeomorphic s) t) ==> ((compact s) = (compact t)))))"}
{"_id": "812", "title": "ANR_OPEN_IN", "text": "|- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\\ (ANR t)) ==> (ANR s))))"}
{"_id": "813", "title": "HOMOTOPY_EQUIVALENT_EMPTY", "text": "|- ((!s. (((homotopy_equivalent s) EMPTY) = (s = EMPTY))) /\\ (!t. (((homotopy_equivalent EMPTY) t) = (t = EMPTY))))"}
{"_id": "814", "title": "REAL_SECOND_MEAN_VALUE_THEOREM_FULL", "text": "|- (!f. (!g. (!a. (!b. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_le x) y))) ==> ((real_le (g x)) (g y))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ (((has_real_integral (\\x. ((real_mul (g x)) (f x)))) ((real_add ((real_mul (g a)) ((real_integral (closed_real_interval ((CONS (a , c)) NIL))) f))) ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f)))) (closed_real_interval ((CONS (a , b)) NIL))))))))))"}
{"_id": "815", "title": "HAS_REAL_DERIVATIVE_WITHIN_SUBSET", "text": "|- (!f. (!s. (!t. (!x. (((((has_real_derivative f) f') ((within (atreal x)) s)) /\\ (t SUBSET s)) ==> (((has_real_derivative f) f') ((within (atreal x)) t)))))))"}
{"_id": "816", "title": "RPOW_ZERO", "text": "|- (!y. (((rpow (real_of_num (NUMERAL _0))) y) = (((COND (y = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))"}
{"_id": "817", "title": "Lp_main_estimate.v_prime_subset", "text": "|- (!V. (!X. (!f. ((V SUBSET X) ==> (((v_prime V) f) SUBSET X)))))"}
{"_id": "818", "title": "Hypermap.lemma_in_subset", "text": "|- (!s. (!t. (!x. (((s SUBSET t) /\\ (x IN s)) ==> (x IN t)))))"}
{"_id": "819", "title": "Hypermap.iterate_map_valuation", "text": "|- (!p. (!n. (!x. ((p (((POWER p) n) x)) = (((POWER p) (SUC n)) x)))))"}
{"_id": "820", "title": "AR_FROM_UNION_AND_INTER", "text": "|- (!s. (!t. (((closed s) /\\ ((closed t) /\\ ((AR (s UNION t)) /\\ (AR (s INTER t))))) ==> ((AR s) /\\ (AR t)))))"}
{"_id": "821", "title": "Hexagons.LE_sqrt8_2h0", "text": "|- ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))) sqrt8)"}
{"_id": "822", "title": "PROPER_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP", "text": "|- (!p. (!c. (!s. (((((IMAGE p) c) = s) /\\ ((!k. (((k SUBSET s) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%2100. (?x. (((SETSPEC GEN%PVAR%2100) ((x IN c) /\\ ((p x) IN k))) x))))))) /\\ (!x. ((x IN c) ==> (?t. (?u. (?q. ((x IN t) /\\ (((open_in ((subtopology euclidean) c)) t) /\\ (((open_in ((subtopology euclidean) s)) u) /\\ ((homeomorphism (t , u)) (p , q)))))))))))) ==> ((covering_space (c , p)) s)))))"}
{"_id": "823", "title": "ARITH_EQ", "text": "|- ((!m. (!n. (((NUMERAL m) = (NUMERAL n)) = (m = n)))) /\\ (((_0 = _0) = T) /\\ ((!n. (((BIT0 n) = _0) = (n = _0))) /\\ ((!n. (((BIT1 n) = _0) = F)) /\\ ((!n. ((_0 = (BIT0 n)) = (_0 = n))) /\\ ((!n. ((_0 = (BIT1 n)) = F)) /\\ ((!m. (!n. (((BIT0 m) = (BIT0 n)) = (m = n)))) /\\ ((!m. (!n. (((BIT0 m) = (BIT1 n)) = F))) /\\ ((!m. (!n. (((BIT1 m) = (BIT0 n)) = F))) /\\ (!m. (!n. (((BIT1 m) = (BIT1 n)) = (m = n)))))))))))))"}
{"_id": "824", "title": "SUBTOPOLOGY_UNIV", "text": "|- (!top. (((subtopology top) UNIV) = top))"}
{"_id": "825", "title": "ANALYTIC_ON_ADD", "text": "|- (!f. (!g. (!s. ((((analytic_on f) s) /\\ ((analytic_on g) s)) ==> ((analytic_on (\\z. ((vector_add (f z)) (g z)))) s)))))"}
{"_id": "826", "title": "Nonlin_def.edge2_126_x", "text": "|- (!d. (!x4. (!x5. ((((edge2_126_x d) x4) x5) = (((((((compose6 edge2_flatD_x1) (constant6 d)) proj_x1) proj_x2) proj_x6) (constant6 x4)) (constant6 x5))))))"}
{"_id": "827", "title": "GE", "text": "|- (!n. (!m. ((m >= n) = (n <= m))))"}
{"_id": "828", "title": "LOCALLY_NEGLIGIBLE_ALT", "text": "|- (!s. ((negligible s) = (!x. ((x IN s) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\\ ((x IN u) /\\ (negligible u))))))))"}
{"_id": "829", "title": "LINEAR_INJECTIVE_IMP_SURJECTIVE", "text": "|- (!f. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (!y. (?x. ((f x) = y)))))"}
{"_id": "830", "title": "CX_TAN", "text": "|- (!x. ((Cx (tan x)) = (ctan (Cx x))))"}
{"_id": "831", "title": "FORALL_IN_INSERT", "text": "|- (!P. (!a. (!s. ((!x. ((x IN (a INSERT s)) ==> (P x))) = ((P a) /\\ (!x. ((x IN s) ==> (P x))))))))"}
{"_id": "832", "title": "ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT", "text": "|- (!op. (!f. (!g. (!s. (((bilinear op) /\\ (((measurable_on f) s) /\\ ((bounded ((IMAGE f) s)) /\\ ((absolutely_integrable_on g) s)))) ==> ((absolutely_integrable_on (\\x. ((op (f x)) (g x)))) s))))))"}
{"_id": "833", "title": "CONTRACTIBLE_TRANSLATION", "text": "|- (!a. (!s. ((contractible ((IMAGE (\\x. ((vector_add a) x))) s)) = (contractible s))))"}
{"_id": "834", "title": "REAL_LT_NEG2", "text": "|- (!x. (!y. (((real_lt (real_neg x)) (real_neg y)) = ((real_lt y) x))))"}
{"_id": "835", "title": "Sphere.quadratic_root_plus", "text": "|- (!b. (!c. (!a. ((quadratic_root_plus (a , (b , c))) = ((real_div ((real_add (real_neg b)) (sqrt ((real_sub ((real_pow b) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul a) c)))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) a))))))"}
{"_id": "836", "title": "NEUTRAL_REAL_ADD", "text": "|- ((neutral real_add) = (real_of_num (NUMERAL _0)))"}
{"_id": "837", "title": "COMPLEX_STONE_WEIERSTRASS_ALT", "text": "|- (!P. (!s. (((compact s) /\\ ((!c. (P (\\x. c))) /\\ ((!f. ((P f) ==> (P (\\x. (cnj (f x)))))) /\\ ((!f. (!g. (((P f) /\\ (P g)) ==> (P (\\x. ((vector_add (f x)) (g x))))))) /\\ ((!f. (!g. (((P f) /\\ (P g)) ==> (P (\\x. ((complex_mul (f x)) (g x))))))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ (~ (x = y)))) ==> (?f. ((P f) /\\ (((continuous_on f) s) /\\ (~ ((f x) = (f y)))))))))))))) ==> (!f. (!e. ((((continuous_on f) s) /\\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?g. ((P g) /\\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))"}
{"_id": "838", "title": "LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE", "text": "|- (!f. (!s. (((!t. ((t SUBSET ((IMAGE f) s)) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%1904. (?x. (((SETSPEC GEN%PVAR%1904) ((x IN s) /\\ ((f x) IN t))) x))))) = ((open_in ((subtopology euclidean) ((IMAGE f) s))) t)))) /\\ ((locally path_connected) s)) ==> ((locally path_connected) ((IMAGE f) s)))))"}
{"_id": "839", "title": "RTC_INDUCT", "text": "|- (!R. (!P. (((!x. (!y. (((R x) y) ==> ((P x) y)))) /\\ ((!x. ((P x) x)) /\\ (!x. (!y. (!z. ((((P x) y) /\\ ((P y) z)) ==> ((P x) z))))))) ==> (!x. (!y. ((((RTC R) x) y) ==> ((P x) y)))))))"}
{"_id": "840", "title": "Kepler_spec.sum_spec", "text": "|- (!f. (!n. ((((sum ((NUMERAL _0) .. (NUMERAL _0))) f) = (f (NUMERAL _0))) /\\ (((sum ((NUMERAL _0) .. (SUC n))) f) = ((real_add ((sum ((NUMERAL _0) .. n)) f)) (f (SUC n)))))))"}
{"_id": "841", "title": "PCROSS", "text": "|- (!s. (!t. ((s PCROSS t) = (GSPEC (\\GEN%PVAR%239. (?x. (?y. (((SETSPEC GEN%PVAR%239) ((x IN s) /\\ (y IN t))) ((pastecart x) y)))))))))"}
{"_id": "842", "title": "Fan.extension_sigma_fan", "text": "|- (!x. (!V. (!E. (!v. (!u. ((((((extension_sigma_fan x) V) E) v) u) = (((COND (~ (u IN (((set_of_edge v) V) E)))) u) (((((sigma_fan x) V) E) v) u))))))))"}
{"_id": "843", "title": "IS_INTERVAL_PATH_CONNECTED_1", "text": "|- (!s. ((is_interval s) = (path_connected s)))"}
{"_id": "844", "title": "HAS_REAL_INTEGRAL_REFLECT", "text": "|- (!f. (!i. (!a. (!b. ((((has_real_integral (\\x. (f (real_neg x)))) i) (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) = (((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "845", "title": "UNIFORM_LIM_COMPLEX_INV", "text": "|- (!net. (!P. (!f. (!l. (!b. (((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) (l n)))) e))))) net))) /\\ (((real_lt (real_of_num (NUMERAL _0))) b) /\\ ((eventually (\\x. (!n. ((P n) ==> ((real_le b) (vector_norm (l n))))))) net))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub (complex_inv ((f n) x))) (complex_inv (l n))))) e))))) net)))))))))"}
{"_id": "846", "title": "DIVMOD_UNIQ", "text": "|- (!m. (!n. (!q. (!r. (((m = ((q * n) + r)) /\\ (r < n)) ==> (((m DIV n) = q) /\\ ((m MOD n) = r)))))))"}
{"_id": "847", "title": "CONNECTED_BALL", "text": "|- (!x. (!e. (connected (ball (x , e)))))"}
{"_id": "848", "title": "BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE", "text": "|- (!f. (!g. (!h. (!s. ((((uniformly_continuous_on f) s) /\\ (((uniformly_continuous_on g) s) /\\ ((bilinear h) /\\ ((bounded ((IMAGE f) s)) /\\ (bounded ((IMAGE g) s)))))) ==> ((uniformly_continuous_on (\\x. ((h (f x)) (g x)))) s))))))"}
{"_id": "849", "title": "REAL_DIFFERENTIABLE_AT_SQRT", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable sqrt) (atreal x))))"}
{"_id": "850", "title": "Misc_defs_and_lemmas.NUM_COUNTABLE", "text": "|- (COUNTABLE UNIV)"}
{"_id": "851", "title": "Nonlin_def.cayleyR6u", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. 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{"_id": "852", "title": "Tame_defs.nextElem", "text": "|- (((((nextElem NIL) b) x) = b) /\\ ((((nextElem ((CONS a) aas)) b) x) = (((COND (x = a)) (((COND (aas = NIL)) b) (HD aas))) (((nextElem aas) b) x))))"}
{"_id": "853", "title": "Hypermap.lemma_edge_subset", "text": "|- (!H. (!x. ((x IN (dart H)) ==> (((edge H) x) SUBSET (dart H)))))"}
{"_id": "854", "title": "HOMOTOPICALLY_TRIVIAL_RETRACTION_NULL_GEN", "text": "|- (!P. (!Q. (!s. (!t. (!u. (!h. (!k. (((((continuous_on h) s) /\\ ((((IMAGE h) s) = t) /\\ (((continuous_on k) t) /\\ ((((IMAGE k) t) SUBSET s) /\\ ((!y. ((y IN t) ==> ((h (k y)) = y))) /\\ ((!f. ((((continuous_on f) u) /\\ ((((IMAGE f) u) SUBSET t) /\\ (Q f))) ==> (P (k o f)))) /\\ ((!f. ((((continuous_on f) u) /\\ ((((IMAGE f) u) SUBSET s) /\\ (P f))) ==> (Q (h o f)))) /\\ (!h. (!k. ((!x. ((x IN u) ==> ((h x) = (k x)))) ==> ((Q h) = (Q k)))))))))))) /\\ (!f. ((((continuous_on f) u) /\\ ((((IMAGE f) u) SUBSET s) /\\ (P f))) ==> (?c. ((((homotopic_with P) (u , s)) f) (\\x. c)))))) ==> (!f. ((((continuous_on f) u) /\\ ((((IMAGE f) u) SUBSET t) /\\ (Q f))) ==> (?c. ((((homotopic_with Q) (u , t)) f) (\\x. c)))))))))))))"}
{"_id": "855", "title": "CONIC_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((conic ((IMAGE f) s)) = (conic s)))))"}
{"_id": "856", "title": "EXISTS_DROP", "text": "|- ((?x. (P x)) = (?x. (P (drop x))))"}
{"_id": "857", "title": "SUBSPACE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (subspace s)) ==> (subspace ((IMAGE f) s)))))"}
{"_id": "858", "title": "SPAN_FINITE", "text": "|- (!s. ((FINITE s) ==> ((span s) = (GSPEC (\\GEN%PVAR%141. (?y. (((SETSPEC GEN%PVAR%141) (?u. (((vsum s) (\\v. ((% (u v)) v))) = y))) y)))))))"}
{"_id": "859", "title": "FUNDAMENTAL_THEOREM_OF_CALCULUS", "text": "|- (!f. (!f'. (!a. (!b. ((((real_le (drop a)) (drop b)) /\\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (((has_vector_derivative f) (f' x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))) ==> (((has_integral f') ((vector_sub (f b)) (f a))) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "860", "title": "CARD_FRONTIER_INTERVAL_1", "text": "|- (!s. ((is_interval s) ==> ((FINITE (frontier s)) /\\ ((CARD (frontier s)) <= (NUMERAL (BIT0 (BIT1 _0)))))))"}
{"_id": "861", "title": "JOINABLE_CONNECTED_COMPONENT_EQ", "text": "|- (!s. (!t. (!x. (!y. (((connected t) /\\ ((t SUBSET s) /\\ ((~ ((((connected_component s) x) INTER t) = EMPTY)) /\\ (~ ((((connected_component s) y) INTER t) = EMPTY))))) ==> (((connected_component s) x) = ((connected_component s) y)))))))"}
{"_id": "862", "title": "REAL_DIFFERENTIABLE_AT_TAN", "text": "|- (!x. ((~ ((cos x) = (real_of_num (NUMERAL _0)))) ==> ((real_differentiable tan) (atreal x))))"}
{"_id": "863", "title": "HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET", "text": "|- (!f. (!s. (!t. (!x. (((((has_vector_derivative f) f') ((within (at x)) s)) /\\ (t SUBSET s)) ==> (((has_vector_derivative f) f') ((within (at x)) t)))))))"}
{"_id": "864", "title": "LOCALLY_CONNECTED_TRANSLATION_EQ", "text": "|- (!a. (!s. (((locally connected) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((locally connected) s))))"}
{"_id": "865", "title": "DIM_EQ_FULL", "text": "|- (!s. (((dim s) = (dimindex UNIV)) = ((span s) = UNIV)))"}
{"_id": "866", "title": "int_sub_th", "text": "|- (!x. (!y. ((real_of_int ((int_sub x) y)) = ((real_sub (real_of_int x)) (real_of_int y)))))"}
{"_id": "867", "title": "CONTINUOUS_ON_EMPTY", "text": "|- (!f. ((continuous_on f) EMPTY))"}
{"_id": "868", "title": "Lp_ineqs_proofs.pack_dist", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!x. (((packing V) /\\ (x IN (dart_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance x))))))))"}
{"_id": "869", "title": "Add_triangle.face_split_fan_face_explicit", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((LET (\\w'. (\\H. (\\H2. (LET_END ((((v , w) IN (dart_of_fan (V , E))) /\\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face H) (v , w))))) ==> (((((face H2) (v , w)) = ((v , w) INSERT ((w , w') INSERT ((w' , v) INSERT EMPTY)))) /\\ (((face H2) (w' , w)) = ((w' , w) INSERT (((face H) (v , w)) DIFF ((v , w) INSERT ((w' , v) INSERT EMPTY)))))) /\\ ((((CARD ((face H2) (v , w))) = (NUMERAL (BIT1 (BIT1 _0)))) /\\ ((CARD ((face H2) (w' , w))) = ((CARD ((face H) (v , w))) - (NUMERAL (BIT1 _0))))) /\\ ((((v , w) IN ((face H) (v , w))) /\\ (((w' , v) IN ((face H) (v , w))) /\\ (~ ((w' , w) IN ((face H) (v , w)))))) /\\ ((~ (w' = w)) /\\ ((~ (v = w)) /\\ (~ (v = w'))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) (hypermap_of_fan (V , E))) (hypermap_of_fan ((split_fan_face (V , E)) (v , w))))))))))"}
{"_id": "870", "title": "DOT_LMUL", "text": "|- (!c. (!x. (!y. (((dot ((% c) x)) y) = ((real_mul c) ((dot x) y))))))"}
{"_id": "871", "title": "CEXP_0", "text": "|- ((cexp (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "872", "title": "int_max_th", "text": "|- (!x. (!y. ((real_of_int ((int_max x) y)) = ((real_max (real_of_int x)) (real_of_int y)))))"}
{"_id": "873", "title": "Constants_approx.CONST_INTERVAL", "text": "|- (!x. ((interval_arith x) (x , x)))"}
{"_id": "874", "title": "HAS_DERIVATIVE_VSUM", "text": "|- (!f. (!net. (!s. (((FINITE s) /\\ (!a. ((a IN s) ==> (((has_derivative (f a)) (f' a)) net)))) ==> (((has_derivative (\\x. ((vsum s) (\\a. ((f a) x))))) (\\h. ((vsum s) (\\a. ((f' a) h))))) net)))))"}
{"_id": "875", "title": "KREIN_MILMAN_MINKOWSKI", "text": "|- (!s. (((convex s) /\\ (compact s)) ==> (s = ((hull convex) (GSPEC (\\GEN%PVAR%2241. (?x. (((SETSPEC GEN%PVAR%2241) ((extreme_point_of x) s)) x))))))))"}
{"_id": "876", "title": "REAL_LOG_CONVEX_IMP_POS", "text": "|- (!f. (!s. (!x. ((((real_log_convex_on f) s) /\\ (x IN s)) ==> ((real_le (real_of_num (NUMERAL _0))) (f x))))))"}
{"_id": "877", "title": "Terminal.FUNLIST_EXPLICIT", "text": "|- ((!data. (!d. (!k. (!i. (!j. ((((((funlist_v39 data) d) k) i) j) = (((COND ((i MOD k) = (j MOD k))) (real_of_num (NUMERAL _0))) (((ASSOCD_v39 ((psort k) (i , j))) ((MAP (GABS (\\f. (!u. (!d. ((GEQ (f (u , d))) (((psort k) u) , d))))))) data)) d)))))))) /\\ ((!data. (!u. (!u'. (!k. (!i. (!j. (((((((funlistA_v39 data) u) u') k) i) j) = (((COND ((i MOD k) = (j MOD k))) u) (((ASSOCD_v39 ((psort k) (i , j))) ((MAP (GABS (\\f. (!u. (!d. ((GEQ (f (u , d))) (((psort k) u) , d))))))) data)) u'))))))))) /\\ ((((NUMERAL _0) MOD (NUMERAL (BIT1 (BIT1 _0)))) = (NUMERAL _0)) /\\ ((((NUMERAL (BIT1 _0)) MOD (NUMERAL (BIT1 (BIT1 _0)))) = (NUMERAL (BIT1 _0))) /\\ ((((NUMERAL (BIT0 (BIT1 _0))) MOD (NUMERAL (BIT1 (BIT1 _0)))) = (NUMERAL (BIT0 (BIT1 _0)))) /\\ ((((NUMERAL (BIT1 (BIT1 _0))) MOD (NUMERAL (BIT1 (BIT1 _0)))) = (NUMERAL _0)) /\\ ((!x. (x = x)) /\\ ((~ ((NUMERAL _0) = (NUMERAL (BIT1 _0)))) /\\ ((~ ((NUMERAL _0) = (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((~ ((NUMERAL (BIT1 _0)) = (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((~ ((NUMERAL _0) = (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((~ ((NUMERAL (BIT1 _0)) = (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((~ ((NUMERAL (BIT0 (BIT1 _0))) = (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((!a. (!b. (!c. (!d. (((a , b) = (c , d)) = ((a = c) /\\ (b = d))))))) /\\ ((!a. (!d. ((((ASSOCD_v39 a) NIL) d) = d))) /\\ ((!a. (!d. (!h. (!t. ((((ASSOCD_v39 a) ((CONS h) t)) d) = (((COND (a = (FST h))) (SND h)) (((ASSOCD_v39 a) t) d))))))) /\\ ((!k. (((MAP (GABS (\\f. (!u. (!d. ((GEQ (f (u , d))) (((psort k) u) , d))))))) NIL) = NIL)) /\\ ((!p. (!d. ((((ASSOCD_v39 p) NIL) d) = d))) /\\ (!k. (!i. (!j. (!r. (!t. (((MAP (GABS (\\f. (!u. (!d. ((GEQ (f (u , d))) (((psort k) u) , d))))))) ((CONS ((i , j) , r)) t)) = ((CONS (((psort k) (i , j)) , r)) ((MAP (GABS (\\f. (!u. (!d. ((GEQ (f (u , d))) (((psort k) u) , d))))))) t))))))))))))))))))))))))))))))))))))))))))"}
{"_id": "878", "title": "Lp_ineqs_def.rho_fan", "text": "|- (!v. ((rho_fan v) = (real_abs ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (ln_fan v)))))))"}
{"_id": "879", "title": "MATRIX_CMUL", "text": "|- (!f. (!c. ((linear f) ==> ((matrix (\\x. ((% c) (f x)))) = ((%% c) (matrix f))))))"}
{"_id": "880", "title": "IN_SEGMENT", "text": "|- (!a. (!b. (!x. (((x IN (closed_segment ((CONS (a , b)) NIL))) = (?u. (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le u) (real_of_num (NUMERAL (BIT1 _0)))) /\\ (x = ((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) a)) ((% u) b))))))) /\\ ((x IN (open_segment (a , b))) = ((~ (a = b)) /\\ (?u. (((real_lt (real_of_num (NUMERAL _0))) u) /\\ (((real_lt u) (real_of_num (NUMERAL (BIT1 _0)))) /\\ (x = ((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) a)) ((% u) b))))))))))))"}
{"_id": "881", "title": "Sphere.rhazim", "text": "|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((rhazim y1) y2) y3) y4) y5) y6) = ((real_mul (rho y1)) ((((((dih_y y1) y2) y3) y4) y5) y6)))))))))"}
{"_id": "882", "title": "HAS_REAL_INTEGRAL_RESTRICT", "text": "|- (!f. (!s. (!t. ((s SUBSET t) ==> ((((has_real_integral (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) i) t) = (((has_real_integral f) i) s))))))"}
{"_id": "883", "title": "BETWEEN_IN_SEGMENT", "text": "|- (!x. (!a. (!b. (((between x) (a , b)) = (x IN (closed_segment ((CONS (a , b)) NIL)))))))"}
{"_id": "884", "title": "HAS_MEASURE_ALMOST", "text": "|- (!s. (!s'. (!t. (!m. ((((has_measure s) m) /\\ ((negligible t) /\\ ((s UNION t) = (s' UNION t)))) ==> ((has_measure s') m))))))"}
{"_id": "885", "title": "VSUM_DELETE", "text": "|- (!f. (!s. (!a. (((FINITE s) /\\ (a IN s)) ==> (((vsum (s DELETE a)) f) = ((vector_sub ((vsum s) f)) (f a)))))))"}
{"_id": "886", "title": "SUM_MULTICOUNT_GEN", "text": "|- (!R. (!s. (!t. (!k. (((FINITE s) /\\ ((FINITE t) /\\ (!j. ((j IN t) ==> ((CARD (GSPEC (\\GEN%PVAR%203. (?i. (((SETSPEC GEN%PVAR%203) ((i IN s) /\\ ((R i) j))) i))))) = (k j)))))) ==> (((sum s) (\\i. (real_of_num (CARD (GSPEC (\\GEN%PVAR%204. (?j. (((SETSPEC GEN%PVAR%204) ((j IN t) /\\ ((R i) j))) j)))))))) = ((sum t) (\\i. (real_of_num (k i))))))))))"}
{"_id": "887", "title": "REAL_LT_ADDNEG2", "text": "|- (!x. (!y. (!z. (((real_lt ((real_add x) (real_neg y))) z) = ((real_lt x) ((real_add z) y))))))"}
{"_id": "888", "title": "ABSORPTION", "text": "|- (!x. (!s. ((x IN s) = ((x INSERT s) = s))))"}
{"_id": "889", "title": "CONVEX_CBALL", "text": "|- (!x. (!e. (convex (cball (x , e)))))"}
{"_id": "890", "title": "CLOG_WORKS", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((cexp (clog z)) = z) /\\ (((real_lt (real_neg pi)) (Im (clog z))) /\\ ((real_le (Im (clog z))) pi)))))"}
{"_id": "891", "title": "REAL_SUB_LT", "text": "|- (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) ((real_sub x) y)) = ((real_lt y) x))))"}
{"_id": "892", "title": "PATH_IMAGE_SUBPATH_GEN", "text": "|- (!u. (!v. (!g. ((path_image (((subpath u) v) g)) = ((IMAGE g) (closed_segment ((CONS (u , v)) NIL)))))))"}
{"_id": "893", "title": "Hypermap_iso.image_set_inj_gen", "text": "|- (!u. (!f. ((!x. (!y. (((x IN u) /\\ ((y IN u) /\\ ((f x) = (f y)))) ==> (x = y)))) ==> (!s. (!t. (((s SUBSET u) /\\ ((t SUBSET u) /\\ (((IMAGE f) s) = ((IMAGE f) t)))) ==> (s = t)))))))"}
{"_id": "894", "title": "vector_derivative", "text": "|- (!f. (!net. (((vector_derivative f) net) = (@f'. (((has_vector_derivative f) f') net)))))"}
{"_id": "895", "title": "CONVEX_CONE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((convex_cone s) /\\ (linear f)) ==> (convex_cone ((IMAGE f) s)))))"}
{"_id": "896", "title": "Ssrnat.subn_add2l", "text": "|- (!p. (!m. (!n. (((p + m) - (p + n)) = (m - n)))))"}
{"_id": "897", "title": "EVENTUALLY_SEQUENTIALLY", "text": "|- (!p. (((eventually p) sequentially) = (?N. (!n. ((N <= n) ==> (p n))))))"}
{"_id": "898", "title": "Sphere.affsign", "text": "|- (!v. (!sgn. (!s. (!t. (((((affsign sgn) s) t) v) = (?f. ((v = ((lin_combo (s UNION t)) f)) /\\ ((!w. ((t w) ==> (sgn (f w)))) /\\ (((sum (s UNION t)) f) = (real_of_num (NUMERAL (BIT1 _0))))))))))))"}
{"_id": "899", "title": "REAL_DIFFERENTIABLE_AT_SIN", "text": "|- (!x. ((real_differentiable sin) (atreal x)))"}
{"_id": "900", "title": "REALLIM_1_OVER_N", "text": "|- (((---> (\\n. (real_inv (real_of_num n)))) (real_of_num (NUMERAL _0))) sequentially)"}
{"_id": "901", "title": "DIAMETER_POS_LE", "text": "|- (!s. ((bounded s) ==> ((real_le (real_of_num (NUMERAL _0))) (diameter s))))"}
{"_id": "902", "title": "HAS_INTEGRAL_SPIKE_INTERIOR", "text": "|- (!f. (!g. (!a. (!b. (!y. (((!x. ((x IN (open_interval (a , b))) ==> ((g x) = (f x)))) /\\ (((has_integral f) y) (closed_interval ((CONS (a , b)) NIL)))) ==> (((has_integral g) y) (closed_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "903", "title": "BERNOULLI_ALT", "text": "|- (!n. (!x. (((sum ((NUMERAL _0) .. n)) (\\k. ((real_mul (real_of_num (binom ((n + (NUMERAL (BIT1 _0))) , k)))) ((bernoulli k) x)))) = ((real_mul ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))) ((real_pow x) n)))))"}
{"_id": "904", "title": "DIST_NZ", "text": "|- (!x. (!y. ((~ (x = y)) = ((real_lt (real_of_num (NUMERAL _0))) (distance (x , y))))))"}
{"_id": "905", "title": "VECTOR_DERIVATIVE_WORKS", "text": "|- (!net. (!f. (((differentiable f) net) = (((has_vector_derivative f) ((vector_derivative f) net)) net))))"}
{"_id": "906", "title": "IN_BALL", "text": "|- (!x. (!y. (!e. ((y IN (ball (x , e))) = ((real_lt (distance (x , y))) e)))))"}
{"_id": "907", "title": "VSUM_SUPERSET", "text": "|- (!f. (!u. (!v. (((u SUBSET v) /\\ (!x. (((x IN v) /\\ (~ (x IN u))) ==> ((f x) = (vec (NUMERAL _0)))))) ==> (((vsum v) f) = ((vsum u) f))))))"}
{"_id": "908", "title": "REAL_POW2_ABS", "text": "|- (!x. (((real_pow (real_abs x)) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))))"}
{"_id": "909", "title": "SUM_LE_NUMSEG", "text": "|- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\\ (i <= n)) ==> ((real_le (f i)) (g i)))) ==> ((real_le ((sum (m .. n)) f)) ((sum (m .. n)) g)))))))"}
{"_id": "910", "title": "LOCALLY_COMPACT_INTER", "text": "|- (!s. (!t. ((((locally compact) s) /\\ ((locally compact) t)) ==> ((locally compact) (s INTER t)))))"}
{"_id": "911", "title": "MAP", "text": "|- ((!f. (((MAP f) NIL) = NIL)) /\\ (!f. (!h. (!t. (((MAP f) ((CONS h) t)) = ((CONS (f h)) ((MAP f) t)))))))"}
{"_id": "912", "title": "ORTHOGONAL_TRANSFORMATION_INVERSE", "text": "|- (!f. ((orthogonal_transformation f) ==> (?g. ((orthogonal_transformation g) /\\ ((!x. ((g (f x)) = x)) /\\ (!y. ((f (g y)) = y)))))))"}
{"_id": "913", "title": "DROP_EQ_0", "text": "|- (!x. (((drop x) = (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))"}
{"_id": "914", "title": "HOMOTOPY_EQUIVALENT_TRANSLATION_LEFT_EQ", "text": "|- (!a. (!s. (!t. (((homotopy_equivalent ((IMAGE (\\x. ((vector_add a) x))) s)) t) = ((homotopy_equivalent s) t)))))"}
{"_id": "915", "title": "Tame_defs2.sub", "text": "|- (!a. (!n. (((sub a) n) = (sub1 (a , n)))))"}
{"_id": "916", "title": "RELATIVE_FRONTIER_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((relative_frontier ((IMAGE f) s)) = ((IMAGE f) (relative_frontier s))))))"}
{"_id": "917", "title": "DIFFERENTIABLE_AT_IMP_DIFFERENTIABLE_ON", "text": "|- (!f. (!s. ((!x. ((x IN s) ==> ((differentiable f) (at x)))) ==> ((differentiable_on f) s))))"}
{"_id": "918", "title": "DIM_SUMS_INTER", "text": "|- (!s. (!t. (((subspace s) /\\ (subspace t)) ==> (((dim (GSPEC (\\GEN%PVAR%205. (?x. (?y. (((SETSPEC GEN%PVAR%205) ((x IN s) /\\ (y IN t))) ((vector_add x) y))))))) + (dim (s INTER t))) = ((dim s) + (dim t))))))"}
{"_id": "919", "title": "REALLIM_WITHINREAL", "text": "|- (!f. (!l. (!a. (!s. ((((---> f) l) ((within (atreal a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (((x IN s) /\\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\\ ((real_lt (real_abs ((real_sub x) a))) d))) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e))))))))))))"}
{"_id": "920", "title": "SPHERE_TRANSLATION", "text": "|- (!a. (!x. (!r. ((sphere (((vector_add a) x) , r)) = ((IMAGE (\\y. ((vector_add a) y))) (sphere (x , r)))))))"}
{"_id": "921", "title": "LIM_Z_TIMES_CLOG", "text": "|- (((--> (\\z. ((complex_mul z) (clog z)))) (Cx (real_of_num (NUMERAL _0)))) (at (Cx (real_of_num (NUMERAL _0)))))"}
{"_id": "922", "title": "orthogonal_transformation", "text": "|- (!f. ((orthogonal_transformation f) = ((linear f) /\\ (!v. (!w. (((dot (f v)) (f w)) = ((dot v) w)))))))"}
{"_id": "923", "title": "COMPLEX_ADD_RID", "text": "|- (!x. (((vector_add x) (Cx (real_of_num (NUMERAL _0)))) = x))"}
{"_id": "924", "title": "RPOW_MINUS1_QUOTIENT_LT", "text": "|- (!a. (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) a) /\\ ((~ (a = (real_of_num (NUMERAL (BIT1 _0))))) /\\ (((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt x) y)))) ==> ((real_lt ((real_div ((real_sub ((rpow a) x)) (real_of_num (NUMERAL (BIT1 _0))))) x)) ((real_div ((real_sub ((rpow a) y)) (real_of_num (NUMERAL (BIT1 _0))))) y))))))"}
{"_id": "925", "title": "AFF_DIM_SING", "text": "|- (!a. ((aff_dim (a INSERT EMPTY)) = (int_of_num (NUMERAL _0))))"}
{"_id": "926", "title": "INSERT_AC", "text": "|- (((x INSERT (y INSERT s)) = (y INSERT (x INSERT s))) /\\ ((x INSERT (x INSERT s)) = (x INSERT s)))"}
{"_id": "927", "title": "CONNECTED_TRANSLATION_EQ", "text": "|- (!a. (!s. ((connected ((IMAGE (\\x. ((vector_add a) x))) s)) = (connected s))))"}
{"_id": "928", "title": "INTEGRABLE_SUB", "text": "|- (!f. (!g. (!s. ((((integrable_on f) s) /\\ ((integrable_on g) s)) ==> ((integrable_on (\\x. ((vector_sub (f x)) (g x)))) s)))))"}
{"_id": "929", "title": "Tame_inequalities.INEQ_ALT", "text": "|- (!A. (!bounds. (((ineq bounds) A) = (((ALL (GABS (\\f. (!a. (!x. (!b. ((GEQ (f (a , (x , b)))) (((real_le a) x) /\\ ((real_le x) b))))))))) bounds) ==> A))))"}
{"_id": "930", "title": "ITERATE_IMAGE_NONZERO", "text": "|- (!op. ((monoidal op) ==> (!g. (!f. (!s. (((FINITE s) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((~ (x = y)) /\\ ((f x) = (f y))))) ==> ((g (f x)) = (neutral op)))))) ==> ((((iterate op) ((IMAGE f) s)) g) = (((iterate op) s) (g o f)))))))))"}
{"_id": "931", "title": "LIM_NULL_COMPLEX", "text": "|- (!net. (!f. ((((--> f) l) net) = (((--> (\\x. ((vector_sub (f x)) l))) (Cx (real_of_num (NUMERAL _0)))) net))))"}
{"_id": "932", "title": "CARD_LE_TRANS", "text": "|- (!s. (!t. (!u. (((s <=_c t) /\\ (t <=_c u)) ==> (s <=_c u)))))"}
{"_id": "933", "title": "SIMPLEX_IMP_POLYTOPE", "text": "|- (!n. (!s. (((simplex n) s) ==> (polytope s))))"}
{"_id": "934", "title": "AFF_DIM_LE_CARD", "text": "|- (!s. ((FINITE s) ==> ((int_le (aff_dim s)) ((int_sub (int_of_num (CARD s))) (int_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "935", "title": "PASTECART_IN_INTERIOR_SUBTOPOLOGY", "text": "|- (!s. (!t. (!u. (!x. (!y. (((((pastecart x) y) IN u) /\\ ((open_in ((subtopology euclidean) (s PCROSS t))) u)) ==> (?v. (?w. (((open_in ((subtopology euclidean) s)) v) /\\ ((x IN v) /\\ (((open_in ((subtopology euclidean) t)) w) /\\ ((y IN w) /\\ ((v PCROSS w) SUBSET u)))))))))))))"}
{"_id": "936", "title": "INSIDE_INSIDE", "text": "|- (!s. (!t. ((s SUBSET (inside t)) ==> (((inside s) DIFF t) SUBSET (inside t)))))"}
{"_id": "937", "title": "Lp_ineqs_proofs.y5_hi_std2", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))"}
{"_id": "938", "title": "Yxionxl2.LINAER_SYM_0", "text": "|- (linear (\\x. (vector_neg x)))"}
{"_id": "939", "title": "dot", "text": "|- (!x. (!y. (((dot x) y) = ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. ((real_mul (x $ i)) (y $ i)))))))"}
{"_id": "940", "title": "LINEAR_FSTCART", "text": "|- (linear fstcart)"}
{"_id": "941", "title": "DIST_TRIANGLE", "text": "|- (!x. (!y. (!z. ((real_le (distance (x , z))) ((real_add (distance (x , y))) (distance (y , z)))))))"}
{"_id": "942", "title": "REAL_MEASURABLE_ON_RESTRICT", "text": "|- (!f. (!s. ((((real_measurable_on f) UNIV) /\\ (real_lebesgue_measurable s)) ==> ((real_measurable_on (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) UNIV))))"}
{"_id": "943", "title": "Tame_lemmas.fully_surrounded_sol", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> (((sol (vec (NUMERAL _0))) (((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w)) = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) ((sum ((face (hypermap_of_fan (V , E))) (v , w))) (\\x. ((real_sub ((azim_dart (V , E)) x)) pi))))))))))))"}
{"_id": "944", "title": "CNJ_POW", "text": "|- (!z. (!n. ((cnj ((complex_pow z) n)) = ((complex_pow (cnj z)) n))))"}
{"_id": "945", "title": "LIM_COMPOSE_AT", "text": "|- (!net. (!f. (!g. (!y. (!z. (((((--> f) y) net) /\\ (((eventually (\\w. (((f w) = y) ==> ((g y) = z)))) net) /\\ (((--> g) z) (at y)))) ==> (((--> (g o f)) z) net)))))))"}
{"_id": "946", "title": "INTEGRABLE_ADD", "text": "|- (!f. (!g. (!s. ((((integrable_on f) s) /\\ ((integrable_on g) s)) ==> ((integrable_on (\\x. ((vector_add (f x)) (g x)))) s)))))"}
{"_id": "947", "title": "REAL_LOG_CONVEX_CONST", "text": "|- (!s. (!a. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((real_log_convex_on (\\x. a)) s))))"}
{"_id": "948", "title": "CONTINUOUS_COMPLEX_DIV_WITHIN", "text": "|- (!f. (!g. (!s. (!a. ((((continuous f) ((within (at a)) s)) /\\ (((continuous g) ((within (at a)) s)) /\\ (~ ((g a) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((continuous (\\x. ((complex_div (f x)) (g x)))) ((within (at a)) s)))))))"}
{"_id": "949", "title": "o_DEF", "text": "|- (!f. (!g. ((f o g) = (\\x. (f (g x))))))"}
{"_id": "950", "title": "REAL_HALF", "text": "|- ((!e. (((real_lt (real_of_num (NUMERAL _0))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_lt (real_of_num (NUMERAL _0))) e))) /\\ ((!e. (((real_add ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = e)) /\\ (!e. (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = e))))"}
{"_id": "951", "title": "Seq.incr_nth", "text": "|- ((((incr_nth ((CONS n) v')) (SUC i)) = ((CONS n) ((incr_nth v') i))) /\\ ((((incr_nth ((CONS n) v')) (NUMERAL _0)) = ((CONS (SUC n)) v')) /\\ (((incr_nth NIL) i) = (((ncons i) (NUMERAL _0)) ((CONS (NUMERAL (BIT1 _0))) NIL)))))"}
{"_id": "952", "title": "OUTSIDE_SUBSET_CONVEX", "text": "|- (!s. (!t. (((convex t) /\\ (s SUBSET t)) ==> ((UNIV DIFF t) SUBSET (outside s)))))"}
{"_id": "953", "title": "FINITE_POLYHEDRON_EXTREME_POINTS", "text": "|- (!s. ((polyhedron s) ==> (FINITE (GSPEC (\\GEN%PVAR%2354. (?v. (((SETSPEC GEN%PVAR%2354) ((extreme_point_of v) s)) v)))))))"}
{"_id": "954", "title": "BIJ", "text": "|- (!f. (!s. (!t. ((((BIJ f) s) t) = ((((INJ f) s) t) /\\ (((SURJ f) s) t))))))"}
{"_id": "955", "title": "FORALL_3", "text": "|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (NUMERAL (BIT1 (BIT1 _0))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\\ ((P (NUMERAL (BIT0 (BIT1 _0)))) /\\ (P (NUMERAL (BIT1 (BIT1 _0))))))))"}
{"_id": "956", "title": "SUM_POS_BOUND", "text": "|- (!f. (!b. (!s. (((FINITE s) /\\ ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) /\\ ((real_le ((sum s) f)) b))) ==> (!x. ((x IN s) ==> ((real_le (f x)) b)))))))"}
{"_id": "957", "title": "Planarity.aff_ge1_subset_aff_ge", "text": "|- (!x. (!v. (!u. (!v1. ((((DISJOINT (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY))) /\\ ((~ (collinear (x INSERT (v1 INSERT (u INSERT EMPTY))))) /\\ (v1 IN ((aff_ge (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY)))))) ==> (((aff_ge (x INSERT EMPTY)) (v1 INSERT (u INSERT EMPTY))) SUBSET ((aff_ge (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY)))))))))"}
{"_id": "958", "title": "Tame_list.lower_case", "text": "|- ((MAP = map) /\\ ((LENGTH = sizel) /\\ ((FILTER = filter) /\\ ((REVERSE = rev) /\\ ((!s. (!x0. ((~ (NIL = s)) ==> ((LAST s) = ((last x0) s))))) /\\ ((APPEND = cat) /\\ (ALL = all)))))))"}
{"_id": "959", "title": "Pack1.sum_measure_voronoi_le_ball", "text": "|- (!S. (!p. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\\ ((packing S) /\\ (saturated S))) ==> ((real_le ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\\v. (measure ((voronoi_open S) v))))) (measure (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))"}
{"_id": "960", "title": "Dih2k_hypermap.DART_OF_HYP_SY_EQ", "text": "|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\\ (((NUMERAL (BIT1 _0)) < (dimindex UNIV)) /\\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ ((j <= (dimindex UNIV)) /\\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (((darts_of_hyp (E_SY (vecmats l))) (V_SY (vecmats l))) = ((F_SY (vecmats l)) UNION ((IMAGE (nn_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (F_SY (vecmats l))))))"}
{"_id": "961", "title": "REAL_EQ_NEG2", "text": "|- (!x. (!y. (((real_neg x) = (real_neg y)) = (x = y))))"}
{"_id": "962", "title": "HAS_COMPLEX_DERIVATIVE_CACS", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative cacs) (vector_neg (complex_inv (csin (cacs z))))) (at z))))"}
{"_id": "963", "title": "int_pow_th", "text": "|- (!x. (!n. ((real_of_int ((int_pow x) n)) = ((real_pow (real_of_int x)) n))))"}
{"_id": "964", "title": "LE_ADD2", "text": "|- (!m. (!n. (!p. (!q. (((m <= p) /\\ (n <= q)) ==> ((m + n) <= (p + q)))))))"}
{"_id": "965", "title": "AFFINE_HULL_HALFSPACE_LT", "text": "|- (!a. (!b. (((hull affine) (GSPEC (\\GEN%PVAR%1543. (?x. (((SETSPEC GEN%PVAR%1543) ((real_lt ((dot a) x)) b)) x))))) = (((COND ((a = (vec (NUMERAL _0))) /\\ ((real_le b) (real_of_num (NUMERAL _0))))) EMPTY) UNIV))))"}
{"_id": "966", "title": "AFF_DIM_DIM_SUBSPACE", "text": "|- (!s. ((subspace s) ==> ((aff_dim s) = (int_of_num (dim s)))))"}
{"_id": "967", "title": "PERMUTATION_COMPOSE", "text": "|- (!p. (!q. (((permutation p) /\\ (permutation q)) ==> (permutation (p o q)))))"}
{"_id": "968", "title": "Nonlinear_lemma.dih_x_sym", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih_x x1) x2) x3) x4) x5) x6) = ((((((dih_x x1) x3) x2) x4) x6) x5))))))))"}
{"_id": "969", "title": "REAL_LE_LMUL_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le ((real_mul z) x)) ((real_mul z) y)) = ((real_le x) y))))))"}
{"_id": "970", "title": "NOT_CONS_NIL", "text": "|- (!h. (!t. (~ (((CONS h) t) = NIL))))"}
{"_id": "971", "title": "Seq.eqseq_cons", "text": "|- (!x1. (!x2. (!s1. (!s2. ((((CONS x1) s1) = ((CONS x2) s2)) = ((x1 = x2) /\\ (s1 = s2)))))))"}
{"_id": "972", "title": "REAL_INTEGRABLE_INTEGRAL", "text": "|- (!f. (!i. (((real_integrable_on f) i) ==> (((has_real_integral f) ((real_integral i) f)) i))))"}
{"_id": "973", "title": "PRE_ELIM_THM", "text": "|- ((P (PRE n)) = (!m. (((n = (SUC m)) \\/ ((m = (NUMERAL _0)) /\\ (n = (NUMERAL _0)))) ==> (P m))))"}
{"_id": "974", "title": "HAS_REAL_FRECHET_DERIVATIVE_WITHIN", "text": "|- ((((has_real_derivative f) f') ((within (atreal x)) s)) = (((has_derivative (lift o (f o drop))) (\\x. ((% f') x))) ((within (at (lift x))) ((IMAGE lift) s))))"}
{"_id": "975", "title": "INTERVAL_NE_EMPTY_1", "text": "|- ((!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) = ((real_le (drop a)) (drop b))))) /\\ (!a. (!b. ((~ ((open_interval (a , b)) = EMPTY)) = ((real_lt (drop a)) (drop b))))))"}
{"_id": "976", "title": "VSUM_CLAUSES_LEFT", "text": "|- (!f. (!m. (!n. ((m <= n) ==> (((vsum (m .. n)) f) = ((vector_add (f m)) ((vsum ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))"}
{"_id": "977", "title": "REAL_INTERVAL_NE_EMPTY", "text": "|- ((!a. (!b. ((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) = ((real_le a) b)))) /\\ (!a. (!b. ((~ ((open_real_interval (a , b)) = EMPTY)) = ((real_lt a) b)))))"}
{"_id": "978", "title": "Seq2.mem_flatten", "text": "|- (!x. (!L. (((MEM x) (flatten L)) = (?l. (((MEM l) L) /\\ ((MEM x) l))))))"}
{"_id": "979", "title": "VSUM_COMPONENT", "text": "|- (!s. (!f. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((((vsum s) f) $ i) = ((sum s) (\\x. ((f x) $ i))))))))"}
{"_id": "980", "title": "REAL_NOT_LT", "text": "|- (!x. (!y. ((~ ((real_lt x) y)) = ((real_le y) x))))"}
{"_id": "981", "title": "CONVEX_HULL_EMPTY", "text": "|- (((hull convex) EMPTY) = EMPTY)"}
{"_id": "982", "title": "MATRIX_VECTOR_MUL_ADD_RDISTRIB", "text": "|- (!A. (!B. (!x. (((matrix_vector_mul ((matrix_add A) B)) x) = ((vector_add ((matrix_vector_mul A) x)) ((matrix_vector_mul B) x))))))"}
{"_id": "983", "title": "OPEN_INTERVAL", "text": "|- (!a. (!b. (open (open_interval (a , b)))))"}
{"_id": "984", "title": "Collect_geom2.NOT_COPLANAR_IMP_RADV_PROPERTIES", "text": "|- ((~ (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) ==> (!w. (((v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))) w) ==> ((radV (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = (distance ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) , w))))))"}
{"_id": "985", "title": "BASIS_INJ_EQ", "text": "|- (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ (j <= (dimindex UNIV))))) ==> (((basis i) = (basis j)) = (i = j)))))"}
{"_id": "986", "title": "SERIES_SUB", "text": "|- (!x. (!x0. (!y. (!y0. (!s. (((((sums x) x0) s) /\\ (((sums y) y0) s)) ==> (((sums (\\n. ((vector_sub (x n)) (y n)))) ((vector_sub x0) y0)) s)))))))"}
{"_id": "987", "title": "MULT_SYM", "text": "|- (!m. (!n. ((m * n) = (n * m))))"}
{"_id": "988", "title": "REAL_CONTINUOUS_ON_UNION", "text": "|- (!f. (!s. (!t. (((real_closed s) /\\ ((real_closed t) /\\ (((real_continuous_on f) s) /\\ ((real_continuous_on f) t)))) ==> ((real_continuous_on f) (s UNION t))))))"}
{"_id": "989", "title": "Fan.sum4_azim_fan", "text": "|- (!x. (!v. (!u. (!w1. (!w2. ((((real_le ((((azim x) v) u) w1)) ((((azim x) v) u) w2)) /\\ ((~ (collinear (x INSERT (v INSERT (w1 INSERT EMPTY))))) /\\ ((~ (collinear (x INSERT (v INSERT (w2 INSERT EMPTY))))) /\\ (~ (collinear (x INSERT (v INSERT (u INSERT EMPTY)))))))) ==> (((((azim x) v) u) w2) = ((real_add ((((azim x) v) u) w1)) ((((azim x) v) w1) w2)))))))))"}
{"_id": "990", "title": "HAS_REAL_MEASURE_MEASURE", "text": "|- (!s. ((real_measurable s) = ((has_real_measure s) (real_measure s))))"}
{"_id": "991", "title": "real_polynomial_function_RULES", "text": "|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (real_polynomial_function (\\x. (x $ i))))) /\\ ((!c. (real_polynomial_function (\\x. c))) /\\ ((!f. (!g. (((real_polynomial_function f) /\\ (real_polynomial_function g)) ==> (real_polynomial_function (\\x. ((real_add (f x)) (g x))))))) /\\ (!f. (!g. (((real_polynomial_function f) /\\ (real_polynomial_function g)) ==> (real_polynomial_function (\\x. ((real_mul (f x)) (g x))))))))))"}
{"_id": "992", "title": "LE_EXISTS", "text": "|- (!m. (!n. ((m <= n) = (?d. (n = (m + d))))))"}
{"_id": "993", "title": "SETDIST_LE_SING", "text": "|- (!s. (!t. (!x. ((x IN s) ==> ((real_le (setdist (s , t))) (setdist ((x INSERT EMPTY) , t)))))))"}
{"_id": "994", "title": "HOMOTOPIC_INVERTIBLE_LINEAR_MAPS_ALT", "text": "|- (!f. (!g. (((((homotopic_with (\\h. ((linear h) /\\ (invertible (matrix h))))) (UNIV , UNIV)) f) g) = ((linear f) /\\ ((linear g) /\\ ((real_lt (real_of_num (NUMERAL _0))) ((real_mul (real_sgn (det (matrix f)))) (real_sgn (det (matrix g))))))))))"}
{"_id": "995", "title": "LEBESGUE_MEASURABLE_COMPL", "text": "|- (!s. ((lebesgue_measurable (UNIV DIFF s)) = (lebesgue_measurable s)))"}
{"_id": "996", "title": "ENR_IMP_FSGIMA", "text": "|- (!s. ((ENR s) ==> (fsigma s)))"}
{"_id": "997", "title": "Seq.nth0", "text": "|- (!x0. (!s. ((((nth x0) s) (NUMERAL _0)) = ((headl x0) s))))"}
{"_id": "998", "title": "Ssrfun.injective", "text": "|- (!f. ((injective f) = (!x1. (!x2. (((f x1) = (f x2)) ==> (x1 = x2))))))"}
{"_id": "999", "title": "Ssrbool.negbK", "text": "|- (!b. ((~ (~ b)) = b))"}
{"_id": "1000", "title": "HOMOTOPIC_WITH_TRANS", "text": "|- (!P. (!X. (!Y. (!f. (!g. (!h. ((((((homotopic_with P) (X , Y)) f) g) /\\ ((((homotopic_with P) (X , Y)) g) h)) ==> ((((homotopic_with P) (X , Y)) f) h))))))))"}
{"_id": "1001", "title": "Ssrbool.addbb", "text": "|- (!b. (((XOR b) b) = F))"}
{"_id": "1002", "title": "CONNECTED_OPEN_PATH_CONNECTED", "text": "|- (!s. (((open s) /\\ (connected s)) ==> (path_connected s)))"}
{"_id": "1003", "title": "ANR_COVERING_SPACE_EQ", "text": "|- (!p. (!s. (!c. (((covering_space (c , p)) s) ==> ((ANR s) = (ANR c))))))"}
{"_id": "1004", "title": "NOT_EXISTS_THM", "text": "|- (!P. ((~ (?x. (P x))) = (!x. (~ (P x)))))"}
{"_id": "1005", "title": "REAL_LE_SUP", "text": "|- (!s. (!a. (!b. (!y. (((y IN s) /\\ (((real_le a) y) /\\ (!x. ((x IN s) ==> ((real_le x) b))))) ==> ((real_le a) (sup s)))))))"}
{"_id": "1006", "title": "REALLIM_TRANSFORM_EVENTUALLY", "text": "|- (!net. (!f. (!g. (!l. ((((eventually (\\x. ((f x) = (g x)))) net) /\\ (((---> f) l) net)) ==> (((---> g) l) net))))))"}
{"_id": "1007", "title": "EXISTS_PATH_SUBPATH_TO_FRONTIER", "text": "|- (!g. (!s. (((path g) /\\ (((pathstart g) IN s) /\\ (~ ((pathfinish g) IN s)))) ==> (?h. ((path h) /\\ (((pathstart h) = (pathstart g)) /\\ (((path_image h) SUBSET (path_image g)) /\\ ((((path_image h) DELETE (pathfinish h)) SUBSET (interior s)) /\\ ((pathfinish h) IN (frontier s))))))))))"}
{"_id": "1008", "title": "AZIM_EXISTS", "text": "|- (!v. (!w. (!w1. (!w2. (?theta. (((real_le (real_of_num (NUMERAL _0))) theta) /\\ (((real_lt theta) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\\ (?h1. (?h2. (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\\ ((((% (distance (w , v))) e3) = ((vector_sub w) v)) /\\ (~ (w = v)))) ==> (?psi. (?r1. (?r2. ((((vector_sub w1) v) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub w) v))))) /\\ ((((vector_sub w2) v) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) theta)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) theta)))) e2)) ((% h2) ((vector_sub w) v))))) /\\ (((~ (collinear (v INSERT (w INSERT (w1 INSERT EMPTY))))) ==> ((real_lt (real_of_num (NUMERAL _0))) r1)) /\\ ((~ (collinear (v INSERT (w INSERT (w2 INSERT EMPTY))))) ==> ((real_lt (real_of_num (NUMERAL _0))) r2)))))))))))))))))))))"}
{"_id": "1009", "title": "ITERATE_EQ_GENERAL_INVERSES", "text": "|- (!op. ((monoidal op) ==> (!s. (!t. (!f. (!g. (!h. (!k. (((!y. ((y IN t) ==> (((k y) IN s) /\\ ((h (k y)) = y)))) /\\ (!x. ((x IN s) ==> (((h x) IN t) /\\ (((k (h x)) = x) /\\ ((g (h x)) = (f x))))))) ==> ((((iterate op) s) f) = (((iterate op) t) g)))))))))))"}
{"_id": "1010", "title": "List_hypermap.fst_n_list_ext_power", "text": "|- (!x. (!y. (!L. (!n. (((good_list L) /\\ ((x , y) IN (darts_of_list L))) ==> ((FST (((POWER (n_list_ext L)) n) (x , y))) = x))))))"}
{"_id": "1011", "title": "CROSS_RMUL", "text": "|- (!c. (!x. (!y. (((cross x) ((% c) y)) = ((% c) ((cross x) y))))))"}
{"_id": "1012", "title": "Ssrnat.iter", "text": "|- (((((iter (SUC n)) f) x) = (f (((iter n) f) x))) /\\ ((((iter (NUMERAL _0)) f) x) = x))"}
{"_id": "1013", "title": "LIFT_IN_IMAGE_LIFT", "text": "|- (!x. (!s. (((lift x) IN ((IMAGE lift) s)) = (x IN s))))"}
{"_id": "1014", "title": "Ssrnat.half0", "text": "|- ((half (NUMERAL _0)) = (NUMERAL _0))"}
{"_id": "1015", "title": "Trigonometry.CFXEKKP1", "text": "|- ((cos ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0)))"}
{"_id": "1016", "title": "Hypermap_iso.ext_dart_eq_inv_contracted_dart", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((ext_dart (V , E)) d) = (((res_inv contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E)))) d)))))))"}
{"_id": "1017", "title": "CONTINUOUS_ON_UPPERHALF_ARG", "text": "|- ((continuous_on (Cx o Arg)) ((GSPEC (\\GEN%PVAR%4840. (?z. (((SETSPEC GEN%PVAR%4840) ((real_le (real_of_num (NUMERAL _0))) (Im z))) z)))) DIFF ((Cx (real_of_num (NUMERAL _0))) INSERT EMPTY)))"}
{"_id": "1018", "title": "Packing3.BARV_INITIAL_SUBLIST", "text": "|- (!V. (!k. (!ul. (!vl. (((((barV V) k) ul) /\\ (((initial_sublist vl) ul) /\\ ((NUMERAL _0) < (LENGTH vl)))) ==> (((barV V) ((LENGTH vl) - (NUMERAL (BIT1 _0)))) vl))))))"}
{"_id": "1019", "title": "INT_OF_NUM_LE", "text": "|- (!m. (!n. (((int_le (int_of_num m)) (int_of_num n)) = (m <= n))))"}
{"_id": "1020", "title": "REAL_LEBESGUE_MEASURABLE_COUNTABLE_INTERS", "text": "|- (!f. (((COUNTABLE f) /\\ (!s. ((s IN f) ==> (real_lebesgue_measurable s)))) ==> (real_lebesgue_measurable (INTERS f))))"}
{"_id": "1021", "title": "Lp_ineqs_proofs.ys_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> (((y1_fan (v , w)) = (vector_norm v)) /\\ (((y2_fan (v , w)) = (vector_norm w)) /\\ ((((y3_fan (V , E)) (v , w)) = (vector_norm (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))) /\\ ((((y4'_fan (V , E)) (v , w)) = (distance (w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))) /\\ ((((y5_fan (V , E)) (v , w)) = (distance (v , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))) /\\ ((y6_fan (v , w)) = (distance (v , w)))))))))))))))))"}
{"_id": "1022", "title": "Trigonometry2.POW2_1", "text": "|- (((real_pow (real_of_num (NUMERAL (BIT1 _0)))) (NUMERAL (BIT0 (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "1023", "title": "eventually", "text": "|- (!net. (!p. (((eventually p) net) = ((trivial_limit net) \\/ (?y. ((?x. (((netord net) x) y)) /\\ (!x. ((((netord net) x) y) ==> (p x)))))))))"}
{"_id": "1024", "title": "VECTOR_MUL_LZERO", "text": "|- (!x. (((% (real_of_num (NUMERAL _0))) x) = (vec (NUMERAL _0))))"}
{"_id": "1025", "title": "Seq.nth", "text": "|- (!x0. (!h. (!t. (!n. (((((nth x0) NIL) n) = x0) /\\ (((((nth x0) ((CONS h) t)) (NUMERAL _0)) = h) /\\ ((((nth x0) ((CONS h) t)) (SUC n)) = (((nth x0) t) n))))))))"}
{"_id": "1026", "title": "Sphere.rad2_y", "text": "|- (rad2_y = (y_of_x rad2_x))"}
{"_id": "1027", "title": "Hypermap_and_fan.COMPONENTS_HYPERMAP_OF_FAN", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((dart (hypermap_of_fan (V , E))) = (dart_of_fan (V , E))) /\\ (((edge_map (hypermap_of_fan (V , E))) = (e_fan_pair_ext (V , E))) /\\ (((node_map (hypermap_of_fan (V , E))) = (n_fan_pair_ext (V , E))) /\\ ((face_map (hypermap_of_fan (V , E))) = (f_fan_pair_ext (V , E)))))))))"}
{"_id": "1028", "title": "EVEN", "text": "|- (((EVEN (NUMERAL _0)) = T) /\\ (!n. ((EVEN (SUC n)) = (~ (EVEN n)))))"}
{"_id": "1029", "title": "POSITIVE_SEMIDEFINITE_DIAGONAL_INEQUALITY", "text": "|- (!A. (!i. (!j. (((positive_semidefinite A) /\\ (((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ (j <= (dimindex UNIV)))))) ==> ((real_le ((real_pow ((A $ i) $ j)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ i) $ i)) ((A $ j) $ j)))))))"}
{"_id": "1030", "title": "int_of_num_th", "text": "|- (!n. ((real_of_int (int_of_num n)) = (real_of_num n)))"}
{"_id": "1031", "title": "MATRIX_INV_MUL_INNER", "text": "|- (!A. (((matrix_mul A) ((matrix_mul (matrix_inv A)) A)) = A))"}
{"_id": "1032", "title": "CSQRT_UNIQUE", "text": "|- (!s. (!z. (((((complex_pow s) (NUMERAL (BIT0 (BIT1 _0)))) = z) /\\ (((real_lt (real_of_num (NUMERAL _0))) (Re s)) \\/ (((Re s) = (real_of_num (NUMERAL _0))) /\\ ((real_le (real_of_num (NUMERAL _0))) (Im s))))) ==> ((csqrt z) = s))))"}
{"_id": "1033", "title": "REAL_ADD", "text": "|- (!w. (!z. (((real w) /\\ (real z)) ==> (real ((vector_add w) z)))))"}
{"_id": "1034", "title": "CARD_INTSEG_INT", "text": "|- (!a. (!b. (((integer a) /\\ (integer b)) ==> ((CARD (GSPEC (\\GEN%PVAR%85. (?x. (((SETSPEC GEN%PVAR%85) ((integer x) /\\ (((real_le a) x) /\\ ((real_le x) b)))) x))))) = (((COND ((real_lt b) a)) (NUMERAL _0)) (num_of_int (int_of_real ((real_add ((real_sub b) a)) (real_of_num (NUMERAL (BIT1 _0)))))))))))"}
{"_id": "1035", "title": "INCREASING_BOUNDED_VARIATION", "text": "|- (!f. (!a. (!b. ((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f x))) (drop (f y)))))) ==> ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "1036", "title": "real_ge", "text": "|- (!y. (!x. (((real_ge x) y) = ((real_le y) x))))"}
{"_id": "1037", "title": "Hypermap.orbit_one_point", "text": "|- (!f. (!x. (((f x) = x) = (((orbit_map f) x) = (x INSERT EMPTY)))))"}
{"_id": "1038", "title": "num_INDUCTION", "text": "|- (!P. (((P (NUMERAL _0)) /\\ (!n. ((P n) ==> (P (SUC n))))) ==> (!n. (P n))))"}
{"_id": "1039", "title": "RPOW_SQRT", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((rpow x) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (sqrt x))))"}
{"_id": "1040", "title": "COUNTABLE_INSERT", "text": "|- (!x. (!s. ((COUNTABLE (x INSERT s)) = (COUNTABLE s))))"}
{"_id": "1041", "title": "REAL_LE_LT", "text": "|- (!x. (!y. (((real_le x) y) = (((real_lt x) y) \\/ (x = y)))))"}
{"_id": "1042", "title": "REAL_LT_BETWEEN", "text": "|- (!a. (!b. (((real_lt a) b) = (?x. (((real_lt a) x) /\\ ((real_lt x) b))))))"}
{"_id": "1043", "title": "REAL_ABS_SUB", "text": "|- (!x. (!y. ((real_abs ((real_sub x) y)) = (real_abs ((real_sub y) x)))))"}
{"_id": "1044", "title": "Tame_defs2.sub1", "text": "|- (!n. (!xs. ((sub1 (xs , n)) = ((EL n) xs))))"}
{"_id": "1045", "title": "CONTINUOUS_COMPLEX_POW", "text": "|- (!net. (!f. (!n. (((continuous f) net) ==> ((continuous (\\x. ((complex_pow (f x)) n))) net)))))"}
{"_id": "1046", "title": "EXTENSION", "text": "|- (!s. (!t. ((s = t) = (!x. ((x IN s) = (x IN t))))))"}
{"_id": "1047", "title": "holomorphic_on", "text": "|- (!f. (!s. (((holomorphic_on f) s) = (!x. ((x IN s) ==> (?f'. (((has_complex_derivative f) f') ((within (at x)) s))))))))"}
{"_id": "1048", "title": "Planarity.DARTSET_LEADS_INTO_FAN", "text": "|- (!x. (!V. (!E. (!ds. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((fan80 (x , (V , E))) /\\ (ds IN (face_set (hypermap1_of_fanx (x , (V , E)))))))) ==> (!y. ((y IN ds) ==> (((((dartset_leads_into_fan x) V) E) ds) = (((((dart_leads_into x) V) E) (pr2 y)) (pr3 y))))))))))"}
{"_id": "1049", "title": "Bump.MCELL_PARAM_UL", "text": "|- (!V. (!ul. (!vl. (!X. (!k. (((k <= (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\\ ((packing V) /\\ ((saturated V) /\\ ((X = (((mcell k) V) ul)) /\\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((~ (negligible X)) /\\ (vl = (SND ((cell_params V) X))))))))) ==> ((X = (((mcell k) V) vl)) /\\ (vl IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))))))))))"}
{"_id": "1050", "title": "Lp_gen_theory.mem_list_dart_pairs", "text": "|- (!L. (!p. (((MEM p) (list_dart_pairs L)) = (((MEM (FST p)) (list_of_darts L)) /\\ ((MEM (SND p)) (list_of_darts L))))))"}
{"_id": "1051", "title": "COMPLEX_POW_MUL", "text": "|- (!x. (!y. (!n. (((complex_pow ((complex_mul x) y)) n) = ((complex_mul ((complex_pow x) n)) ((complex_pow y) n))))))"}
{"_id": "1052", "title": "VSUM_EQ", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((vsum s) f) = ((vsum s) g))))))"}
{"_id": "1053", "title": "CLOSED_HALFSPACE_COMPONENT_GE", "text": "|- (!a. (!k. (closed (GSPEC (\\GEN%PVAR%775. (?x. (((SETSPEC GEN%PVAR%775) ((real_ge (x $ k)) a)) x)))))))"}
{"_id": "1054", "title": "CLOSED_IN_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((closed_in ((subtopology euclidean) ((IMAGE f) t))) ((IMAGE f) s)) = ((closed_in ((subtopology euclidean) t)) s))))))"}
{"_id": "1055", "title": "COLLINEAR_LEMMA_ALT", "text": "|- (!x. (!y. ((collinear ((vec (NUMERAL _0)) INSERT (x INSERT (y INSERT EMPTY)))) = ((x = (vec (NUMERAL _0))) \\/ (?c. (y = ((% c) x)))))))"}
{"_id": "1056", "title": "CLOSED_IN_SING", "text": "|- (!u. (!x. (((closed_in ((subtopology euclidean) u)) (x INSERT EMPTY)) = (x IN u))))"}
{"_id": "1057", "title": "Lp_ineqs_def.rhazim_fan", "text": "|- (!V. (!E. (!x. (((rhazim_fan (V , E)) x) = ((real_mul (real_abs ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (lmfun (h_dart x))))))) ((azim_dart (V , E)) x))))))"}
{"_id": "1058", "title": "NPRODUCT_CONST", "text": "|- (!c. (!s. ((FINITE s) ==> (((nproduct s) (\\x. c)) = (c EXP (CARD s))))))"}
{"_id": "1059", "title": "Seq.constant", "text": "|- (((constant NIL) = T) /\\ ((constant ((CONS x) s')) = ((all (pred1 x)) s')))"}
{"_id": "1060", "title": "LIM_TRANSFORM_WITHIN_SET_IMP", "text": "|- (!f. (!l. (!a. (!s. (!t. ((((eventually (\\x. ((x IN t) ==> (x IN s)))) (at a)) /\\ (((--> f) l) ((within (at a)) s))) ==> (((--> f) l) ((within (at a)) t))))))))"}
{"_id": "1061", "title": "RSTC_TRANS", "text": "|- (!R. (!x. (!y. (!z. (((((RSTC R) x) y) /\\ (((RSTC R) y) z)) ==> (((RSTC R) x) z))))))"}
{"_id": "1062", "title": "CONIC_HULL_AS_IMAGE", "text": "|- (!s. (((hull conic) s) = ((IMAGE (\\z. ((% (drop (fstcart z))) (sndcart z)))) ((GSPEC (\\GEN%PVAR%1458. (?t. (((SETSPEC GEN%PVAR%1458) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t)))) PCROSS s))))"}
{"_id": "1063", "title": "MTOPOLOGY_REAL_EUCLIDEAN_METRIC", "text": "|- ((mtopology real_euclidean_metric) = euclideanreal)"}
{"_id": "1064", "title": "COMPONENTS_EQ_EMPTY", "text": "|- (!s. (((components s) = EMPTY) = (s = EMPTY)))"}
{"_id": "1065", "title": "LOCALLY_COMPACT_TRANSLATION_EQ", "text": "|- (!a. (!s. (((locally compact) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((locally compact) s))))"}
{"_id": "1066", "title": "REAL_LT_INV_EQ", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) (real_inv x)) = ((real_lt (real_of_num (NUMERAL _0))) x)))"}
{"_id": "1067", "title": "DOT_POS_LE", "text": "|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((dot x) x)))"}
{"_id": "1068", "title": "AFFSIGN_ALT", "text": "|- ((((affsign sgn) s) t) = (GSPEC (\\GEN%PVAR%5090. (?y. (((SETSPEC GEN%PVAR%5090) (?f. ((!w. ((w IN (s UNION t)) ==> ((w IN t) ==> (sgn (f w))))) /\\ ((((sum (s UNION t)) f) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((vsum (s UNION t)) (\\v. ((% (f v)) v))) = y))))) y)))))"}
{"_id": "1069", "title": "POLYTOPE_LINEAR_IMAGE", "text": "|- (!f. (!p. (((linear f) /\\ (polytope p)) ==> (polytope ((IMAGE f) p)))))"}
{"_id": "1070", "title": "HAS_REAL_MEASURE_0", "text": "|- (!s. (((has_real_measure s) (real_of_num (NUMERAL _0))) = (real_negligible s)))"}
{"_id": "1071", "title": "List_hypermap.face_set_k", "text": "|- (!H. (!k. (((face_set_k k) H) = (GSPEC (\\GEN%PVAR%8596. (?f. (((SETSPEC GEN%PVAR%8596) ((f IN (face_set H)) /\\ ((CARD f) = k))) f)))))))"}
{"_id": "1072", "title": "LIM_NORM_LBOUND", "text": "|- (!net. (!f. (!l. (!b. (((~ (trivial_limit net)) /\\ ((((--> f) l) net) /\\ ((eventually (\\x. ((real_le b) (vector_norm (f x))))) net))) ==> ((real_le b) (vector_norm l)))))))"}
{"_id": "1073", "title": "REAL_SUMMABLE_FROM_ELSEWHERE", "text": "|- (!f. (!m. (!n. (((real_summable (from m)) f) ==> ((real_summable (from n)) f)))))"}
{"_id": "1074", "title": "RE_CACS_BOUNDS", "text": "|- (!z. (((real_lt (real_neg pi)) (Re (cacs z))) /\\ ((real_le (Re (cacs z))) pi)))"}
{"_id": "1075", "title": "SPHERE_EQ_EMPTY", "text": "|- (!a. (!r. (((sphere (a , r)) = EMPTY) = ((real_lt r) (real_of_num (NUMERAL _0))))))"}
{"_id": "1076", "title": "HAS_REAL_INTEGRAL_NEGLIGIBLE", "text": "|- (!f. (!s. (!t. (((real_negligible s) /\\ (!x. ((x IN (t DIFF s)) ==> ((f x) = (real_of_num (NUMERAL _0)))))) ==> (((has_real_integral f) (real_of_num (NUMERAL _0))) t)))))"}
{"_id": "1077", "title": "sum", "text": "|- (sum = (iterate real_add))"}
{"_id": "1078", "title": "MEASURE_COUNTABLE_UNIONS_LE", "text": "|- (!d. (!B. (((!n. (measurable (d n))) /\\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\\k. (measure (d k))))) B))) ==> ((measurable (UNIONS (GSPEC (\\GEN%PVAR%3383. (?n. (((SETSPEC GEN%PVAR%3383) (n IN UNIV)) (d n))))))) /\\ ((real_le (measure (UNIONS (GSPEC (\\GEN%PVAR%3384. (?n. (((SETSPEC GEN%PVAR%3384) (n IN UNIV)) (d n)))))))) B)))))"}
{"_id": "1079", "title": "IN_CBALL", "text": "|- (!x. (!y. (!e. ((y IN (cball (x , e))) = ((real_le (distance (x , y))) e)))))"}
{"_id": "1080", "title": "Seq.cat0s", "text": "|- (!s. (((cat NIL) s) = s))"}
{"_id": "1081", "title": "Tame_defs.tame_8", "text": "|- (!H. ((tame_8 H) = ((number_of_faces H) >= (NUMERAL (BIT1 (BIT1 _0))))))"}
{"_id": "1082", "title": "MEASURABLE_LEBESGUE_MEASURABLE_SUBSET", "text": "|- (!s. (!t. (((lebesgue_measurable s) /\\ ((measurable t) /\\ (s SUBSET t))) ==> (measurable s))))"}
{"_id": "1083", "title": "Hypermap_iso.finite_surj_eq_bij", "text": "|- (!s. (!t. ((FINITE s) ==> ((FINITE t) ==> (((CARD s) = (CARD t)) ==> (!f. ((((SURJ f) s) t) = (((BIJ f) s) t))))))))"}
{"_id": "1084", "title": "INTEGRABLE_COMPLEX_LMUL_EQ", "text": "|- (!f. (!s. (!c. (((integrable_on (\\x. ((complex_mul c) (f x)))) s) = ((c = (Cx (real_of_num (NUMERAL _0)))) \\/ ((integrable_on f) s))))))"}
{"_id": "1085", "title": "Fan.remark1_fan", "text": "|- (!x. (!V. (!E. (!u. (!v. ((FAN (x , (V , E))) ==> ((FINITE (((set_of_edge v) V) E)) /\\ ((((v INSERT (u INSERT EMPTY)) IN E) = (u IN (((set_of_edge v) V) E))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) ==> ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) /\\ ((~ (x = v)) /\\ ((~ (x = u)) /\\ ((~ (v = u)) /\\ (((DISJOINT (x INSERT (v INSERT EMPTY))) (u INSERT EMPTY)) /\\ (((DISJOINT (x INSERT (u INSERT EMPTY))) (v INSERT EMPTY)) /\\ (((DISJOINT (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY))) /\\ ((~ (u IN (aff (x INSERT (v INSERT EMPTY))))) /\\ ((v IN V) /\\ ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) = (~ (u IN (aff (x INSERT (v INSERT EMPTY))))))))))))))))))))))))"}
{"_id": "1086", "title": "LOCALLY_PATH_CONNECTED_SPHERE_GEN", "text": "|- (!s. (((bounded s) /\\ (convex s)) ==> ((locally path_connected) (relative_frontier s))))"}
{"_id": "1087", "title": "INFSUM_UNIQUE", "text": "|- (!f. (!l. (!s. ((((sums f) l) s) ==> (((infsum s) f) = l)))))"}
{"_id": "1088", "title": "closure", "text": "|- (!s. ((closure s) = (s UNION (GSPEC (\\GEN%PVAR%568. (?x. (((SETSPEC GEN%PVAR%568) ((limit_point_of x) s)) x)))))))"}
{"_id": "1089", "title": "CONNECTED_COMPONENT_MAXIMAL", "text": "|- (!s. (!t. (!x. (((x IN t) /\\ ((connected t) /\\ (t SUBSET s))) ==> (t SUBSET ((connected_component s) x))))))"}
{"_id": "1090", "title": "REAL_DIFFERENTIABLE_COMPOSE_WITHIN", "text": "|- (!f. (!g. (!x. (!s. ((((real_differentiable f) ((within (atreal x)) s)) /\\ ((real_differentiable g) ((within (atreal (f x))) ((IMAGE f) s)))) ==> ((real_differentiable (g o f)) ((within (atreal x)) s)))))))"}
{"_id": "1091", "title": "VECTOR_VARIATION_CONTINUOUS", "text": "|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (a , b)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (a , b)) NIL))))))))))"}
{"_id": "1092", "title": "Oxl_2012.CARD_EMPTY", "text": "|- ((CARD EMPTY) = (NUMERAL _0))"}
{"_id": "1093", "title": "Tame_lemmas.reflection", "text": "|- (!x. (!n. (((reflection n) x) = ((vector_sub x) ((% ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div ((dot x) n)) ((dot n) n)))) n)))))"}
{"_id": "1094", "title": "Tame_defs.rotate_to", "text": "|- (!v. (!vs. (((rotate_to vs) v) = ((CONS v) ((APPEND (SND ((split_at v) vs))) (FST ((split_at v) vs)))))))"}
{"_id": "1095", "title": "ORTHOGONAL_MATRIX_TRANSP", "text": "|- (!A. ((orthogonal_matrix (transp A)) = (orthogonal_matrix A)))"}
{"_id": "1096", "title": "POLYHEDRON_SUMS", "text": "|- (!s. (!t. (((polyhedron s) /\\ (polyhedron t)) ==> (polyhedron (GSPEC (\\GEN%PVAR%2421. (?x. (?y. (((SETSPEC GEN%PVAR%2421) ((x IN s) /\\ (y IN t))) ((vector_add x) y))))))))))"}
{"_id": "1097", "title": "int_mul_th", "text": "|- (!x. (!y. ((real_of_int ((int_mul x) y)) = ((real_mul (real_of_int x)) (real_of_int y)))))"}
{"_id": "1098", "title": "CONNECTED_COMPONENT_EQ_SELF", "text": "|- (!s. (!x. (((connected s) /\\ (x IN s)) ==> (((connected_component s) x) = s))))"}
{"_id": "1099", "title": "support", "text": "|- (!s. (!f. (!op. ((((support op) f) s) = (GSPEC (\\GEN%PVAR%158. (?x. (((SETSPEC GEN%PVAR%158) ((x IN s) /\\ (~ ((f x) = (neutral op))))) x))))))))"}
{"_id": "1100", "title": "Functional_equation.proj_x6", "text": "|- (((((((proj_x6 x1) x2) x3) x4) x5) x6) = x6)"}
{"_id": "1101", "title": "DIAMETER_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((diameter ((IMAGE f) s)) = (diameter s)))))"}
{"_id": "1102", "title": "Ssrnat.eqn_addr", "text": "|- (!p. (!m. (!n. (((m + p) = (n + p)) = (m = n)))))"}
{"_id": "1103", "title": "LENGTH_EQ_NIL", "text": "|- (!l. (((LENGTH l) = (NUMERAL _0)) = (l = NIL)))"}
{"_id": "1104", "title": "INTERVAL_NE_EMPTY", "text": "|- (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (a $ i)) (b $ i))))) /\\ ((~ ((open_interval (a , b)) = EMPTY)) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i))))))"}
{"_id": "1105", "title": "division_of", "text": "|- (!s. (!i. (((division_of s) i) = ((FINITE s) /\\ ((!k. ((k IN s) ==> ((k SUBSET i) /\\ ((~ (k = EMPTY)) /\\ (?a. (?b. (k = (closed_interval ((CONS (a , b)) NIL))))))))) /\\ ((!k1. (!k2. (((k1 IN s) /\\ ((k2 IN s) /\\ (~ (k1 = k2)))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))) /\\ ((UNIONS s) = i)))))))"}
{"_id": "1106", "title": "IN_CROSS", "text": "|- (!x. (!y. (!s. (!t. (((x , y) IN (s CROSS t)) = ((x IN s) /\\ (y IN t)))))))"}
{"_id": "1107", "title": "Real_ext.REAL_PROP_ZERO_POW", "text": "|- (!x. (!n. ((((real_pow x) n) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) /\\ (~ (n = (NUMERAL _0)))))))"}
{"_id": "1108", "title": "VECTOR_VARIATION_MONOTONE", "text": "|- (!f. (!s. (!t. ((((has_bounded_variation_on f) s) /\\ (t SUBSET s)) ==> ((real_le ((vector_variation t) f)) ((vector_variation s) f))))))"}
{"_id": "1109", "title": "Hypermap.edge_set", "text": "|- (!H. ((edge_set H) = ((set_of_orbits (dart H)) (edge_map H))))"}
{"_id": "1110", "title": "ADD1", "text": "|- (!m. ((SUC m) = (m + (NUMERAL (BIT1 _0)))))"}
{"_id": "1111", "title": "VECTOR_MUL_EQ_0", "text": "|- (!a. (!x. ((((% a) x) = (vec (NUMERAL _0))) = ((a = (real_of_num (NUMERAL _0))) \\/ (x = (vec (NUMERAL _0)))))))"}
{"_id": "1112", "title": "nproduct", "text": "|- (nproduct = (iterate *))"}
{"_id": "1113", "title": "Lp_ineqs_proofs.y2_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (y2_fan d))))))))))"}
{"_id": "1114", "title": "CARD_EXP_0", "text": "|- (!s. (!c. (((^_c s) EMPTY) =_c (c INSERT EMPTY))))"}
{"_id": "1115", "title": "SUM_EQ_0_NUMSEG", "text": "|- (!f. (!m. (!n. ((!i. (((m <= i) /\\ (i <= n)) ==> ((f i) = (real_of_num (NUMERAL _0))))) ==> (((sum (m .. n)) f) = (real_of_num (NUMERAL _0)))))))"}
{"_id": "1116", "title": "REAL_ABS_POS", "text": "|- (!x. ((real_le (real_of_num (NUMERAL _0))) (real_abs x)))"}
{"_id": "1117", "title": "INSIDE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> ((inside ((IMAGE f) s)) = ((IMAGE f) (inside s))))))"}
{"_id": "1118", "title": "VSUM_CASES", "text": "|- (!s. (!P. (!f. (!g. ((FINITE s) ==> (((vsum s) (\\x. (((COND (P x)) (f x)) (g x)))) = ((vector_add ((vsum (GSPEC (\\GEN%PVAR%112. (?x. (((SETSPEC GEN%PVAR%112) ((x IN s) /\\ (P x))) x))))) f)) ((vsum (GSPEC (\\GEN%PVAR%113. (?x. (((SETSPEC GEN%PVAR%113) ((x IN s) /\\ (~ (P x)))) x))))) g))))))))"}
{"_id": "1119", "title": "REAL_CONTINUOUS_ON_EQ", "text": "|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((real_continuous_on f) s)) ==> ((real_continuous_on g) s)))))"}
{"_id": "1120", "title": "CLOSED_PATH_IMAGE", "text": "|- (!g. ((path g) ==> (closed (path_image g))))"}
{"_id": "1121", "title": "STARLIKE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((starlike ((IMAGE f) s)) = (starlike s)))))"}
{"_id": "1122", "title": "HOMOTOPIC_ON_COMPONENTS", "text": "|- (!s. (!t. (!f. (!g. ((((locally connected) s) /\\ (!c. ((c IN (components s)) ==> ((((homotopic_with (\\x. T)) (c , t)) f) g)))) ==> ((((homotopic_with (\\x. T)) (s , t)) f) g))))))"}
{"_id": "1123", "title": "VECTOR_NEG_NEG", "text": "|- ((vector_neg (vector_neg x)) = x)"}
{"_id": "1124", "title": "REAL_ABS_DIV", "text": "|- (!x. (!y. ((real_abs ((real_div x) y)) = ((real_div (real_abs x)) (real_abs y)))))"}
{"_id": "1125", "title": "DISJOINT", "text": "|- (!s. (!t. (((DISJOINT s) t) = ((s INTER t) = EMPTY))))"}
{"_id": "1126", "title": "ABSOLUTELY_SETCONTINUOUS_INDEFINITE_INTEGRAL", "text": "|- (!f. (!s. ((((absolutely_integrable_on f) s) /\\ (lebesgue_measurable s)) ==> ((absolutely_setcontinuous_on (\\k. ((integral k) f))) s))))"}
{"_id": "1127", "title": "Hypermap.lemmaZHQCZLX", "text": "|- (!H. (((simple_hypermap H) /\\ ((plain_hypermap H) /\\ (!x. ((x IN (dart H)) ==> ((NUMERAL (BIT1 (BIT1 _0))) <= (CARD ((face H) x))))))) ==> (!x. ((x IN (dart H)) ==> (~ (((node_map H) x) = x))))))"}
{"_id": "1128", "title": "APPEND", "text": "|- ((!l. (((APPEND NIL) l) = l)) /\\ (!h. (!t. (!l. (((APPEND ((CONS h) t)) l) = ((CONS h) ((APPEND t) l)))))))"}
{"_id": "1129", "title": "BALL_TRANSLATION", "text": "|- (!a. (!x. (!r. ((ball (((vector_add a) x) , r)) = ((IMAGE (\\y. ((vector_add a) y))) (ball (x , r)))))))"}
{"_id": "1130", "title": "MAT_0_COMPONENT", "text": "|- (!i. (!j. ((((mat (NUMERAL _0)) $ i) $ j) = (real_of_num (NUMERAL _0)))))"}
{"_id": "1131", "title": "drop", "text": "|- (!x. ((drop x) = (x $ (NUMERAL (BIT1 _0)))))"}
{"_id": "1132", "title": "SUBSPACE_0", "text": "|- ((subspace s) ==> ((vec (NUMERAL _0)) IN s))"}
{"_id": "1133", "title": "real_continuous_atreal", "text": "|- (((real_continuous f) (atreal x)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x'. (((real_lt (real_abs ((real_sub x') x))) d) ==> ((real_lt (real_abs ((real_sub (f x')) (f x)))) e))))))))"}
{"_id": "1134", "title": "real_summable", "text": "|- (!f. (!s. (((real_summable s) f) = (?l. (((real_sums f) l) s)))))"}
{"_id": "1135", "title": "ROOT_MONO_LE_EQ", "text": "|- (!n. (!x. (!y. ((~ (n = (NUMERAL _0))) ==> (((real_le ((root n) x)) ((root n) y)) = ((real_le x) y))))))"}
{"_id": "1136", "title": "VSUM_0", "text": "|- (((vsum s) (\\x. (vec (NUMERAL _0)))) = (vec (NUMERAL _0)))"}
{"_id": "1137", "title": "Add_triangle.fully_surrounded_imp_fan80", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (fan80 ((vec (NUMERAL _0)) , (V , E)))))))"}
{"_id": "1138", "title": "RELPOW_R", "text": "|- ((((((RELPOW (NUMERAL _0)) R) x) y) = (x = y)) /\\ (((((RELPOW (SUC n)) R) x) y) = (?z. (((R x) z) /\\ ((((RELPOW n) R) z) y)))))"}
{"_id": "1139", "title": "NORM_REAL", "text": "|- (!x. ((vector_norm x) = (real_abs (x $ (NUMERAL (BIT1 _0))))))"}
{"_id": "1140", "title": "Seq.sumn0", "text": "|- ((sumn NIL) = (NUMERAL _0))"}
{"_id": "1141", "title": "CONVEX_CLOSURE_RELATIVE_INTERIOR", "text": "|- (!s. ((convex s) ==> ((closure (relative_interior s)) = (closure s))))"}
{"_id": "1142", "title": "LE_ANTISYM", "text": "|- (!m. (!n. (((m <= n) /\\ (n <= m)) = (m = n))))"}
{"_id": "1143", "title": "COMPLEX_POLY_NEG_CLAUSES", "text": "|- ((!x. ((vector_neg x) = ((complex_mul (Cx (real_neg (real_of_num (NUMERAL (BIT1 _0)))))) x))) /\\ (!x. (!y. (((vector_sub x) y) = ((vector_add x) ((complex_mul (Cx (real_neg (real_of_num (NUMERAL (BIT1 _0)))))) y))))))"}
{"_id": "1144", "title": "MEM_LINEAR_IMAGE", "text": "|- (!f. (!x. (!l. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((MEM (f x)) ((MAP f) l)) = ((MEM x) l))))))"}
{"_id": "1145", "title": "PATH_LENGTH_TRANSLATION", "text": "|- (!a. (!g. ((path_length ((\\x. ((vector_add a) x)) o g)) = (path_length g))))"}
{"_id": "1146", "title": "INT_LE_TRANS", "text": "|- (!x. (!y. (!z. ((((int_le x) y) /\\ ((int_le y) z)) ==> ((int_le x) z)))))"}
{"_id": "1147", "title": "HAS_REAL_INTEGRAL_REFL", "text": "|- (!f. (!a. (((has_real_integral f) (real_of_num (NUMERAL _0))) (closed_real_interval ((CONS (a , a)) NIL)))))"}
{"_id": "1148", "title": "EXISTS_SUBSET_IMAGE_INJ", "text": "|- (!P. (!f. (!s. ((?t. ((t SUBSET ((IMAGE f) s)) /\\ (P t))) = (?t. ((t SUBSET s) /\\ ((!x. (!y. (((x IN t) /\\ (y IN t)) ==> (((f x) = (f y)) = (x = y))))) /\\ (P ((IMAGE f) t)))))))))"}
{"_id": "1149", "title": "HAS_REAL_MEASURE_DISJOINT_UNIONS", "text": "|- (!m. (!f. (((FINITE f) /\\ ((!s. ((s IN f) ==> ((has_real_measure s) (m s)))) /\\ (!s. (!t. (((s IN f) /\\ ((t IN f) /\\ (~ (s = t)))) ==> ((DISJOINT s) t)))))) ==> ((has_real_measure (UNIONS f)) ((sum f) m)))))"}
{"_id": "1150", "title": "DIMINDEX_2", "text": "|- ((dimindex UNIV) = (NUMERAL (BIT0 (BIT1 _0))))"}
{"_id": "1151", "title": "Miqmcsn.scs_3M1_prime", "text": "|- (scs_3M1_prime = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) NIL)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) NIL)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))))"}
{"_id": "1152", "title": "HOLOMORPHIC_ON_ADD", "text": "|- (!f. (!g. (!s. ((((holomorphic_on f) s) /\\ ((holomorphic_on g) s)) ==> ((holomorphic_on (\\z. ((vector_add (f z)) (g z)))) s)))))"}
{"_id": "1153", "title": "INT_OF_NUM_LT", "text": "|- (!m. (!n. (((int_lt (int_of_num m)) (int_of_num n)) = (m < n))))"}
{"_id": "1154", "title": "Ssrbool.andTb", "text": "|- (!b. ((T /\\ b) = b))"}
{"_id": "1155", "title": "REAL_ARCH_SIMPLE", "text": "|- (!x. (?n. ((real_le x) (real_of_num n))))"}
{"_id": "1156", "title": "real_max", "text": "|- (!n. (!m. (((real_max m) n) = (((COND ((real_le m) n)) n) m))))"}
{"_id": "1157", "title": "REAL_UNIFORMLY_CONTINUOUS_ON", "text": "|- (!f. (!s. (((real_uniformly_continuous_on f) s) = ((uniformly_continuous_on (lift o (f o drop))) ((IMAGE lift) s)))))"}
{"_id": "1158", "title": "Asfutbf.IN_NEG", "text": "|- (!v. ((v IN ((IMAGE vector_neg) V)) = ((vector_neg v) IN V)))"}
{"_id": "1159", "title": "ITER_ALT", "text": "|- ((!f. (!x. ((((ITER (NUMERAL _0)) f) x) = x))) /\\ (!f. (!n. (!x. ((((ITER (SUC n)) f) x) = (((ITER n) f) (f x)))))))"}
{"_id": "1160", "title": "SPAN_SUPERSET", "text": "|- (!x. ((x IN s) ==> (x IN (span s))))"}
{"_id": "1161", "title": "CARD_LT_CONG", "text": "|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\\ (t =_c t')) ==> ((s <_c t) = (s' <_c t')))))))"}
{"_id": "1162", "title": "SUBSET_EMPTY", "text": "|- (!s. ((s SUBSET EMPTY) = (s = EMPTY)))"}
{"_id": "1163", "title": "ANR_INTERVAL", "text": "|- ((!a. (!b. (ANR (closed_interval ((CONS (a , b)) NIL))))) /\\ (!a. (!b. (ANR (open_interval (a , b))))))"}
{"_id": "1164", "title": "TRUTH", "text": "|- T"}
{"_id": "1165", "title": "Ssrbool.contraNN", "text": "|- (!c. (!b. ((c ==> b) ==> ((~ b) ==> (~ c)))))"}
{"_id": "1166", "title": "CLOSED_UNIV", "text": "|- (closed UNIV)"}
{"_id": "1167", "title": "Ssrnat.mulnn", "text": "|- (!m. ((m * m) = (m EXP (NUMERAL (BIT0 (BIT1 _0))))))"}
{"_id": "1168", "title": "TAGGED_PARTIAL_DIVISION_OF_TRIVIAL", "text": "|- (!p. (((tagged_partial_division_of p) EMPTY) = (p = EMPTY)))"}
{"_id": "1169", "title": "LIM_SEQUENTIALLY", "text": "|- (!s. (!l. ((((--> s) l) sequentially) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (distance ((s n) , l))) e)))))))))"}
{"_id": "1170", "title": "REAL_SERIES_0", "text": "|- (!s. (((real_sums (\\n. (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) s))"}
{"_id": "1171", "title": "COMPONENTS_EQ_SING_EXISTS", "text": "|- (!s. ((?a. ((components s) = (a INSERT EMPTY))) = ((connected s) /\\ (~ (s = EMPTY)))))"}
{"_id": "1172", "title": "SEGMENT_1", "text": "|- ((!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = (((COND ((real_le (drop a)) (drop b))) (closed_interval ((CONS (a , b)) NIL))) (closed_interval ((CONS (b , a)) NIL)))))) /\\ (!a. (!b. ((open_segment (a , b)) = (((COND ((real_le (drop a)) (drop b))) (open_interval (a , b))) (open_interval (b , a)))))))"}
{"_id": "1173", "title": "HOLOMORPHIC_ON_IMP_CONTINUOUS_ON", "text": "|- (!f. (!s. (((holomorphic_on f) s) ==> ((continuous_on f) s))))"}
{"_id": "1174", "title": "summable", "text": "|- (!f. (!s. (((summable s) f) = (?l. (((sums f) l) s)))))"}
{"_id": "1175", "title": "BOUNDED_EMPTY", "text": "|- (bounded EMPTY)"}
{"_id": "1176", "title": "Sphere.dihatn_x", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dihatn_x x1) x2) x3) x4) x5) x6) = ((LET (\\d_x4. (LET_END ((LET (\\d. (LET_END ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (atn ((real_div (real_neg d_x4)) (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x1) d))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) ((((((delta_x4 x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "1177", "title": "less", "text": "|- (!l. (!x. (!y. (((less l) (x , y)) = ((l (x , y)) /\\ (~ (x = y)))))))"}
{"_id": "1178", "title": "Real_ext.REAL_PROP_POS_RADD", "text": "|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))"}
{"_id": "1179", "title": "IS_REALINTERVAL_UNIV", "text": "|- (is_realinterval UNIV)"}
{"_id": "1180", "title": "INTEGRAL_INTERVALS_INCLUSION_EXCLUSION", "text": "|- (!f. (!a. (!b. (!c. (!d. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((c IN (closed_interval ((CONS (a , b)) NIL))) /\\ (d IN (closed_interval ((CONS (a , b)) NIL))))) ==> (((integral (closed_interval ((CONS (a , d)) NIL))) f) = ((vsum (GSPEC (\\GEN%PVAR%2926. (?s. (((SETSPEC GEN%PVAR%2926) (s SUBSET ((NUMERAL (BIT1 _0)) .. (dimindex UNIV)))) s))))) (\\s. ((% ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (CARD (GSPEC (\\GEN%PVAR%2927. (?i. (((SETSPEC GEN%PVAR%2927) ((i IN s) /\\ ((real_lt (d $ i)) (c $ i)))) i))))))) ((integral (closed_interval ((CONS ((lambdai. (((COND (i IN s)) ((real_min (c $ i)) (d $ i))) (a $ i))) , (lambdai. (((COND (i IN s)) ((real_max (c $ i)) (d $ i))) (c $ i))))) NIL))) f)))))))))))"}
{"_id": "1181", "title": "Seq.undup", "text": "|- (((undup NIL) = NIL) /\\ ((undup ((CONS x) s')) = (((COND ((MEM x) s')) (undup s')) ((CONS x) (undup s')))))"}
{"_id": "1182", "title": "Hypermap.number_of_edges", "text": "|- (!H. ((number_of_edges H) = (CARD (edge_set H))))"}
{"_id": "1183", "title": "Functional_equation.gamma2_x1_div_a", "text": "|- (!m. ((gamma2_x1_div_a m) = (promote1_to_6 (gamma2_x_div_azim m))))"}
{"_id": "1184", "title": "REAL_SERIES_NEG", "text": "|- (!x. (!x0. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\\n. (real_neg (x n)))) (real_neg x0)) s)))))"}
{"_id": "1185", "title": "FINITE_IMP_BOUNDED", "text": "|- (!s. ((FINITE s) ==> (bounded s)))"}
{"_id": "1186", "title": "DISJOINT_INSERT", "text": "|- (!x. (!s. (!t. (((DISJOINT (x INSERT s)) t) = (((DISJOINT s) t) /\\ (~ (x IN t)))))))"}
{"_id": "1187", "title": "ABSOLUTELY_REAL_INTEGRABLE_DECREASING_PRODUCT", "text": "|- (!f. (!g. (!a. (!b. (((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_le x) y))) ==> ((real_le (f y)) (f x))))) /\\ ((absolutely_real_integrable_on g) (closed_real_interval ((CONS (a , b)) NIL)))) ==> ((absolutely_real_integrable_on (\\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "1188", "title": "HAS_COMPLEX_DERIVATIVE_CCOS", "text": "|- (!z. (((has_complex_derivative ccos) (vector_neg (csin z))) (at z)))"}
{"_id": "1189", "title": "ABSOLUTELY_INTEGRABLE_CONVOLUTION_SYM", "text": "|- (!bop. (!f. (!g. (!x. (((absolutely_integrable_on (\\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV) = ((absolutely_integrable_on (\\y. ((bop (f y)) (g ((vector_sub x) y))))) UNIV))))))"}
{"_id": "1190", "title": "INDEPENDENT_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((independent ((IMAGE f) s)) = (independent s)))))"}
{"_id": "1191", "title": "CONNECTED_INSERT", "text": "|- (!s. (!a. ((connected s) ==> ((connected (a INSERT s)) = ((s = EMPTY) \\/ (a IN (closure s)))))))"}
{"_id": "1192", "title": "LT_IMP_LE", "text": "|- (!m. (!n. ((m < n) ==> (m <= n))))"}
{"_id": "1193", "title": "Sphere.sqrt8", "text": "|- (sqrt8 = (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))))"}
{"_id": "1194", "title": "REAL_SUM_INTEGRAL_UBOUND_INCREASING", "text": "|- (!f. (!g. (!m. (!n. (((m <= n) /\\ ((!x. ((x IN (closed_real_interval ((CONS ((real_of_num m) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL))) ==> (((has_real_derivative g) (f x)) ((within (atreal x)) (closed_real_interval ((CONS ((real_of_num m) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) NIL)))))) /\\ (!x. (!y. ((((real_le (real_of_num m)) x) /\\ (((real_le x) y) /\\ ((real_le y) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((real_le (f x)) (f y))))))) ==> ((real_le ((sum (m .. n)) (\\k. (f (real_of_num k))))) ((real_sub (g ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) (g (real_of_num m)))))))))"}
{"_id": "1195", "title": "COMPACT_SIMPLEX", "text": "|- (!n. (!s. (((simplex n) s) ==> (compact s))))"}
{"_id": "1196", "title": "Hypermap_iso.bij_contracted_dart2", "text": "|- (!V. (!E. (((BIJ contracted_dart) (d20_fan ((vec (NUMERAL _0)) , (V , E)))) (GSPEC (\\GEN%PVAR%7631. (?v. (((SETSPEC GEN%PVAR%7631) ((v IN V) /\\ ((((set_of_edge v) V) E) = EMPTY))) (v , v))))))))"}
{"_id": "1197", "title": "Lp_ineqs_proofs.azim_fan_eq_dih_y", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((azim_dart (V , E)) d) = ((((((dih_y (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)))))))))))"}
{"_id": "1198", "title": "Sphere.packing_lt", "text": "|- ((packing V) = (!u. (!v. (((u IN V) /\\ ((v IN V) /\\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v)))))"}
{"_id": "1199", "title": "DIFF", "text": "|- (!s. (!t. ((s DIFF t) = (GSPEC (\\GEN%PVAR%4. (?x. (((SETSPEC GEN%PVAR%4) ((x IN s) /\\ (~ (x IN t)))) x)))))))"}
{"_id": "1200", "title": "LINEAR_ROTATE2D", "text": "|- (!t. (linear (rotate2d t)))"}
{"_id": "1201", "title": "CX_POW", "text": "|- (!x. (!n. ((Cx ((real_pow x) n)) = ((complex_pow (Cx x)) n))))"}
{"_id": "1202", "title": "CONTINUOUS_ON_CLOSED_GEN", "text": "|- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> (((continuous_on f) s) = (!u. (((closed_in ((subtopology euclidean) t)) u) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%616. (?x. (((SETSPEC GEN%PVAR%616) ((x IN s) /\\ ((f x) IN u))) x))))))))))))"}
{"_id": "1203", "title": "REAL_CONTINUOUS_ATREAL_WITHINREAL", "text": "|- (!f. (!s. (!x. (((real_continuous f) (atreal x)) ==> ((real_continuous f) ((within (atreal x)) s))))))"}
{"_id": "1204", "title": "LOCALLY_CONNECTED_PCROSS", "text": "|- (!s. (!t. ((((locally connected) s) /\\ ((locally connected) t)) ==> ((locally connected) (s PCROSS t)))))"}
{"_id": "1205", "title": "REAL_SUMS", "text": "|- (((real_sums f) l) = ((sums (lift o f)) (lift l)))"}
{"_id": "1206", "title": "REAL_LT_MIN", "text": "|- (!x. (!y. (!z. (((real_lt z) ((real_min x) y)) = (((real_lt z) x) /\\ ((real_lt z) y))))))"}
{"_id": "1207", "title": "FACE_OF_LINEAR_IMAGE", "text": "|- (!f. (!c. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((face_of ((IMAGE f) c)) ((IMAGE f) s)) = ((face_of c) s))))))"}
{"_id": "1208", "title": "NEGLIGIBLE_SUBSET", "text": "|- (!s. (!t. (((negligible s) /\\ (t SUBSET s)) ==> (negligible t))))"}
{"_id": "1209", "title": "IN_IMAGE", "text": "|- (!y. (!s. (!f. ((y IN ((IMAGE f) s)) = (?x. ((y = (f x)) /\\ (x IN s)))))))"}
{"_id": "1210", "title": "OPEN_EXISTS_IN", "text": "|- (!P. (!Q. ((!a. ((P a) ==> (open (GSPEC (\\GEN%PVAR%542. (?x. (((SETSPEC GEN%PVAR%542) ((Q a) x)) x))))))) ==> (open (GSPEC (\\GEN%PVAR%543. (?x. (((SETSPEC GEN%PVAR%543) (?a. ((P a) /\\ ((Q a) x)))) x))))))))"}
{"_id": "1211", "title": "Hypermap_and_fan.IN_DART_OF_FAN", "text": "|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (x IN (dart_of_fan (V , E)))) ==> (?v. (?w. ((x = (v , w)) /\\ (((v , w) IN (dart_of_fan (V , E))) /\\ ((v IN V) /\\ (w IN V))))))))))"}
{"_id": "1212", "title": "compact_in", "text": "|- (!top. (!s. (((compact_in s) top) = ((s SUBSET (topspace top)) /\\ (!U. (((!u. ((u IN U) ==> ((open_in top) u))) /\\ (s SUBSET (UNIONS U))) ==> (?V. ((FINITE V) /\\ ((V SUBSET U) /\\ (s SUBSET (UNIONS V)))))))))))"}
{"_id": "1213", "title": "Collect_geom2.ETA_Y_POW2_EQ", "text": "|- (((real_div ((real_mul ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0))))))) (((ups_x ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_pow (((eta_y ((dist3 v2) v3)) ((dist3 v1) v3)) ((dist3 v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))))"}
{"_id": "1214", "title": "REAL_INTEGRABLE_BY_PARTS", "text": "|- (!f. (!g. (!f'. (!g'. (!a. (!b. (!c. (((COUNTABLE c) /\\ (((real_continuous_on (\\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((!x. ((x IN ((open_real_interval (a , b)) DIFF c)) ==> ((((has_real_derivative f) (f' x)) (atreal x)) /\\ (((has_real_derivative g) (g' x)) (atreal x))))) /\\ ((real_integrable_on (\\x. ((real_mul (f x)) (g' x)))) (closed_real_interval ((CONS (a , b)) NIL)))))) ==> ((real_integrable_on (\\x. ((real_mul (f' x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL)))))))))))"}
{"_id": "1215", "title": "RELATIVE_BOUNDARY_OF_TRIANGLE", "text": "|- (!a. (!b. (!c. ((~ (collinear (a INSERT (b INSERT (c INSERT EMPTY))))) ==> ((((hull convex) (a INSERT (b INSERT (c INSERT EMPTY)))) DIFF (relative_interior ((hull convex) (a INSERT (b INSERT (c INSERT EMPTY)))))) = ((closed_segment ((CONS (a , b)) NIL)) UNION ((closed_segment ((CONS (b , c)) NIL)) UNION (closed_segment ((CONS (c , a)) NIL)))))))))"}
{"_id": "1216", "title": "ANR_SING", "text": "|- (!a. (ANR (a INSERT EMPTY)))"}
{"_id": "1217", "title": "SC_RULES", "text": "|- (!R. ((!x. (!y. (((R x) y) ==> (((SC R) x) y)))) /\\ (!x. (!y. ((((SC R) x) y) ==> (((SC R) y) x))))))"}
{"_id": "1218", "title": "INTERVAL_REAL_INTERVAL", "text": "|- (((closed_interval ((CONS (a , b)) NIL)) = ((IMAGE lift) (closed_real_interval ((CONS ((drop a) , (drop b))) NIL)))) /\\ ((open_interval (a , b)) = ((IMAGE lift) (open_real_interval ((drop a) , (drop b))))))"}
{"_id": "1219", "title": "VECTOR_VARIATION_TRANSLATION2", "text": "|- (!a. (!f. (!s. (((vector_variation ((IMAGE (\\x. ((vector_add (vector_neg a)) x))) s)) (\\x. (f ((vector_add a) x)))) = ((vector_variation s) f)))))"}
{"_id": "1220", "title": "Seq.in_cons", "text": "|- (!y. (!s. (!x. (((MEM x) ((CONS y) s)) = ((x = y) \\/ ((MEM x) s))))))"}
{"_id": "1221", "title": "TRACE_COVARIANCE_POS_LE", "text": "|- (!A. ((real_le (real_of_num (NUMERAL _0))) (trace ((matrix_mul (transp A)) A))))"}
{"_id": "1222", "title": "MCBALL_SUBSET_MSPACE", "text": "|- (!m. (!x. (!r. (((mcball m) (x , r)) SUBSET (mspace m)))))"}
{"_id": "1223", "title": "CONIC_RELATIVE_INTERIOR_INSERT", "text": "|- (!s. ((conic s) ==> (conic ((vec (NUMERAL _0)) INSERT (relative_interior s)))))"}
{"_id": "1224", "title": "EXTREME_POINTS_OF_CONVEX_HULL", "text": "|- (!s. ((GSPEC (\\GEN%PVAR%2195. (?x. (((SETSPEC GEN%PVAR%2195) ((extreme_point_of x) ((hull convex) s))) x)))) SUBSET s))"}
{"_id": "1225", "title": "ADD_CLAUSES", "text": "|- ((!n. (((NUMERAL _0) + n) = n)) /\\ ((!m. ((m + (NUMERAL _0)) = m)) /\\ ((!m. (!n. (((SUC m) + n) = (SUC (m + n))))) /\\ (!m. (!n. ((m + (SUC n)) = (SUC (m + n))))))))"}
{"_id": "1226", "title": "ONORM_TRIANGLE", "text": "|- (!f. (!g. (((linear f) /\\ (linear g)) ==> ((real_le (onorm (\\x. ((vector_add (f x)) (g x))))) ((real_add (onorm f)) (onorm g))))))"}
{"_id": "1227", "title": "SETDIST_LE_DIST", "text": "|- (!s. (!t. (!x. (!y. (((x IN s) /\\ (y IN t)) ==> ((real_le (setdist (s , t))) (distance (x , y))))))))"}
{"_id": "1228", "title": "ABSOLUTELY_INTEGRABLE_TRANSLATION", "text": "|- (!f. (!s. (!a. (((absolutely_integrable_on (\\x. (f ((vector_add a) x)))) s) = ((absolutely_integrable_on f) ((IMAGE (\\x. ((vector_add a) x))) s))))))"}
{"_id": "1229", "title": "AFFINE_TRANSLATION_EQ", "text": "|- (!a. (!s. ((affine ((IMAGE (\\x. ((vector_add a) x))) s)) = (affine s))))"}
{"_id": "1230", "title": "LOCALLY_CONNECTED_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((locally connected) ((IMAGE f) s)) = ((locally connected) s)))))"}
{"_id": "1231", "title": "Nonlin_def.delta_x1", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((delta_x1 x1) x2) x3) x4) x5) x6) = ((real_add ((real_mul (real_neg x1)) x4)) ((real_add ((real_sub ((real_sub ((real_mul x2) x5)) ((real_mul x3) x5))) ((real_mul x2) x6))) ((real_add ((real_mul x3) x6)) ((real_mul x4) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6)))))))))))))))"}
{"_id": "1232", "title": "Packing3.BARV_CONS", "text": "|- (!V. (!k. (!ul. ((((barV V) k) ul) ==> (?h. (?t. ((ul = ((CONS h) t)) /\\ (h = (HD ul)))))))))"}
{"_id": "1233", "title": "AFFINE_HULL_FINITE_STEP_GEN", "text": "|- (!P. (((?u. ((!x. ((x IN EMPTY) ==> ((P x) (u x)))) /\\ ((((sum EMPTY) u) = w) /\\ (((vsum EMPTY) (\\x. ((% (u x)) x))) = y)))) = ((w = (real_of_num (NUMERAL _0))) /\\ (y = (vec (NUMERAL _0))))) /\\ (((FINITE s) /\\ ((!y. (((a IN s) /\\ ((P a) y)) ==> ((P a) ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) /\\ (!x. (!y. (((a IN s) /\\ (((P a) x) /\\ ((P a) y))) ==> ((P a) ((real_add x) y))))))) ==> ((?u. ((!x. ((x IN (a INSERT s)) ==> ((P x) (u x)))) /\\ ((((sum (a INSERT s)) u) = w) /\\ (((vsum (a INSERT s)) (\\x. ((% (u x)) x))) = y)))) = (?v. (?u. (((P a) v) /\\ ((!x. ((x IN s) ==> ((P x) (u x)))) /\\ ((((sum s) u) = ((real_sub w) v)) /\\ (((vsum s) (\\x. ((% (u x)) x))) = ((vector_sub y) ((% v) a))))))))))))"}
{"_id": "1234", "title": "MAP_REVERSE", "text": "|- (!f. (!l. ((REVERSE ((MAP f) l)) = ((MAP f) (REVERSE l)))))"}
{"_id": "1235", "title": "ORDER_EXISTENCE_CARD", "text": "|- (!R. (!f. (!z. (!k. (((FINITE (GSPEC (\\GEN%PVAR%74. (?n. (((SETSPEC GEN%PVAR%74) (n IN UNIV)) (R (((ITER n) f) z))))))) /\\ (((CARD (GSPEC (\\GEN%PVAR%75. (?n. (((SETSPEC GEN%PVAR%75) (n IN UNIV)) (R (((ITER n) f) z))))))) <= k) /\\ (((R z) z) /\\ ((!x. (!y. (((R x) y) ==> ((R y) x)))) /\\ ((!x. (!y. (!z. ((((R x) y) /\\ ((R y) z)) ==> ((R x) z))))) /\\ (!x. (!y. (((R (f x)) (f y)) = ((R x) y))))))))) ==> (?d. (((NUMERAL _0) < d) /\\ ((d <= k) /\\ (!n. (((R (((ITER n) f) z)) z) = ((num_divides d) n)))))))))))"}
{"_id": "1236", "title": "SERIES_GOESTOZERO", "text": "|- (!s. (!x. (((summable s) x) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\n. ((n IN s) ==> ((real_lt (vector_norm (x n))) e)))) sequentially))))))"}
{"_id": "1237", "title": "MAP_APPEND", "text": "|- (!f. (!l1. (!l2. (((MAP f) ((APPEND l1) l2)) = ((APPEND ((MAP f) l1)) ((MAP f) l2))))))"}
{"_id": "1238", "title": "TC_CASES_L", "text": "|- (!R. (!x. (!z. ((((TC R) x) z) = (((R x) z) \\/ (?y. ((((TC R) x) y) /\\ ((R y) z))))))))"}
{"_id": "1239", "title": "Sphere.dih5atn_x", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih5atn_x x1) x2) x3) x4) x5) x6) = ((LET (\\e. (LET_END ((LET (\\d. (LET_END ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (atn ((real_div (real_neg e)) (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x5) d))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) (((((((rotate5 delta_x4) x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "1240", "title": "Ssrnat.leq_pmul2r", "text": "|- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> (((n1 * m) <= (n2 * m)) = (n1 <= n2))))))"}
{"_id": "1241", "title": "IN_SET_OF_LIST", "text": "|- (!x. (!l. ((x IN (set_of_list l)) = ((MEM x) l))))"}
{"_id": "1242", "title": "IN_FROM", "text": "|- (!m. (!n. ((m IN (from n)) = (n <= m))))"}
{"_id": "1243", "title": "OPEN_IN_OPEN_INTER", "text": "|- (!u. (!s. ((open s) ==> ((open_in ((subtopology euclidean) u)) (u INTER s)))))"}
{"_id": "1244", "title": "LT_MULT_LCANCEL", "text": "|- (!m. (!n. (!p. (((m * n) < (m * p)) = ((~ (m = (NUMERAL _0))) /\\ (n < p))))))"}
{"_id": "1245", "title": "CONTRACTIBLE_SING", "text": "|- (!a. (contractible (a INSERT EMPTY)))"}
{"_id": "1246", "title": "CLOG_NEG_1", "text": "|- ((clog (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) = ((complex_mul ii) (Cx pi)))"}
{"_id": "1247", "title": "ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS", "text": "|- (!f. (!s. ((((absolutely_continuous_on f) s) /\\ (is_interval s)) ==> ((continuous_on f) s))))"}
{"_id": "1248", "title": "FRONTIER_INTERIORS", "text": "|- (!s. ((frontier s) = ((UNIV DIFF (interior s)) DIFF (interior (UNIV DIFF s)))))"}
{"_id": "1249", "title": "MBALL_SUBSET_MCBALL", "text": "|- (!m. (!x. (!r. (((mball m) (x , r)) SUBSET ((mcball m) (x , r))))))"}
{"_id": "1250", "title": "Fnjlbxs.vector_of_list_left_inverse", "text": "|- (!d. (!list. (((dimindex UNIV) = d) ==> (((d * (LENGTH list)) <= (dimindex UNIV)) ==> ((((list_of_vector d) (LENGTH list)) ((vector_of_list d) list)) = list)))))"}
{"_id": "1251", "title": "real_log_convex_on", "text": "|- (!s. (!f. (((real_log_convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\\ ((y IN s) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((real_le (real_of_num (NUMERAL _0))) (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) /\\ ((real_le (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) ((real_mul ((rpow (f x)) u)) ((rpow (f y)) v))))))))))))"}
{"_id": "1252", "title": "COMPACT_PATH_IMAGE", "text": "|- (!g. ((path g) ==> (compact (path_image g))))"}
{"_id": "1253", "title": "GDELTA_TRANSLATION", "text": "|- (!a. (!s. ((gdelta ((IMAGE (\\x. ((vector_add a) x))) s)) = (gdelta s))))"}
{"_id": "1254", "title": "Topology.complement_set", "text": "|- (!x. (!v. ((complement_set (x INSERT (v INSERT EMPTY))) = (GSPEC (\\GEN%PVAR%5482. (?y. (((SETSPEC GEN%PVAR%5482) (~ (y IN (aff (x INSERT (v INSERT EMPTY)))))) y)))))))"}
{"_id": "1255", "title": "Seq.nth_cat", "text": "|- (!x0. (!s1. (!s2. (!n. ((((nth x0) ((cat s1) s2)) n) = (((COND (n < (sizel s1))) (((nth x0) s1) n)) (((nth x0) s2) (n - (sizel s1)))))))))"}
{"_id": "1256", "title": "CONVEX_INTERVAL", "text": "|- (!a. (!b. ((convex (closed_interval ((CONS (a , b)) NIL))) /\\ (convex (open_interval (a , b))))))"}
{"_id": "1257", "title": "TC_CASES_R", "text": "|- (!R. (!x. (!z. ((((TC R) x) z) = (((R x) z) \\/ (?y. (((R x) y) /\\ (((TC R) y) z))))))))"}
{"_id": "1258", "title": "AFFINE_AFFINE_HULL", "text": "|- (!s. (affine ((hull affine) s)))"}
{"_id": "1259", "title": "AFF_DIM_DIM", "text": "|- (!s. ((aff_dim s) = (((COND ((vec (NUMERAL _0)) IN ((hull affine) s))) (int_of_num (dim s))) ((int_sub (int_of_num (dim s))) (int_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "1260", "title": "CONTINUOUS_CLOSED_IN_PREIMAGE_GEN", "text": "|- (!f. (!s. (!t. (!u. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ ((closed_in ((subtopology euclidean) t)) u))) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%618. (?x. (((SETSPEC GEN%PVAR%618) ((x IN s) /\\ ((f x) IN u))) x))))))))))"}
{"_id": "1261", "title": "aff_ge_def", "text": "|- (aff_ge = (affsign sgn_ge))"}
{"_id": "1262", "title": "METRIC_TOPCONTINUOUS", "text": "|- (!m. (!m'. (!f. ((((topcontinuous (mtopology m)) (mtopology m')) f) = ((!x. ((x IN (mspace m)) ==> ((f x) IN (mspace m')))) /\\ (!a. (!e. ((((real_lt (real_of_num (NUMERAL _0))) e) /\\ (a IN (mspace m))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (((x IN (mspace m)) /\\ ((real_lt ((mdist m) (a , x))) d)) ==> ((real_lt ((mdist m') ((f a) , (f x)))) e)))))))))))))"}
{"_id": "1263", "title": "FACE_OF_SIMPLEX_SUBSET", "text": "|- (!n. (!s. (!f. ((((simplex n) s) /\\ ((face_of f) s)) ==> (?c. ((c SUBSET (GSPEC (\\GEN%PVAR%2470. (?x. (((SETSPEC GEN%PVAR%2470) ((extreme_point_of x) s)) x))))) /\\ (f = ((hull convex) c))))))))"}
{"_id": "1264", "title": "EMPTY_SUBSET", "text": "|- (!s. (EMPTY SUBSET s))"}
{"_id": "1265", "title": "Nonlin_def.sqrt_x2", "text": "|- (((((((sqrt_x2 x1) x2) x3) x4) x5) x6) = (sqrt x2))"}
{"_id": "1266", "title": "measurable_on", "text": "|- (!s. (!f. (((measurable_on f) s) = (?k. (?g. ((negligible k) /\\ ((!n. ((continuous_on (g n)) UNIV)) /\\ (!x. ((~ (x IN k)) ==> (((--> (\\n. ((g n) x))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) sequentially))))))))))"}
{"_id": "1267", "title": "VECTOR_VARIATION_CONTINUOUS_RIGHT", "text": "|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (c , b)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (c , b)) NIL))))))))))"}
{"_id": "1268", "title": "SUBSPACE_IMP_AFFINE", "text": "|- (!s. ((subspace s) ==> (affine s)))"}
{"_id": "1269", "title": "Nonlin_def.sqrt_x1", "text": "|- (((((((sqrt_x1 x1) x2) x3) x4) x5) x6) = (sqrt x1))"}
{"_id": "1270", "title": "CONTINUOUS_LIFT_ABS_COMPONENT", "text": "|- (!net. (!f. (!k. (((continuous f) net) ==> ((continuous (\\x. (lift (real_abs ((f x) $ k))))) net)))))"}
{"_id": "1271", "title": "Nonlin_def.uni", "text": "|- (!f. (!x. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((uni (f , x)) x1) x2) x3) x4) x5) x6) = (f ((((((x x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "1272", "title": "NORM_BASIS", "text": "|- (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV))) ==> ((vector_norm (basis k)) = (real_of_num (NUMERAL (BIT1 _0))))))"}
{"_id": "1273", "title": "TRACE_COVARIANCE_POSITIVE_SEMIDEFINITE_LE", "text": "|- (!A. ((positive_semidefinite A) ==> ((real_le (trace ((matrix_mul (transp A)) A))) ((real_pow (trace A)) (NUMERAL (BIT0 (BIT1 _0)))))))"}
{"_id": "1274", "title": "RC_REFL", "text": "|- (!R. (!x. (((RC R) x) x)))"}
{"_id": "1275", "title": "DEPENDENT_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((dependent ((IMAGE f) s)) = (dependent s)))))"}
{"_id": "1276", "title": "Seq.has_count", "text": "|- (!a. (!s. (((has a) s) = ((NUMERAL _0) < ((count a) s)))))"}
{"_id": "1277", "title": "REAL_LE_ADDR", "text": "|- (!x. (!y. (((real_le x) ((real_add x) y)) = ((real_le (real_of_num (NUMERAL _0))) y))))"}
{"_id": "1278", "title": "arc", "text": "|- (!g. ((arc g) = ((path g) /\\ (!x. (!y. (((x IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ ((y IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ ((g x) = (g y)))) ==> (x = y)))))))"}
{"_id": "1279", "title": "NEGLIGIBLE_IMP_MEASURABLE", "text": "|- (!s. ((negligible s) ==> (measurable s)))"}
{"_id": "1280", "title": "POLYTOPE_EQ_BOUNDED_POLYHEDRON", "text": "|- (!s. ((polytope s) = ((polyhedron s) /\\ (bounded s))))"}
{"_id": "1281", "title": "INT_GT", "text": "|- (!x. (!y. (((int_gt x) y) = ((int_lt y) x))))"}
{"_id": "1282", "title": "INTEGER_ADD", "text": "|- (!x. (!y. (((integer x) /\\ (integer y)) ==> (integer ((real_add x) y)))))"}
{"_id": "1283", "title": "UNBOUNDED_OUTSIDE", "text": "|- (!s. ((bounded s) ==> (~ (bounded (outside s)))))"}
{"_id": "1284", "title": "TAYLOR_CCOS", "text": "|- (!n. (!z. ((real_le (vector_norm ((vector_sub (ccos z)) ((vsum ((NUMERAL _0) .. n)) (\\k. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) k)) ((complex_div ((complex_pow z) ((NUMERAL (BIT0 (BIT1 _0))) * k))) (Cx (real_of_num (FACT ((NUMERAL (BIT0 (BIT1 _0))) * k))))))))))) ((real_mul (exp (real_abs (Im z)))) ((real_div ((real_pow (vector_norm z)) (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (FACT (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))))))))"}
{"_id": "1285", "title": "Ssrnat.distn", "text": "|- (!m. (!n. (((distn m) n) = ((m - n) + (n - m)))))"}
{"_id": "1286", "title": "VECTOR_ADD_SUB", "text": "|- (((vector_sub ((vector_add x) y)) x) = y)"}
{"_id": "1287", "title": "CEXP_CONVERGES", "text": "|- (!z. (((sums (\\n. ((complex_div ((complex_pow z) n)) (Cx (real_of_num (FACT n)))))) (cexp z)) (from (NUMERAL _0))))"}
{"_id": "1288", "title": "REAL", "text": "|- (!z. ((real z) = ((Cx (Re z)) = z)))"}
{"_id": "1289", "title": "UNION_ACI", "text": "|- (((p UNION q) = (q UNION p)) /\\ ((((p UNION q) UNION r) = (p UNION (q UNION r))) /\\ (((p UNION (q UNION r)) = (q UNION (p UNION r))) /\\ (((p UNION p) = p) /\\ ((p UNION (p UNION q)) = (p UNION q))))))"}
{"_id": "1290", "title": "LINEAR_CX_IM", "text": "|- (linear (Cx o Im))"}
{"_id": "1291", "title": "CONNECTED_DISJOINT_UNIONS_OPEN_UNIQUE", "text": "|- (!f. (!f'. ((((pairwise DISJOINT) f) /\\ (((pairwise DISJOINT) f') /\\ ((!s. ((s IN f) ==> ((open s) /\\ ((connected s) /\\ (~ (s = EMPTY)))))) /\\ ((!s. ((s IN f') ==> ((open s) /\\ ((connected s) /\\ (~ (s = EMPTY)))))) /\\ ((UNIONS f) = (UNIONS f')))))) ==> (f = f'))))"}
{"_id": "1292", "title": "EVENTUALLY_AT_INFINITY", "text": "|- (!p. (((eventually p) at_infinity) = (?b. (!x. (((real_ge (vector_norm x)) b) ==> (p x))))))"}
{"_id": "1293", "title": "CONNECTED_INTERVAL", "text": "|- ((!a. (!b. (connected (closed_interval ((CONS (a , b)) NIL))))) /\\ (!a. (!b. (connected (open_interval (a , b))))))"}
{"_id": "1294", "title": "Miqmcsn.SCS_4M8_02_SLICE_02", "text": "|- ((scs_arrow_v39 (scs_4M8_02 INSERT EMPTY)) (((scs_prop_equ_v39 (scs_opp_v39 scs_3T7)) (NUMERAL (BIT0 (BIT1 _0)))) INSERT EMPTY))"}
{"_id": "1295", "title": "CBALL_TRIVIAL", "text": "|- (!x. ((cball (x , (real_of_num (NUMERAL _0)))) = (x INSERT EMPTY)))"}
{"_id": "1296", "title": "Lp_ineqs_proofs.f_fan_pair_ext_in_darts_k", "text": "|- (!V. (!E. (!k. (!d. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (d IN ((darts_k k) (hypermap_of_fan (V , E))))) ==> ((((f_fan_pair_ext (V , E)) d) IN ((darts_k k) (hypermap_of_fan (V , E)))) /\\ (((inverse (f_fan_pair_ext (V , E))) d) IN ((darts_k k) (hypermap_of_fan (V , E))))))))))"}
{"_id": "1297", "title": "MATRIX_MUL_LID", "text": "|- (!A. (((matrix_mul (mat (NUMERAL (BIT1 _0)))) A) = A))"}
{"_id": "1298", "title": "HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homeomorphic s) ((IMAGE f) t)) = ((homeomorphic s) t))))))"}
{"_id": "1299", "title": "BARYCENTRE_TRANSLATION", "text": "|- (!a. (!s. ((barycentre ((IMAGE (\\x. ((vector_add a) x))) s)) = ((vector_add (((COND ((FINITE s) /\\ (~ (s = EMPTY)))) a) (vec (NUMERAL _0)))) (barycentre s)))))"}
{"_id": "1300", "title": "NORM_TRIANGLE_LE", "text": "|- (!x. (!y. (((real_le ((real_add (vector_norm x)) (vector_norm y))) e) ==> ((real_le (vector_norm ((vector_add x) y))) e))))"}
{"_id": "1301", "title": "PATHSTART_TRANSLATION", "text": "|- (!a. (!g. ((pathstart ((\\x. ((vector_add a) x)) o g)) = ((vector_add a) (pathstart g)))))"}
{"_id": "1302", "title": "CLOG_II", "text": "|- ((clog ii) = ((complex_mul ii) (Cx ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "1303", "title": "NSUM_CONST", "text": "|- (!c. (!s. ((FINITE s) ==> (((nsum s) (\\n. c)) = ((CARD s) * c)))))"}
{"_id": "1304", "title": "CONTINUOUS_ON_PASTECART", "text": "|- (!f. (!g. (!s. ((((continuous_on f) s) /\\ ((continuous_on g) s)) ==> ((continuous_on (\\x. ((pastecart (f x)) (g x)))) s)))))"}
{"_id": "1305", "title": "Hypermap_iso.hyp_iso_comm", "text": "|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((((edge_map G) (f x)) = (f ((edge_map H) x))) /\\ ((((node_map G) (f x)) = (f ((node_map H) x))) /\\ (((face_map G) (f x)) = (f ((face_map H) x)))))))))))"}
{"_id": "1306", "title": "HAS_COMPLEX_DERIVATIVE_CLOG", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> (((has_complex_derivative clog) (complex_inv z)) (at z))))"}
{"_id": "1307", "title": "JACOBIAN_WORKS", "text": "|- (!f. (!net. (((differentiable f) net) = (((has_derivative f) (\\h. ((matrix_vector_mul ((jacobian f) net)) h))) net))))"}
{"_id": "1308", "title": "HOMOTOPY_EQUIVALENT_TRANSLATION_RIGHT_EQ", "text": "|- (!a. (!s. (!t. (((homotopy_equivalent s) ((IMAGE (\\x. ((vector_add a) x))) t)) = ((homotopy_equivalent s) t)))))"}
{"_id": "1309", "title": "Add_triangle.f_fan_pair_eq_inv_n_o_e", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((f_fan_pair (V , E)) d) = ((inverse (n_fan_pair_ext (V , E))) ((e_fan_pair (V , E)) d))))))))"}
{"_id": "1310", "title": "CONTINUOUS_ON_EQ_CONTINUOUS_AT", "text": "|- (!f. (!s. ((open s) ==> (((continuous_on f) s) = (!x. ((x IN s) ==> ((continuous f) (at x))))))))"}
{"_id": "1311", "title": "Ssrnat.odd_add", "text": "|- (!m. (!n. ((odd (m + n)) = ((XOR (odd m)) (odd n)))))"}
{"_id": "1312", "title": "REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_0", "text": "|- (!f. (!g. (!s. ((((real_measurable_on f) s) /\\ (((real_continuous_on g) UNIV) /\\ ((g (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0))))) ==> ((real_measurable_on (g o f)) s)))))"}
{"_id": "1313", "title": "COMPACT_CONTINUOUS_IMAGE", "text": "|- (!f. (!s. ((((continuous_on f) s) /\\ (compact s)) ==> (compact ((IMAGE f) s)))))"}
{"_id": "1314", "title": "Seq2.list_sum_perm_eq", "text": "|- (!s1. (!s2. (!f. (((perm_eq s1) s2) ==> (((list_sum s1) f) = ((list_sum s2) f))))))"}
{"_id": "1315", "title": "FLOOR_POS", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (?n. ((floor x) = (real_of_num n)))))"}
{"_id": "1316", "title": "CONTINUOUS_CLOSED_IN_PREIMAGE_EQ", "text": "|- (!f. (!s. (((continuous_on f) s) = (!t. ((closed t) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%630. (?x. (((SETSPEC GEN%PVAR%630) ((x IN s) /\\ ((f x) IN t))) x))))))))))"}
{"_id": "1317", "title": "Seq.sumn", "text": "|- (sumn = ((foldr +) (NUMERAL _0)))"}
{"_id": "1318", "title": "TRACE_POSITIVE_SEMIDEFINITE", "text": "|- (!A. ((positive_semidefinite A) ==> ((real_le (real_of_num (NUMERAL _0))) (trace A))))"}
{"_id": "1319", "title": "Nonlin_def.unit6", "text": "|- (((((((unit6 x1) x2) x3) x4) x5) x6) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "1320", "title": "COLLINEAR_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((collinear ((IMAGE f) s)) = (collinear s)))))"}
{"_id": "1321", "title": "Hypermap.lemma_power_loop_map_in_loop", "text": "|- (!L. (!x. (!k. (((in_dart_of_loop x) L) ==> ((in_dart_of_loop (((POWER (loop_map L)) k) x)) L)))))"}
{"_id": "1322", "title": "Ssrnat.sub0n", "text": "|- (!n. (((NUMERAL _0) - n) = (NUMERAL _0)))"}
{"_id": "1323", "title": "Nonlinear_lemma.halfbump_x4", "text": "|- (!x1. (!x2. (!x3. (!x5. (!x6. (!x4. (((((((halfbump_x4 x1) x2) x3) x4) x5) x6) = (halfbump_x x4))))))))"}
{"_id": "1324", "title": "Seq.constant_nseq", "text": "|- (!n. (!x. (constant ((nseq n) x))))"}
{"_id": "1325", "title": "REAL_OF_NUM_SUC", "text": "|- (!n. (((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (SUC n))))"}
{"_id": "1326", "title": "Lp_ineqs_def.list_defs_th", "text": "|- ((!fan. (!g. ((azim_list (g , fan)) = ((azim_dart fan) o g)))) /\\ ((!fan. (!g. ((azim2_list (g , fan)) = ((azim2_fan fan) o g)))) /\\ ((!fan. (!g. ((azim3_list (g , fan)) = ((azim3_fan fan) o g)))) /\\ ((!h. ((yn_list h) = (yn_fan o h))) /\\ ((!h. ((ln_list h) = (ln_fan o h))) /\\ ((!h. ((rho_list h) = (rho_fan o h))) /\\ ((!fan. (!g. ((rhazim_list (g , fan)) = ((rhazim_fan fan) o g)))) /\\ ((!fan. (!g. ((rhazim2_list (g , fan)) = ((rhazim2_fan fan) o g)))) /\\ ((!fan. (!g. ((rhazim3_list (g , fan)) = ((rhazim3_fan fan) o g)))) /\\ ((!g. ((ye_list g) = (ye_fan o g))) /\\ ((!fan. (!g. ((sol_list (g , fan)) = ((sol_fan fan) o (IMAGE g))))) /\\ ((!fan. (!g. ((tau_list (g , fan)) = ((tau_fan fan) o (IMAGE g))))) /\\ ((!g. ((y1_list g) = (y1_fan o g))) /\\ ((!g. ((y2_list g) = (y2_fan o g))) /\\ ((!fan. (!g. ((y3_list (g , fan)) = ((y3_fan fan) o g)))) /\\ ((!fan. (!g. ((y4_list (g , fan)) = ((y4_fan fan) o g)))) /\\ ((!fan. (!g. ((y5_list (g , fan)) = ((y5_fan fan) o g)))) /\\ ((!g. ((y6_list g) = (y6_fan o g))) /\\ ((!fan. (!g. ((y8_list (g , fan)) = ((y8_fan fan) o g)))) /\\ ((!fan. (!g. ((y9_list (g , fan)) = ((y9_fan fan) o g)))) /\\ (!fan. (!g. ((y4'_list (g , fan)) = ((y4'_fan fan) o g))))))))))))))))))))))))"}
{"_id": "1327", "title": "Fan.n_fan", "text": "|- (!x. (!V. (!E. ((((n_fan x) V) E) = (GABS (\\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (v , ((((((sigma_fan x) V) E) v) w) , (((((sigma_fan x) V) E) v) (((((sigma_fan x) V) E) v) w))))))))))))))))"}
{"_id": "1328", "title": "Sphere.rotate5", "text": "|- (!f. (!x5. (!x3. (!x4. (!x2. (!x6. (!x1. ((((((((rotate5 f) x1) x2) x3) x4) x5) x6) = ((((((f x5) x3) x4) x2) x6) x1)))))))))"}
{"_id": "1329", "title": "INVERTIBLE_FIXPOINT_PROPERTY", "text": "|- (!s. (!t. (!i. (!r. ((((continuous_on i) t) /\\ ((((IMAGE i) t) SUBSET s) /\\ (((continuous_on r) s) /\\ ((((IMAGE r) s) SUBSET t) /\\ (!y. ((y IN t) ==> ((r (i y)) = y))))))) ==> ((!f. ((((continuous_on f) s) /\\ (((IMAGE f) s) SUBSET s)) ==> (?x. ((x IN s) /\\ ((f x) = x))))) ==> (!g. ((((continuous_on g) t) /\\ (((IMAGE g) t) SUBSET t)) ==> (?y. ((y IN t) /\\ ((g y) = y)))))))))))"}
{"_id": "1330", "title": "RC_TC", "text": "|- (!R. ((RC (TC R)) = (TC (RC R))))"}
{"_id": "1331", "title": "DIST_0", "text": "|- (!x. (((distance (x , (vec (NUMERAL _0)))) = (vector_norm x)) /\\ ((distance ((vec (NUMERAL _0)) , x)) = (vector_norm x))))"}
{"_id": "1332", "title": "VECTOR_ANGLE_RANGE", "text": "|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) ((vector_angle x) y)) /\\ ((real_le ((vector_angle x) y)) pi))))"}
{"_id": "1333", "title": "REAL_MUL_LID", "text": "|- (!x. (((real_mul (real_of_num (NUMERAL (BIT1 _0)))) x) = x))"}
{"_id": "1334", "title": "Ssrnat.dist0n", "text": "|- (!n. (((distn (NUMERAL _0)) n) = n))"}
{"_id": "1335", "title": "Hypermap.lemma_dart_inveriant_under_inverse_maps", "text": "|- (!H. (!x. ((x IN (dart H)) ==> ((((inverse (edge_map H)) x) IN (dart H)) /\\ ((((inverse (node_map H)) x) IN (dart H)) /\\ (((inverse (face_map H)) x) IN (dart H)))))))"}
{"_id": "1336", "title": "ordinal", "text": "|- (!l. ((ordinal l) = ((woset l) /\\ (!x. (((fl l) x) ==> (x = (@y. (~ ((less l) (y , x))))))))))"}
{"_id": "1337", "title": "LOG_MUL", "text": "|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((log ((real_mul x) y)) = ((real_add (log x)) (log y))))))"}
{"_id": "1338", "title": "SUBSET_DIFF", "text": "|- (!s. (!t. ((s DIFF t) SUBSET s)))"}
{"_id": "1339", "title": "CONNECTED_IMP_PERFECT", "text": "|- (!s. (!x. (((connected s) /\\ ((~ (?a. (s = (a INSERT EMPTY)))) /\\ (x IN s))) ==> ((limit_point_of x) s))))"}
{"_id": "1340", "title": "Imjxphr.HYPER_MM_COLLINEAR", "text": "|- ((((scs_k_v39 s) = k) /\\ ((is_scs_v39 s) /\\ (((MMs_v39 s) w) /\\ (((NUMERAL (BIT1 (BIT1 _0))) < k) /\\ (((w l) = v2) /\\ (((w (SUC l)) = v1) /\\ ((((real_pow (vector_norm v1)) (NUMERAL (BIT0 (BIT1 _0)))) = x1) /\\ ((((real_pow (vector_norm v2)) (NUMERAL (BIT0 (BIT1 _0)))) = x2) /\\ (((real_pow (vector_norm ((vector_sub v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))) = x6))))))))) ==> (((real_lt (real_of_num (NUMERAL _0))) x1) /\\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\\ (((real_lt (real_of_num (NUMERAL _0))) x6) /\\ (~ (collinear ((vec (NUMERAL _0)) INSERT (v1 INSERT (v2 INSERT EMPTY)))))))))"}
{"_id": "1341", "title": "Fnjlbxs.lim_not_equal", "text": "|- (!a. (!b. (!f. (!g. (((~ (a = b)) /\\ ((((--> f) a) sequentially) /\\ (((--> g) b) sequentially))) ==> (?N. (!n. ((N <= n) ==> (~ ((f n) = (g n)))))))))))"}
{"_id": "1342", "title": "Ssrnat.distnEl", "text": "|- (!m. (!n. ((n <= m) ==> (((distn m) n) = (m - n)))))"}
{"_id": "1343", "title": "HOMEOMORPHISM_GRAPH_EXPLICIT", "text": "|- (!f. (!s. (((homeomorphism (s , (GSPEC (\\GEN%PVAR%1033. (?x. (((SETSPEC GEN%PVAR%1033) (x IN s)) ((pastecart x) (f x)))))))) ((\\x. ((pastecart x) (f x))) , fstcart)) = ((continuous_on f) s))))"}
{"_id": "1344", "title": "COUNTABLE_EMPTY", "text": "|- (COUNTABLE EMPTY)"}
{"_id": "1345", "title": "HAS_COMPLEX_DERIVATIVE_CTAN", "text": "|- (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> (((has_complex_derivative ctan) (complex_inv ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0)))))) (at z))))"}
{"_id": "1346", "title": "outside", "text": "|- (!s. ((outside s) = (GSPEC (\\GEN%PVAR%2007. (?x. (((SETSPEC GEN%PVAR%2007) ((~ (x IN s)) /\\ (~ (bounded ((connected_component (UNIV DIFF s)) x))))) x))))))"}
{"_id": "1347", "title": "INT_FORALL_ABS", "text": "|- (!P. ((!n. (P (int_of_num n))) = (!x. (P (int_abs x)))))"}
{"_id": "1348", "title": "DIST_LIFT", "text": "|- (!x. (!y. ((distance ((lift x) , (lift y))) = (real_abs ((real_sub x) y)))))"}
{"_id": "1349", "title": "Collect_geom.DELTA_X34", "text": "|- (!x12. (!x13. (!x14. (!x23. (!x24. (!x34. (((((((delta x12) x13) x14) x23) x24) x34) = ((real_add ((real_mul (real_neg x12)) ((real_pow x34) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul ((real_add ((real_mul (real_neg x13)) x14)) ((real_add ((real_mul (real_neg x23)) x24)) ((real_add ((real_mul x13) x24)) ((real_add ((real_mul x14) x23)) ((real_add ((real_mul (real_neg x12)) x12)) ((real_add ((real_mul x12) x14)) ((real_add ((real_mul x12) x24)) ((real_add ((real_mul x12) x13)) ((real_mul x12) x23)))))))))) x34)) ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24)))))))))))))))))))))"}
{"_id": "1350", "title": "PATH_COMPONENT_IN", "text": "|- (!s. (!x. (!y. ((((path_component s) x) y) ==> ((x IN s) /\\ (y IN s))))))"}
{"_id": "1351", "title": "Seq.rotr", "text": "|- (!n. (!s. (((rotr n) s) = ((rot ((sizel s) - n)) s))))"}
{"_id": "1352", "title": "CARD_LE_CONNECTED_COMPONENTS", "text": "|- (!f. (!s. (((continuous_on f) s) ==> ((GSPEC (\\GEN%PVAR%1819. (?y. (((SETSPEC GEN%PVAR%1819) (y IN ((IMAGE f) s))) ((connected_component ((IMAGE f) s)) y))))) <=_c (GSPEC (\\GEN%PVAR%1820. (?x. (((SETSPEC GEN%PVAR%1820) (x IN s)) ((connected_component s) x)))))))))"}
{"_id": "1353", "title": "REAL_LE_MUL2", "text": "|- (!w. (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) w) /\\ (((real_le w) x) /\\ (((real_le (real_of_num (NUMERAL _0))) y) /\\ ((real_le y) z)))) ==> ((real_le ((real_mul w) y)) ((real_mul x) z)))))))"}
{"_id": "1354", "title": "HOMEOMORPHIC_TRANSLATION_LEFT_EQ", "text": "|- (!a. (!s. (!t. (((homeomorphic ((IMAGE (\\x. ((vector_add a) x))) s)) t) = ((homeomorphic s) t)))))"}
{"_id": "1355", "title": "rcone_gt", "text": "|- (rcone_gt = (rconesgn real_gt))"}
{"_id": "1356", "title": "SPAN_LINEAR_IMAGE", "text": "|- (!f. (!s. ((linear f) ==> ((span ((IMAGE f) s)) = ((IMAGE f) (span s))))))"}
{"_id": "1357", "title": "LIFT_NUM", "text": "|- (!n. ((lift (real_of_num n)) = (vec n)))"}
{"_id": "1358", "title": "HAS_INTEGRAL_INTEGRAL", "text": "|- (!f. (!s. (((integrable_on f) s) = (((has_integral f) ((integral s) f)) s))))"}
{"_id": "1359", "title": "MULT_EQ_0", "text": "|- (!m. (!n. (((m * n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) \\/ (n = (NUMERAL _0))))))"}
{"_id": "1360", "title": "HAS_REAL_DERIVATIVE_LMUL_ATREAL", "text": "|- (!f. (!f'. (!c. (!x. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative (\\x. ((real_mul c) (f x)))) ((real_mul c) f')) (atreal x)))))))"}
{"_id": "1361", "title": "REAL_COMPACT_CONTINUOUS_IMAGE", "text": "|- (!f. (!s. ((((real_continuous_on f) s) /\\ (real_compact s)) ==> (real_compact ((IMAGE f) s)))))"}
{"_id": "1362", "title": "INTER_COMM", "text": "|- (!s. (!t. ((s INTER t) = (t INTER s))))"}
{"_id": "1363", "title": "Ssrnat.addn_eq0", "text": "|- (!m. (!n. (((m + n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) /\\ (n = (NUMERAL _0))))))"}
{"_id": "1364", "title": "LOCALLY_PATH_CONNECTED_SPHERE", "text": "|- (!a. (!r. ((locally path_connected) (sphere (a , r)))))"}
{"_id": "1365", "title": "COMPACT_EMPTY", "text": "|- (compact EMPTY)"}
{"_id": "1366", "title": "Fnjlbxs.list_of_vector", "text": "|- (!v. (!d. (!n. ((((list_of_vector d) n) v) = ((TABLE (\\k. (lambdai. (v $ ((k * d) + i))))) n)))))"}
{"_id": "1367", "title": "COMPLEX_SUB_REFL", "text": "|- (!x. (((vector_sub x) x) = (Cx (real_of_num (NUMERAL _0)))))"}
{"_id": "1368", "title": "LT_SUC_LE", "text": "|- (!m. (!n. ((m < (SUC n)) = (m <= n))))"}
{"_id": "1369", "title": "FRONTIER_SUBSET_EQ", "text": "|- (!s. (((frontier s) SUBSET s) = (closed s)))"}
{"_id": "1370", "title": "Hypermap_iso.iso_face_set", "text": "|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> ((face_set G) = ((IMAGE (IMAGE f)) (face_set H)))))))"}
{"_id": "1371", "title": "MEASURABLE_SCALING", "text": "|- (!s. (!c. ((measurable s) ==> (measurable ((IMAGE (\\x. ((% c) x))) s)))))"}
{"_id": "1372", "title": "SPAN_MONO", "text": "|- (!s. (!t. ((s SUBSET t) ==> ((span s) SUBSET (span t)))))"}
{"_id": "1373", "title": "More_list_hypermap.node_hypermap_of_list_explicit", "text": "|- (!L. (!n. (!d. (((good_list_nodes L) /\\ (((MEM n) (list_of_nodes L)) /\\ ((MEM d) n))) ==> (((node (hypermap_of_list L)) d) = (set_of_list n))))))"}
{"_id": "1374", "title": "CLOSED_UNION", "text": "|- (!s. (!t. (((closed s) /\\ (closed t)) ==> (closed (s UNION t)))))"}
{"_id": "1375", "title": "IMAGE_CLAUSES", "text": "|- ((((IMAGE f) EMPTY) = EMPTY) /\\ (((IMAGE f) (x INSERT s)) = ((f x) INSERT ((IMAGE f) s))))"}
{"_id": "1376", "title": "HAS_REAL_DERIVATIVE_ID", "text": "|- (!net. (((has_real_derivative (\\x. x)) (real_of_num (NUMERAL (BIT1 _0)))) net))"}
{"_id": "1377", "title": "BOUNDED_UNION", "text": "|- (!s. (!t. ((bounded (s UNION t)) = ((bounded s) /\\ (bounded t)))))"}
{"_id": "1378", "title": "AFFINE_SCALING", "text": "|- (!s. (!c. ((affine s) ==> (affine ((IMAGE (\\x. ((% c) x))) s)))))"}
{"_id": "1379", "title": "CARD_LT_IMP_LE", "text": "|- (!s. (!t. ((s <_c t) ==> (s <=_c t))))"}
{"_id": "1380", "title": "Ssrbool.contraLR", "text": "|- (!c. (!b. (((~ c) ==> (~ b)) ==> (b ==> c))))"}
{"_id": "1381", "title": "JOINPATHS_LINEAR_IMAGE", "text": "|- (!f. (!g1. (!g2. ((linear f) ==> (((++ (f o g1)) (f o g2)) = (f o ((++ g1) g2)))))))"}
{"_id": "1382", "title": "AFFINE_DEPENDENT_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((affine_dependent ((IMAGE f) s)) = (affine_dependent s)))))"}
{"_id": "1383", "title": "MULT_AC", "text": "|- (((m * n) = (n * m)) /\\ ((((m * n) * p) = (m * (n * p))) /\\ ((m * (n * p)) = (n * (m * p)))))"}
{"_id": "1384", "title": "ORTHOGONAL_TRANSFORMATION_NEG", "text": "|- (!f. ((orthogonal_transformation (\\x. (vector_neg (f x)))) = (orthogonal_transformation f)))"}
{"_id": "1385", "title": "Miqmcsn.PSORT_5_EXPLICIT", "text": "|- ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL _0))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL _0))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT1 (BIT1 _0)))) ((NUMERAL (BIT1 _0)) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL _0))) = ((NUMERAL _0) , (NUMERAL _0))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL _0))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) = ((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT0 (BIT1 _0)))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = ((NUMERAL _0) , (NUMERAL (BIT1 _0)))) /\\ ((((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) = ((NUMERAL (BIT1 _0)) , (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((psort (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((NUMERAL (BIT1 (BIT1 _0))) , (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) = ((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))"}
{"_id": "1386", "title": "RESTRICTION_IN_EXTENSIONAL", "text": "|- (!s. (!f. (((RESTRICTION s) f) IN (EXTENSIONAL s))))"}
{"_id": "1387", "title": "NOT_EVEN", "text": "|- (!n. ((~ (EVEN n)) = (ODD n)))"}
{"_id": "1388", "title": "between", "text": "|- (!a. (!x. (!b. (((between x) (a , b)) = ((distance (a , b)) = ((real_add (distance (a , x))) (distance (x , b))))))))"}
{"_id": "1389", "title": "PATHFINISH_TRANSLATION", "text": "|- (!a. (!g. ((pathfinish ((\\x. ((vector_add a) x)) o g)) = ((vector_add a) (pathfinish g)))))"}
{"_id": "1390", "title": "CARD_EXP_POWERSET", "text": "|- (!s. (((^_c UNIV) s) =_c (GSPEC (\\GEN%PVAR%40. (?t. (((SETSPEC GEN%PVAR%40) (t SUBSET s)) t))))))"}
{"_id": "1391", "title": "RPOW_POW", "text": "|- (!x. (!n. (((rpow x) (real_of_num n)) = ((real_pow x) n))))"}
{"_id": "1392", "title": "Tame_general.CONTRAVENING_FAN", "text": "|- (!V. ((contravening V) ==> (FAN ((vec (NUMERAL _0)) , (V , (ESTD V))))))"}
{"_id": "1393", "title": "VECTOR_MUL_COMPONENT", "text": "|- (!c. (!x. (!i. ((((% c) x) $ i) = ((real_mul c) (x $ i))))))"}
{"_id": "1394", "title": "LOCALLY_EMPTY", "text": "|- (!P. ((locally P) EMPTY))"}
{"_id": "1395", "title": "ABSOLUTELY_REAL_INTEGRABLE_INCREASING_PRODUCT", "text": "|- (!f. (!g. (!a. (!b. (((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_le x) y))) ==> ((real_le (f x)) (f y))))) /\\ ((absolutely_real_integrable_on g) (closed_real_interval ((CONS (a , b)) NIL)))) ==> ((absolutely_real_integrable_on (\\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "1396", "title": "MEMBER_NOT_EMPTY", "text": "|- (!s. ((?x. (x IN s)) = (~ (s = EMPTY))))"}
{"_id": "1397", "title": "LAMBDA_ETA", "text": "|- (!g. ((lambdai. (g $ i)) = g))"}
{"_id": "1398", "title": "CARD_LE_INJ", "text": "|- (!s. (!t. (((FINITE s) /\\ ((FINITE t) /\\ ((CARD s) <= (CARD t)))) ==> (?f. ((((IMAGE f) s) SUBSET t) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))))))))"}
{"_id": "1399", "title": "CARD_EQ_CONG", "text": "|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\\ (t =_c t')) ==> ((s =_c t) = (s' =_c t')))))))"}
{"_id": "1400", "title": "midpoint", "text": "|- (!a. (!b. ((midpoint (a , b)) = ((% (real_inv (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((vector_add a) b)))))"}
{"_id": "1401", "title": "CONTINUOUS_ON_COMPOSE", "text": "|- (!f. (!g. (!s. ((((continuous_on f) s) /\\ ((continuous_on g) ((IMAGE f) s))) ==> ((continuous_on (g o f)) s)))))"}
{"_id": "1402", "title": "SYMMETRIC_MATRIX_INV", "text": "|- (!A. (((transp (matrix_inv A)) = (matrix_inv A)) = ((transp A) = A)))"}
{"_id": "1403", "title": "AFFINE_PCROSS", "text": "|- (!s. (!t. (((affine s) /\\ (affine t)) ==> (affine (s PCROSS t)))))"}
{"_id": "1404", "title": "Nonlin_def.delta_x4", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((delta_x4 x1) x2) x3) x4) x5) x6) = ((real_add ((real_sub ((real_mul (real_neg x2)) x3)) ((real_mul x1) x4))) ((real_add ((real_mul x2) x5)) ((real_add ((real_sub ((real_mul x3) x6)) ((real_mul x5) x6))) ((real_mul x1) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6)))))))))))))))"}
{"_id": "1405", "title": "HAS_BOUNDED_VARIATION_ON_COMPONENTWISE", "text": "|- (!f. (!s. (((has_bounded_variation_on f) s) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((has_bounded_variation_on (\\x. (lift ((f x) $ i)))) s))))))"}
{"_id": "1406", "title": "INSIDE_EMPTY", "text": "|- ((inside EMPTY) = EMPTY)"}
{"_id": "1407", "title": "CONTINUOUS_ON_CX_LIFT", "text": "|- (!f. (!s. (((continuous_on (\\x. (Cx (f x)))) s) = ((continuous_on (\\x. (lift (f x)))) s))))"}
{"_id": "1408", "title": "HOMOTOPIC_PATHS_IMP_PATH", "text": "|- (!s. (!p. (!q. ((((homotopic_paths s) p) q) ==> ((path p) /\\ (path q))))))"}
{"_id": "1409", "title": "CONVEX_HALFSPACE_LE", "text": "|- (!a. (!b. (convex (GSPEC (\\GEN%PVAR%1428. (?x. (((SETSPEC GEN%PVAR%1428) ((real_le ((dot a) x)) b)) x)))))))"}
{"_id": "1410", "title": "ORTHOGONAL_MUL", "text": "|- ((!a. (!x. (!y. (((orthogonal ((% a) x)) y) = ((a = (real_of_num (NUMERAL _0))) \\/ ((orthogonal x) y)))))) /\\ (!a. (!x. (!y. (((orthogonal x) ((% a) y)) = ((a = (real_of_num (NUMERAL _0))) \\/ ((orthogonal x) y)))))))"}
{"_id": "1411", "title": "Seq.rot_add_mod", "text": "|- (!m. (!n. (!s. ((n <= (sizel s)) ==> ((m <= (sizel s)) ==> (((rot m) ((rot n) s)) = ((rot (((COND ((m + n) <= (sizel s))) (m + n)) ((m + n) - (sizel s)))) s)))))))"}
{"_id": "1412", "title": "REAL_MUL_RINV", "text": "|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (((real_mul x) (real_inv x)) = (real_of_num (NUMERAL (BIT1 _0))))))"}
{"_id": "1413", "title": "UNION_COMM", "text": "|- (!s. (!t. ((s UNION t) = (t UNION s))))"}
{"_id": "1414", "title": "REAL_OF_NUM_EQ", "text": "|- (!m. (!n. (((real_of_num m) = (real_of_num n)) = (m = n))))"}
{"_id": "1415", "title": "CONTINUOUS_ADD", "text": "|- (!f. (!g. (!net. ((((continuous f) net) /\\ ((continuous g) net)) ==> ((continuous (\\x. ((vector_add (f x)) (g x)))) net)))))"}
{"_id": "1416", "title": "Ssrbool.predU", "text": "|- (!p1. (!p2. (((predU p1) p2) = (\\x. ((p1 x) \\/ (p2 x))))))"}
{"_id": "1417", "title": "HOMOTOPIC_PATHS_JOIN", "text": "|- (!s. (!p. (!q. (!p'. (!q'. (((((homotopic_paths s) p) p') /\\ ((((homotopic_paths s) q) q') /\\ ((pathfinish p) = (pathstart q)))) ==> (((homotopic_paths s) ((++ p) q)) ((++ p') q'))))))))"}
{"_id": "1418", "title": "REAL_SUM_INTEGRAL_LBOUND_INCREASING", "text": "|- (!f. (!g. (!m. (!n. (((m <= n) /\\ ((!x. ((x IN (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , (real_of_num n))) NIL))) ==> (((has_real_derivative g) (f x)) ((within (atreal x)) (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , (real_of_num n))) NIL)))))) /\\ (!x. (!y. ((((real_le ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) x) /\\ (((real_le x) y) /\\ ((real_le y) (real_of_num n)))) ==> ((real_le (f x)) (f y))))))) ==> ((real_le ((real_sub (g (real_of_num n))) (g ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))) ((sum (m .. n)) (\\k. (f (real_of_num k))))))))))"}
{"_id": "1419", "title": "DIMINDEX_NONZERO", "text": "|- (!s. (~ ((dimindex s) = (NUMERAL _0))))"}
{"_id": "1420", "title": "Seq2.butlast", "text": "|- (!h. (!h2. (!t. (((butlast NIL) = NIL) /\\ (((butlast ((CONS h) NIL)) = NIL) /\\ ((butlast ((CONS h) ((CONS h2) t))) = ((CONS h) (butlast ((CONS h2) t)))))))))"}
{"_id": "1421", "title": "Ssrnat.leq_eqVlt", "text": "|- (!m. (!n. ((m <= n) = ((m = n) \\/ (m < n)))))"}
{"_id": "1422", "title": "Seq.size", "text": "|- (sizel = LENGTH)"}
{"_id": "1423", "title": "FINITE_NUMSEG", "text": "|- (!m. (!n. (FINITE (m .. n))))"}
{"_id": "1424", "title": "Ssrnat.sqrn_sub", "text": "|- (!m. (!n. ((n <= m) ==> (((m - n) EXP (NUMERAL (BIT0 (BIT1 _0)))) = (((m EXP (NUMERAL (BIT0 (BIT1 _0)))) + (n EXP (NUMERAL (BIT0 (BIT1 _0))))) - ((NUMERAL (BIT0 (BIT1 _0))) * (m * n)))))))"}
{"_id": "1425", "title": "FORALL_COMPLEX", "text": "|- ((!z. (P z)) = (!x. (!y. (P (complex (x , y))))))"}
{"_id": "1426", "title": "Add_triangle.fan_split_fan_face", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((FAN ((vec (NUMERAL _0)) , ((split_fan_face (V , E)) (v , w)))) /\\ (((dart_of_fan ((split_fan_face (V , E)) (v , w))) = ((dart1_of_fan (V , E)) UNION (((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , w) INSERT ((w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) INSERT EMPTY)))) /\\ (fully_surrounded ((split_fan_face (V , E)) (v , w))))))))))))"}
{"_id": "1427", "title": "AFFINE_HULL_EQ_EMPTY", "text": "|- (!s. ((((hull affine) s) = EMPTY) = (s = EMPTY)))"}
{"_id": "1428", "title": "CEXP_CLOG", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((cexp (clog z)) = z)))"}
{"_id": "1429", "title": "Appendix.periodic_empty", "text": "|- (!n. ((periodic EMPTY) n))"}
{"_id": "1430", "title": "Collect_geom.DELTA_RRR_INTERPRETE", "text": "|- (((((((delta r) r) r) a) b) c) = ((real_add ((real_mul (real_neg a)) ((real_mul b) c))) ((real_mul r) (((ups_x a) b) c))))"}
{"_id": "1431", "title": "MEASURE_TRANSLATION", "text": "|- (!a. (!s. ((measure ((IMAGE (\\x. ((vector_add a) x))) s)) = (measure s))))"}
{"_id": "1432", "title": "REAL_INV_0", "text": "|- ((real_inv (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))"}
{"_id": "1433", "title": "MBOUNDED", "text": "|- (!m. (!s. (((mbounded m) s) = ((s = EMPTY) \\/ ((!x. ((x IN s) ==> (x IN (mspace m)))) /\\ (?c. (?b. ((c IN (mspace m)) /\\ (!x. ((x IN s) ==> ((real_le ((mdist m) (c , x))) b)))))))))))"}
{"_id": "1434", "title": "CLOSED_CLOSURE", "text": "|- (!s. (closed (closure s)))"}
{"_id": "1435", "title": "LIM_TRANSFORM_WITHIN_OPEN", "text": "|- (!f. (!g. (!s. (!a. (!l. (((open s) /\\ ((a IN s) /\\ ((!x. (((x IN s) /\\ (~ (x = a))) ==> ((f x) = (g x)))) /\\ (((--> f) l) (at a))))) ==> (((--> g) l) (at a))))))))"}
{"_id": "1436", "title": "ADD_0", "text": "|- (!m. ((m + (NUMERAL _0)) = m))"}
{"_id": "1437", "title": "Hypermap.face_refl", "text": "|- (!H. (!x. (x IN ((face H) x))))"}
{"_id": "1438", "title": "SUPPORTING_HYPERPLANE_FRONTIER", "text": "|- (!s. (!x. (((convex s) /\\ (x IN (frontier s))) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\\ (!y. ((y IN (closure s)) ==> ((real_le ((dot a) x)) ((dot a) y)))))))))"}
{"_id": "1439", "title": "LIM_WITHIN", "text": "|- (!f. (!l. (!a. (!s. ((((--> f) l) ((within (at a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (((x IN s) /\\ (((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\\ ((real_lt (distance (x , a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))"}
{"_id": "1440", "title": "PASTECART_ADD", "text": "|- (!x1. (!y1. (!x2. (!y2. (((vector_add ((pastecart x1) y1)) ((pastecart x2) y2)) = ((pastecart ((vector_add x1) x2)) ((vector_add y1) y2)))))))"}
{"_id": "1441", "title": "RIGHT_FORALL_OR_THM", "text": "|- (!P. (!Q. ((!x. (P \\/ (Q x))) = (P \\/ (!x. (Q x))))))"}
{"_id": "1442", "title": "VECTOR_ADD_AC", "text": "|- ((((vector_add m) n) = ((vector_add n) m)) /\\ ((((vector_add ((vector_add m) n)) p) = ((vector_add m) ((vector_add n) p))) /\\ (((vector_add m) ((vector_add n) p)) = ((vector_add n) ((vector_add m) p)))))"}
{"_id": "1443", "title": "SETDIST_POS_LE", "text": "|- (!s. (!t. ((real_le (real_of_num (NUMERAL _0))) (setdist (s , t)))))"}
{"_id": "1444", "title": "SERIES_DROP_LE", "text": "|- (!f. (!g. (!s. (!a. (!b. (((((sums f) a) s) /\\ ((((sums g) b) s) /\\ (!x. ((x IN s) ==> ((real_le (drop (f x))) (drop (g x))))))) ==> ((real_le (drop a)) (drop b))))))))"}
{"_id": "1445", "title": "IN_INTERVAL", "text": "|- ((!x. ((x IN (open_interval (a , b))) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (((real_lt (a $ i)) (x $ i)) /\\ ((real_lt (x $ i)) (b $ i))))))) /\\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (((real_le (a $ i)) (x $ i)) /\\ ((real_le (x $ i)) (b $ i))))))))"}
{"_id": "1446", "title": "has_derivative", "text": "|- (!f. (!f'. (!net. ((((has_derivative f) f') net) = ((linear f') /\\ (((--> (\\y. ((% (real_inv (vector_norm ((vector_sub y) (netlimit net))))) ((vector_sub (f y)) ((vector_add (f (netlimit net))) (f' ((vector_sub y) (netlimit net)))))))) (vec (NUMERAL _0))) net))))))"}
{"_id": "1447", "title": "INTEGER_SUB", "text": "|- (!x. (!y. (((integer x) /\\ (integer y)) ==> (integer ((real_sub x) y)))))"}
{"_id": "1448", "title": "rconesgn", "text": "|- (!sgn. (!w. (!v. (!h. (((((rconesgn sgn) v) w) h) = (GSPEC (\\GEN%PVAR%5166. (?x. (((SETSPEC GEN%PVAR%5166) ((sgn ((dot ((vector_sub x) v)) ((vector_sub w) v))) ((real_mul (distance (x , v))) ((real_mul (distance (w , v))) h)))) x)))))))))"}
{"_id": "1449", "title": "OPERATIVE_FUNCTION_ENDPOINT_DIFF", "text": "|- (!f. ((operative vector_add) (\\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))))"}
{"_id": "1450", "title": "Kepler_spec.atn_1_series", "text": "|- (!n. ((real_le (real_abs ((real_sub ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) ((sum ((NUMERAL _0) .. n)) (\\i. ((real_div ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) i)) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 (BIT1 _0))))))))"}
{"_id": "1451", "title": "REAL_INTEGRABLE_SPIKE", "text": "|- (!f. (!g. (!s. (!t. (((real_negligible s) /\\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((real_integrable_on f) t) ==> ((real_integrable_on g) t)))))))"}
{"_id": "1452", "title": "Seq2.mem_delete1_uniq", "text": "|- (!x. (!s. (((uniq s) /\\ ((MEM x) s)) ==> (!y. (((MEM y) ((delete1 x) s)) = ((~ (y = x)) /\\ ((MEM y) s)))))))"}
{"_id": "1453", "title": "Ssrbool.andbA", "text": "|- (!b. (!c. (!p. ((b /\\ (c /\\ p)) = ((b /\\ c) /\\ p)))))"}
{"_id": "1454", "title": "Ssrnat.subn_eq0", "text": "|- (!m. (!n. (((m - n) = (NUMERAL _0)) = (m <= n))))"}
{"_id": "1455", "title": "REAL_LE_LSQRT", "text": "|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ (((real_le (real_of_num (NUMERAL _0))) y) /\\ ((real_le x) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((real_le (sqrt x)) y))))"}
{"_id": "1456", "title": "REAL_NEG_ADD", "text": "|- (!x. (!y. ((real_neg ((real_add x) y)) = ((real_add (real_neg x)) (real_neg y)))))"}
{"_id": "1457", "title": "INTEGRAL_COMBINE", "text": "|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\\ (((real_le (drop c)) (drop b)) /\\ ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> (((vector_add ((integral (closed_interval ((CONS (a , c)) NIL))) f)) ((integral (closed_interval ((CONS (c , b)) NIL))) f)) = ((integral (closed_interval ((CONS (a , b)) NIL))) f)))))))"}
{"_id": "1458", "title": "Pack2.PACKING", "text": "|- (!s. ((packing s) = (!u. (!v. (((u IN s) /\\ ((v IN s) /\\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v))))))"}
{"_id": "1459", "title": "COMPLEX_CMUL", "text": "|- (!c. (!x. (((% c) x) = ((complex_mul (Cx c)) x))))"}
{"_id": "1460", "title": "Kepler_spec.abs_spec", "text": "|- (!x. ((((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_abs x) = x)) /\\ (((real_lt x) (real_of_num (NUMERAL _0))) ==> ((real_abs x) = (real_neg x)))))"}
{"_id": "1461", "title": "VECTOR_VARIATION_COMBINE", "text": "|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\\ (((real_le (drop c)) (drop b)) /\\ ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> (((real_add ((vector_variation (closed_interval ((CONS (a , c)) NIL))) f)) ((vector_variation (closed_interval ((CONS (c , b)) NIL))) f)) = ((vector_variation (closed_interval ((CONS (a , b)) NIL))) f)))))))"}
{"_id": "1462", "title": "OUTSIDE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> ((outside ((IMAGE f) s)) = ((IMAGE f) (outside s))))))"}
{"_id": "1463", "title": "PERMUTATION_PERMUTES", "text": "|- (!p. ((permutation p) = (?s. ((FINITE s) /\\ ((permutes p) s)))))"}
{"_id": "1464", "title": "Appendix.scs_b_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_b_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = b))))))))))"}
{"_id": "1465", "title": "Functional_equation.sub6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((sub6 f) g) x1) x2) x3) x4) x5) x6) = ((real_sub ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "1466", "title": "PATH_CONNECTED_CONTINUOUS_IMAGE", "text": "|- (!f. (!s. ((((continuous_on f) s) /\\ (path_connected s)) ==> (path_connected ((IMAGE f) s)))))"}
{"_id": "1467", "title": "FSIGMA_PCROSS", "text": "|- (!s. (!t. (((fsigma s) /\\ (fsigma t)) ==> (fsigma (s PCROSS t)))))"}
{"_id": "1468", "title": "REAL_EXP_MONO_LE", "text": "|- (!x. (!y. (((real_le (exp x)) (exp y)) = ((real_le x) y))))"}
{"_id": "1469", "title": "TAN_PERIODIC_PI", "text": "|- (!x. ((tan ((real_add x) pi)) = (tan x)))"}
{"_id": "1470", "title": "SUBSPACE_UNIV", "text": "|- (subspace UNIV)"}
{"_id": "1471", "title": "HAS_BOUNDED_REAL_VARIATION_REFLECT_EQ", "text": "|- (!f. (!s. (((has_bounded_real_variation_on (\\x. (f (real_neg x)))) s) = ((has_bounded_real_variation_on f) ((IMAGE real_neg) s)))))"}
{"_id": "1472", "title": "MDIST", "text": "|- (!s. (!d. ((is_metric_space (s , d)) ==> ((mdist (metric (s , d))) = d))))"}
{"_id": "1473", "title": "CONTINUOUS_ON_LIFT_DOT", "text": "|- (!s. ((continuous_on (lift o (\\y. ((dot a) y)))) s))"}
{"_id": "1474", "title": "Fan.POWER_0", "text": "|- (!f. (((POWER f) (NUMERAL _0)) = I))"}
{"_id": "1475", "title": "COMPLEX_DIFFERENTIABLE_CPOW_RIGHT", "text": "|- (!w. (!z. ((complex_differentiable (\\z. ((cpow w) z))) (at z))))"}
{"_id": "1476", "title": "ENDS_IN_REAL_INTERVAL", "text": "|- ((!a. (!b. ((a IN (closed_real_interval ((CONS (a , b)) NIL))) = (~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\\ ((!a. (!b. ((b IN (closed_real_interval ((CONS (a , b)) NIL))) = (~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\\ ((!a. (!b. (~ (a IN (open_real_interval (a , b)))))) /\\ (!a. (!b. (~ (b IN (open_real_interval (a , b)))))))))"}
{"_id": "1477", "title": "Add_triangle.fully_surrounded_dart_of_fan_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((dart_of_fan (V , E)) = (dart1_of_fan (V , E)))))))"}
{"_id": "1478", "title": "LIMPT_APPROACHABLE", "text": "|- (!x. (!s. (((limit_point_of x) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?x'. ((x' IN s) /\\ ((~ (x' = x)) /\\ ((real_lt (distance (x' , x))) e)))))))))"}
{"_id": "1479", "title": "URYSOHN_LOCAL", "text": "|- (!s. (!t. (!u. (!a. (!b. ((((closed_in ((subtopology euclidean) u)) s) /\\ (((closed_in ((subtopology euclidean) u)) t) /\\ ((s INTER t) = EMPTY))) ==> (?f. (((continuous_on f) u) /\\ ((!x. ((x IN u) ==> ((f x) IN (closed_segment ((CONS (a , b)) NIL))))) /\\ ((!x. ((x IN s) ==> ((f x) = a))) /\\ (!x. ((x IN t) ==> ((f x) = b)))))))))))))"}
{"_id": "1480", "title": "REAL_LT_SUB_RADD", "text": "|- (!x. (!y. (!z. (((real_lt ((real_sub x) y)) z) = ((real_lt x) ((real_add z) y))))))"}
{"_id": "1481", "title": "inside", "text": "|- (!s. ((inside s) = (GSPEC (\\GEN%PVAR%2006. (?x. (((SETSPEC GEN%PVAR%2006) ((~ (x IN s)) /\\ (bounded ((connected_component (UNIV DIFF s)) x)))) x))))))"}
{"_id": "1482", "title": "COMPONENT_LE_NORM", "text": "|- (!x. (!i. ((real_le (real_abs (x $ i))) (vector_norm x))))"}
{"_id": "1483", "title": "OPEN_IN_PCROSS_EQ", "text": "|- (!s. (!s'. (!t. (!t'. (((open_in ((subtopology euclidean) (s PCROSS t))) (s' PCROSS t')) = ((s' = EMPTY) \\/ ((t' = EMPTY) \\/ (((open_in ((subtopology euclidean) s)) s') /\\ ((open_in ((subtopology euclidean) t)) t')))))))))"}
{"_id": "1484", "title": "ABSOLUTELY_SETCONTINUOUS_COMPARISON", "text": "|- (!f. (!g. (!s. ((((absolutely_setcontinuous_on f) s) /\\ (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((real_le (vector_norm (g (closed_interval ((CONS (a , b)) NIL))))) (vector_norm (f (closed_interval ((CONS (a , b)) NIL))))))))) ==> ((absolutely_setcontinuous_on g) s)))))"}
{"_id": "1485", "title": "Seq.belast", "text": "|- (!x. (!h. (!t. ((((belast x) NIL) = NIL) /\\ (((belast x) ((CONS h) t)) = ((CONS x) ((belast h) t)))))))"}
{"_id": "1486", "title": "SUM_0", "text": "|- (!s. (((sum s) (\\n. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0))))"}
{"_id": "1487", "title": "AGM_2", "text": "|- (!x. (!y. (!u. (!v. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ (((real_le (real_of_num (NUMERAL _0))) y) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le ((real_mul ((rpow x) u)) ((rpow y) v))) ((real_add ((real_mul u) x)) ((real_mul v) y))))))))"}
{"_id": "1488", "title": "ALL2", "text": "|- (((((ALL2 P) NIL) NIL) = T) /\\ (((((ALL2 P) ((CONS h1) t1)) NIL) = F) /\\ (((((ALL2 P) NIL) ((CONS h2) t2)) = F) /\\ ((((ALL2 P) ((CONS h1) t1)) ((CONS h2) t2)) = (((P h1) h2) /\\ (((ALL2 P) t1) t2))))))"}
{"_id": "1489", "title": "RPOW_LE2", "text": "|- (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ (((real_le x) y) /\\ ((real_le (real_of_num (NUMERAL _0))) z))) ==> ((real_le ((rpow x) z)) ((rpow y) z))))))"}
{"_id": "1490", "title": "CONIC_HULL_EXPLICIT", "text": "|- (!s. (((hull conic) s) = (GSPEC (\\GEN%PVAR%1457. (?c. (?x. (((SETSPEC GEN%PVAR%1457) (((real_le (real_of_num (NUMERAL _0))) c) /\\ (x IN s))) ((% c) x))))))))"}
{"_id": "1491", "title": "Ssrnat.add0n", "text": "|- (!n. (((NUMERAL _0) + n) = n))"}
{"_id": "1492", "title": "BALL_SUBSET_CBALL", "text": "|- (!x. (!e. ((ball (x , e)) SUBSET (cball (x , e)))))"}
{"_id": "1493", "title": "ACS_COS", "text": "|- (!x. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_le x) pi)) ==> ((acs (cos x)) = x)))"}
{"_id": "1494", "title": "LENGTH_LIST_OF_SET", "text": "|- (!s. ((FINITE s) ==> ((LENGTH (list_of_set s)) = (CARD s))))"}
{"_id": "1495", "title": "CARD_EXP_ABSORB", "text": "|- (!s. (!t. (((INFINITE t) /\\ ((UNIV <=_c s) /\\ (s <=_c ((^_c UNIV) t)))) ==> (((^_c s) t) =_c ((^_c UNIV) t)))))"}
{"_id": "1496", "title": "HAS_REAL_INTEGRAL_COMBINE", "text": "|- (!f. (!i. (!j. (!a. (!b. (!c. ((((real_le a) c) /\\ (((real_le c) b) /\\ ((((has_real_integral f) i) (closed_real_interval ((CONS (a , c)) NIL))) /\\ (((has_real_integral f) j) (closed_real_interval ((CONS (c , b)) NIL)))))) ==> (((has_real_integral f) ((real_add i) j)) (closed_real_interval ((CONS (a , b)) NIL))))))))))"}
{"_id": "1497", "title": "BOUNDED_INSERT", "text": "|- (!x. (!s. ((bounded (x INSERT s)) = (bounded s))))"}
{"_id": "1498", "title": "CARD_LE_LT", "text": "|- (!s. (!t. ((s <=_c t) = ((s <_c t) \\/ (s =_c t)))))"}
{"_id": "1499", "title": "ABSOLUTELY_INTEGRABLE_CONVOLUTION_L1_LINF", "text": "|- (!bop. (!f. (!g. (!x. (((bilinear bop) /\\ (((absolutely_integrable_on f) UNIV) /\\ (((measurable_on g) UNIV) /\\ (bounded ((IMAGE g) UNIV))))) ==> ((absolutely_integrable_on (\\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))"}
{"_id": "1500", "title": "LINEAR_ID", "text": "|- (linear (\\x. x))"}
{"_id": "1501", "title": "CONTINUOUS_COUNTABLE_RANGE_CONSTANT_EQ", "text": "|- (!s. ((connected s) = (!f. ((((continuous_on f) s) /\\ (COUNTABLE ((IMAGE f) s))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a))))))))"}
{"_id": "1502", "title": "REAL_EQ_LDIV_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> ((((real_div x) z) = y) = (x = ((real_mul y) z)))))))"}
{"_id": "1503", "title": "Conforming.SIGMA_FAN_OF_FANADD_AT_POINT5", "text": "|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\\ ((FAN (x , (V , E1))) /\\ ((fan80 (x , (V , E))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ ((~ (w1 = (((((inverse1_sigma_fan x) V) E) w) u))) /\\ (((w INSERT (w1 INSERT EMPTY)) IN E) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))) ==> ((((((sigma_fan x) V) E1) w) w1) = (((((sigma_fan x) V) E) w) w1))))))))))"}
{"_id": "1504", "title": "DIAMETER_EMPTY", "text": "|- ((diameter EMPTY) = (real_of_num (NUMERAL _0)))"}
{"_id": "1505", "title": "vector_add", "text": "|- (!x. (!y. (((vector_add x) y) = (lambdai. ((real_add (x $ i)) (y $ i))))))"}
{"_id": "1506", "title": "AFFINE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((affine ((IMAGE f) s)) = (affine s)))))"}
{"_id": "1507", "title": "DOT_RZERO", "text": "|- (!x. (((dot x) (vec (NUMERAL _0))) = (real_of_num (NUMERAL _0))))"}
{"_id": "1508", "title": "Hypermap_iso.ext_dart", "text": "|- (!V. (!E. (!v. (!w. (((ext_dart (V , E)) (v , w)) = ((vec (NUMERAL _0)) , (v , (w , (((((extension_sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))"}
{"_id": "1509", "title": "CLOSURE_UNIONS", "text": "|- (!f. ((FINITE f) ==> ((closure (UNIONS f)) = (UNIONS (GSPEC (\\GEN%PVAR%573. (?s. (((SETSPEC GEN%PVAR%573) (s IN f)) (closure s)))))))))"}
{"_id": "1510", "title": "REAL_CONTINUOUS_WITHIN", "text": "|- (!f. (!x. (!s. (((real_continuous f) ((within (at x)) s)) = (((---> f) (f x)) ((within (at x)) s))))))"}
{"_id": "1511", "title": "NADD_LE_LMUL", "text": "|- (!x. (!y. (!z. (((nadd_le y) z) ==> ((nadd_le ((nadd_mul x) y)) ((nadd_mul x) z))))))"}
{"_id": "1512", "title": "Functional_equation.constant6", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((((((((constant6 c) x1) x2) x3) x4) x5) x6) = c))))))))"}
{"_id": "1513", "title": "OPEN_IMP_LOCALLY_PATH_CONNECTED", "text": "|- (!s. ((open s) ==> ((locally path_connected) s)))"}
{"_id": "1514", "title": "Lp_main_estimate.quad_diags_imp", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (!a. (!b. ((LET (\\f. (LET_END ((LET (\\w'. (LET_END ((LET (\\u. (LET_END ((((v , w) IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) /\\ ((~ (a = b)) /\\ ((a IN ((v_prime V) f)) /\\ ((b IN ((v_prime V) f)) /\\ (~ ((a INSERT (b INSERT EMPTY)) IN ((e_prime E) f))))))) ==> ((((a , b) = (w , w')) \\/ ((a , b) = (w' , w))) \\/ (((a , b) = (v , u)) \\/ ((a , b) = (u , v)))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))"}
{"_id": "1515", "title": "SUM_4", "text": "|- (!t. (((sum ((NUMERAL (BIT1 _0)) .. (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) t) = ((real_add (t (NUMERAL (BIT1 _0)))) ((real_add (t (NUMERAL (BIT0 (BIT1 _0))))) ((real_add (t (NUMERAL (BIT1 (BIT1 _0))))) (t (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))"}
{"_id": "1516", "title": "ANALYTIC_ON_COMPOSE", "text": "|- (!f. (!g. (!s. ((((analytic_on f) s) /\\ ((analytic_on g) ((IMAGE f) s))) ==> ((analytic_on (g o f)) s)))))"}
{"_id": "1517", "title": "CLOSED_INJECTIVE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((closed ((IMAGE f) s)) = (closed s)))))"}
{"_id": "1518", "title": "DIM_UNIQUE", "text": "|- (!v. (!b. (((b SUBSET v) /\\ ((v SUBSET (span b)) /\\ ((independent b) /\\ (b HAS_SIZE n)))) ==> ((dim v) = n))))"}
{"_id": "1519", "title": "BOOL_CASES_AX", "text": "|- (!t. ((t = T) \\/ (t = F)))"}
{"_id": "1520", "title": "TRACE_MATRIX_INV_LMUL", "text": "|- (!A. ((trace ((matrix_mul (matrix_inv A)) A)) = (real_of_num (rank A))))"}
{"_id": "1521", "title": "INTEGRABLE_SPIKE", "text": "|- (!f. (!g. (!s. (!t. (((negligible s) /\\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((integrable_on f) t) ==> ((integrable_on g) t)))))))"}
{"_id": "1522", "title": "IMAGE_EQ_EMPTY", "text": "|- (!f. (!s. ((((IMAGE f) s) = EMPTY) = (s = EMPTY))))"}
{"_id": "1523", "title": "MATCH_SEQPATTERN", "text": "|- (((_MATCH x) ((_SEQPATTERN r) s)) = (((COND (?y. ((r x) y))) ((_MATCH x) r)) ((_MATCH x) s)))"}
{"_id": "1524", "title": "CONVEX_IMP_SIMPLY_CONNECTED", "text": "|- (!s. ((convex s) ==> (simply_connected s)))"}
{"_id": "1525", "title": "ORTHOGONAL_TRANSFORMATION", "text": "|- (!f. ((orthogonal_transformation f) = ((linear f) /\\ (!v. ((vector_norm (f v)) = (vector_norm v))))))"}
{"_id": "1526", "title": "Lp_ineqs_proofs.y4'_eq_y4", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((y4'_fan (V , E)) d) = ((y4_fan (V , E)) d))))))))))"}
{"_id": "1527", "title": "Appendix.scs_am_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_am_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = alpha))))))))))"}
{"_id": "1528", "title": "Add_triangle.split_fan_face", "text": "|- (!V. (!v. (!w. (!E. (((split_fan_face (V , E)) (v , w)) = (V , ((w INSERT ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) INSERT EMPTY)) INSERT E)))))))"}
{"_id": "1529", "title": "LIM_AT_INFINITY_WITHIN", "text": "|- (!f. (!l. (!s. ((((--> f) l) ((within at_infinity) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((x IN s) /\\ ((real_ge (vector_norm x)) b)) ==> ((real_lt (distance ((f x) , l))) e))))))))))"}
{"_id": "1530", "title": "ARZELA_ASCOLI", "text": "|- (!f. (!s. (!M. (((compact s) /\\ ((!n. (!x. ((x IN s) ==> ((real_le (vector_norm ((f n) x))) M)))) /\\ (!x. (!e. (((x IN s) /\\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!n. (!y. (((y IN s) /\\ ((real_lt (vector_norm ((vector_sub x) y))) d)) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) e))))))))))) ==> (?g. (((continuous_on g) s) /\\ (?r. ((!m. (!n. ((m < n) ==> ((r m) < (r n))))) /\\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\\ (x IN s)) ==> ((real_lt (vector_norm ((vector_sub ((f (r n)) x)) (g x)))) e)))))))))))))))"}
{"_id": "1531", "title": "real_min", "text": "|- (!m. (!n. (((real_min m) n) = (((COND ((real_le m) n)) m) n))))"}
{"_id": "1532", "title": "Nonlin_def.add6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((add6 f) g) x1) x2) x3) x4) x5) x6) = ((real_add ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "1533", "title": "REAL_LE_ADD2", "text": "|- (!w. (!x. (!y. (!z. ((((real_le w) x) /\\ ((real_le y) z)) ==> ((real_le ((real_add w) y)) ((real_add x) z)))))))"}
{"_id": "1534", "title": "REAL_SGN_NEG", "text": "|- (!x. ((real_sgn (real_neg x)) = (real_neg (real_sgn x))))"}
{"_id": "1535", "title": "COND_ELIM_THM", "text": "|- ((P (((COND c) x) y)) = ((c ==> (P x)) /\\ ((~ c) ==> (P y))))"}
{"_id": "1536", "title": "Lp_ineqs_proofs.y9_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y9_fan (V , E)) d))))))))))"}
{"_id": "1537", "title": "CONTINUOUS_ON_ID", "text": "|- (!s. ((continuous_on (\\x. x)) s))"}
{"_id": "1538", "title": "transp", "text": "|- (!A. ((transp A) = (lambdai. (lambdaj. ((A $ j) $ i)))))"}
{"_id": "1539", "title": "ARC_SIMPLE_PATH_SUBPATH", "text": "|- (!g. (!u. (!v. (((simple_path g) /\\ ((u IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ ((v IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ (~ ((g u) = (g v)))))) ==> (arc (((subpath u) v) g))))))"}
{"_id": "1540", "title": "CONTINUOUS_INJECTIVE_IMAGE_SEGMENT_1", "text": "|- (!f. (!a. (!b. ((((continuous_on f) (closed_segment ((CONS (a , b)) NIL))) /\\ (!x. (!y. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\\ ((y IN (closed_segment ((CONS (a , b)) NIL))) /\\ ((f x) = (f y)))) ==> (x = y))))) ==> (((IMAGE f) (closed_segment ((CONS (a , b)) NIL))) = (closed_segment ((CONS ((f a) , (f b))) NIL)))))))"}
{"_id": "1541", "title": "Hypermap.face_map_and_darts", "text": "|- (!H. ((FINITE (dart H)) /\\ ((permutes (face_map H)) (dart H))))"}
{"_id": "1542", "title": "bounded", "text": "|- (!s. ((bounded s) = (?a. (!x. ((x IN s) ==> ((real_le (vector_norm x)) a))))))"}
{"_id": "1543", "title": "linear", "text": "|- (!f. ((linear f) = ((!x. (!y. ((f ((vector_add x) y)) = ((vector_add (f x)) (f y))))) /\\ (!c. (!x. ((f ((% c) x)) = ((% c) (f x))))))))"}
{"_id": "1544", "title": "RSTC", "text": "|- (!R. ((RSTC R) = (RC (TC (SC R)))))"}
{"_id": "1545", "title": "ABSOLUTELY_REAL_INTEGRABLE_ADD", "text": "|- (!f. (!g. (!s. ((((absolutely_real_integrable_on f) s) /\\ ((absolutely_real_integrable_on g) s)) ==> ((absolutely_real_integrable_on (\\x. ((real_add (f x)) (g x)))) s)))))"}
{"_id": "1546", "title": "EDGE_OF_TRANSLATION_EQ", "text": "|- (!a. (!f. (!s. (((edge_of ((IMAGE (\\x. ((vector_add a) x))) f)) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((edge_of f) s)))))"}
{"_id": "1547", "title": "ITERATE_SUPPORT", "text": "|- (!op. (!f. (!s. ((((iterate op) (((support op) f) s)) f) = (((iterate op) s) f)))))"}
{"_id": "1548", "title": "MEASURABLE_ON_CONVOLUTION", "text": "|- (!bop. (!f. (!g. (!x. (((bilinear bop) /\\ (((measurable_on f) UNIV) /\\ ((measurable_on g) UNIV))) ==> ((measurable_on (\\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))"}
{"_id": "1549", "title": "VECTOR_ADD_COMPONENT", "text": "|- (!x. (!y. (!i. ((((vector_add x) y) $ i) = ((real_add (x $ i)) (y $ i))))))"}
{"_id": "1550", "title": "real_bounded", "text": "|- (!s. ((real_bounded s) = (?B. (!x. ((x IN s) ==> ((real_le (real_abs x)) B))))))"}
{"_id": "1551", "title": "NEGLIGIBLE_COUNTABLE", "text": "|- (!s. ((COUNTABLE s) ==> (negligible s)))"}
{"_id": "1552", "title": "CSIN_CIRCLE", "text": "|- (!z. (((vector_add ((complex_pow (csin z)) (NUMERAL (BIT0 (BIT1 _0))))) ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0))))) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))"}
{"_id": "1553", "title": "SURJECTIVE_IFF_INJECTIVE_GEN", "text": "|- (!s. (!t. (!f. (((FINITE s) /\\ ((FINITE t) /\\ (((CARD s) = (CARD t)) /\\ (((IMAGE f) s) SUBSET t)))) ==> ((!y. ((y IN t) ==> (?x. ((x IN s) /\\ ((f x) = y))))) = (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))))))))"}
{"_id": "1554", "title": "OPEN_TRANSLATION_EQ", "text": "|- (!a. (!s. ((open ((IMAGE (\\x. ((vector_add a) x))) s)) = (open s))))"}
{"_id": "1555", "title": "real_gt", "text": "|- (!y. (!x. (((real_gt x) y) = ((real_lt y) x))))"}
{"_id": "1556", "title": "CONVEX_CONE_PCROSS", "text": "|- (!s. (!t. (((convex_cone s) /\\ (convex_cone t)) ==> (convex_cone (s PCROSS t)))))"}
{"_id": "1557", "title": "HAS_DERIVATIVE_INVERSE_STRONG", "text": "|- (!f. (!g. (!f'. (!g'. (!s. (!x. (((open s) /\\ ((x IN s) /\\ (((continuous_on f) s) /\\ ((!x. ((x IN s) ==> ((g (f x)) = x))) /\\ ((((has_derivative f) f') (at x)) /\\ ((f' o g') = I)))))) ==> (((has_derivative g) g') (at (f x))))))))))"}
{"_id": "1558", "title": "ALTERNATING_SUM_BOUND", "text": "|- (!a. (((!n. ((real_le (real_abs (a (SUC n)))) (real_abs (a n)))) /\\ ((!n. ((EVEN n) ==> ((real_le (real_of_num (NUMERAL _0))) (a n)))) /\\ (!n. ((ODD n) ==> ((real_le (a n)) (real_of_num (NUMERAL _0))))))) ==> (!m. (!n. ((real_le (real_abs ((sum (m .. n)) a))) (real_abs (a m)))))))"}
{"_id": "1559", "title": "Hypermap_iso.bij_res_inv", "text": "|- (!f. (!s. (!t. ((((BIJ f) s) t) ==> (((BIJ ((res_inv f) s)) t) s)))))"}
{"_id": "1560", "title": "Seq.map_rot", "text": "|- (!n0. (!f. (!s. (((map f) ((rot n0) s)) = ((rot n0) ((map f) s))))))"}
{"_id": "1561", "title": "inverse", "text": "|- (!f. ((inverse f) = (\\y. (@x. ((f x) = y)))))"}
{"_id": "1562", "title": "AFFINE_HULL_FINITE", "text": "|- (!s. (((hull affine) s) = (GSPEC (\\GEN%PVAR%1412. (?y. (((SETSPEC GEN%PVAR%1412) (?u. ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((vsum s) (\\v. ((% (u v)) v))) = y)))) y))))))"}
{"_id": "1563", "title": "CEXP_EULER", "text": "|- (!z. ((cexp ((complex_mul ii) z)) = ((vector_add (ccos z)) ((complex_mul ii) (csin z)))))"}
{"_id": "1564", "title": "NOT_BOUNDED_UNIV", "text": "|- (~ (bounded UNIV))"}
{"_id": "1565", "title": "Sphere.rhazim3", "text": "|- (rhazim3 = (node3_y rhazim))"}
{"_id": "1566", "title": "Fan.POWER_1", "text": "|- (!f. (((POWER f) (NUMERAL (BIT1 _0))) = f))"}
{"_id": "1567", "title": "COMPLEX_DERIVATIVE_LMUL", "text": "|- (!f. (!c. (!z. (((complex_differentiable f) (at z)) ==> (((complex_derivative (\\w. ((complex_mul c) (f w)))) z) = ((complex_mul c) ((complex_derivative f) z)))))))"}
{"_id": "1568", "title": "COVERING_SPACE_HOMEOMORPHISM", "text": "|- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\\ ((path_connected c) /\\ (simply_connected s))) ==> (?q. ((homeomorphism (c , s)) (p , q)))))))"}
{"_id": "1569", "title": "SUBSET_INTER", "text": "|- (!s. (!t. (!u. ((s SUBSET (t INTER u)) = ((s SUBSET t) /\\ (s SUBSET u))))))"}
{"_id": "1570", "title": "CFUNSPACE_IMP_BOUNDED2", "text": "|- (!top. (!m. (!f. (!g. (((f IN (mspace ((cfunspace top) m))) /\\ (g IN (mspace ((cfunspace top) m)))) ==> (?b. (!x. ((x IN (topspace top)) ==> ((real_le ((mdist m) ((f x) , (g x)))) b)))))))))"}
{"_id": "1571", "title": "LIPSCHITZ_REAL_POLYNOMIAL_FUNCTION", "text": "|- (!f. (!s. (((real_polynomial_function f) /\\ (bounded s)) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\\ (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (real_abs ((real_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y))))))))))))"}
{"_id": "1572", "title": "PRESERVES_NORM_PRESERVES_DOT", "text": "|- (!f. (!x. (!y. (((linear f) /\\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((dot (f x)) (f y)) = ((dot x) y))))))"}
{"_id": "1573", "title": "Ssrnat.factS", "text": "|- (!n. ((FACT (SUC n)) = ((SUC n) * (FACT n))))"}
{"_id": "1574", "title": "REAL_SUB_ADD", "text": "|- (!x. (!y. (((real_add ((real_sub x) y)) y) = x)))"}
{"_id": "1575", "title": "EVENTUALLY_AT_NEGINFINITY", "text": "|- (!p. (((eventually p) at_neginfinity) = (?b. (!x. (((real_le x) b) ==> (p x))))))"}
{"_id": "1576", "title": "CONNECTED_INTER_FRONTIER", "text": "|- (!s. (!t. (((connected s) /\\ ((~ ((s INTER t) = EMPTY)) /\\ (~ ((s DIFF t) = EMPTY)))) ==> (~ ((s INTER (frontier t)) = EMPTY)))))"}
{"_id": "1577", "title": "PROPER_LOCALLY_INJECTIVE_OPEN_IMP_COVERING_MAP", "text": "|- (!p. (!c. (!s. ((((continuous_on p) c) /\\ ((((IMAGE p) c) = s) /\\ ((!k. (((k SUBSET s) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%2101. (?x. (((SETSPEC GEN%PVAR%2101) ((x IN c) /\\ ((p x) IN k))) x))))))) /\\ ((!u. (((open_in ((subtopology euclidean) c)) u) ==> ((open_in ((subtopology euclidean) s)) ((IMAGE p) u)))) /\\ (!x. ((x IN c) ==> (?t. ((x IN t) /\\ (((open_in ((subtopology euclidean) c)) t) /\\ (!y. (!z. (((y IN t) /\\ ((z IN t) /\\ ((p y) = (p z)))) ==> (y = z))))))))))))) ==> ((covering_space (c , p)) s)))))"}
{"_id": "1578", "title": "REAL_CONVEX_ON_SUBSET", "text": "|- (!f. (!s. (!t. ((((real_convex_on f) t) /\\ (s SUBSET t)) ==> ((real_convex_on f) s)))))"}
{"_id": "1579", "title": "Fan.FAN", "text": "|- (!x. (!V. (!E. ((FAN (x , (V , E))) = (((UNIONS E) SUBSET V) /\\ ((graph E) /\\ ((fan1 (x , (V , E))) /\\ ((fan2 (x , (V , E))) /\\ ((fan6 (x , (V , E))) /\\ (fan7 (x , (V , E))))))))))))"}
{"_id": "1580", "title": "Appendix.is_scs_v39", "text": "|- (!s. ((is_scs_v39 s) = (((real_lt (scs_d_v39 s)) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 _0)))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) /\\ (((NUMERAL (BIT1 (BIT1 _0))) <= (scs_k_v39 s)) /\\ (((scs_k_v39 s) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\\ (((periodic (scs_lo_v39 s)) (scs_k_v39 s)) /\\ (((periodic (scs_hi_v39 s)) (scs_k_v39 s)) /\\ (((periodic (scs_str_v39 s)) (scs_k_v39 s)) /\\ (((periodic (scs_str_v39 s)) (scs_k_v39 s)) /\\ (((periodic2 (scs_a_v39 s)) (scs_k_v39 s)) /\\ (((periodic2 (scs_am_v39 s)) (scs_k_v39 s)) /\\ (((periodic2 (scs_bm_v39 s)) (scs_k_v39 s)) /\\ (((periodic2 (scs_b_v39 s)) (scs_k_v39 s)) /\\ (((periodic2 (scs_J_v39 s)) (scs_k_v39 s)) /\\ ((!i. (!j. (((((scs_a_v39 s) i) j) = (((scs_a_v39 s) j) i)) /\\ (((((scs_am_v39 s) i) j) = (((scs_am_v39 s) j) i)) /\\ (((((scs_bm_v39 s) i) j) = (((scs_bm_v39 s) j) i)) /\\ (((((scs_b_v39 s) i) j) = (((scs_b_v39 s) j) i)) /\\ ((((scs_J_v39 s) i) j) = (((scs_J_v39 s) j) i)))))))) /\\ ((!i. (!j. (((real_le (((scs_a_v39 s) i) j)) (((scs_am_v39 s) i) j)) /\\ (((real_le (((scs_am_v39 s) i) j)) (((scs_bm_v39 s) i) j)) /\\ ((real_le (((scs_bm_v39 s) i) j)) (((scs_b_v39 s) i) j)))))) /\\ ((!i. ((((scs_a_v39 s) i) i) = (real_of_num (NUMERAL _0)))) /\\ ((!i. (!j. (((i < (scs_k_v39 s)) /\\ ((j < (scs_k_v39 s)) /\\ (~ (i = j)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (((scs_a_v39 s) i) j))))) /\\ ((!i. (((scs_k_v39 s) = (NUMERAL (BIT1 (BIT1 _0)))) ==> ((real_lt (((scs_b_v39 s) i) (SUC i))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))) /\\ ((!i. (((NUMERAL (BIT1 (BIT1 _0))) < (scs_k_v39 s)) ==> ((real_le (((scs_b_v39 s) i) (SUC i))) cstab))) /\\ ((!i. (!j. ((((scs_J_v39 s) i) j) ==> (((j MOD (scs_k_v39 s)) = ((SUC i) MOD (scs_k_v39 s))) \\/ ((i MOD (scs_k_v39 s)) = ((SUC j) MOD (scs_k_v39 s))))))) /\\ ((!i. (!j. ((((scs_J_v39 s) i) j) ==> (((((scs_a_v39 s) i) j) = sqrt8) /\\ ((((scs_b_v39 s) i) j) = cstab))))) /\\ (((CARD (GSPEC (\\GEN%PVAR%7668. (?i. (((SETSPEC GEN%PVAR%7668) ((i < (scs_k_v39 s)) /\\ (((real_lt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (((scs_b_v39 s) i) (SUC i))) \\/ ((real_lt (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (((scs_a_v39 s) i) (SUC i)))))) i))))) + (scs_k_v39 s)) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))))))))))"}
{"_id": "1581", "title": "REAL_POW_EQ_ABS", "text": "|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\\ (((real_pow x) n) = ((real_pow y) n))) ==> ((real_abs x) = (real_abs y))))))"}
{"_id": "1582", "title": "CONNECTED_SUBSET_SEGMENT", "text": "|- (!s. (!a. (!b. (((connected s) /\\ ((s SUBSET (closed_segment ((CONS (a , b)) NIL))) /\\ ((a IN s) /\\ (b IN s)))) ==> (s = (closed_segment ((CONS (a , b)) NIL)))))))"}
{"_id": "1583", "title": "Tame_defs.contravening", "text": "|- (!V. ((contravening V) = ((packing V) /\\ ((V SUBSET ball_annulus) /\\ (((real_gt (scriptL V)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) /\\ ((!W. (((packing W) /\\ (W SUBSET ball_annulus)) ==> ((real_le (scriptL W)) (scriptL V)))) /\\ ((((CARD V) = (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 _0)))))) \\/ (((CARD V) = (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 _0)))))) \\/ ((CARD V) = (NUMERAL (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))) /\\ ((!v. ((v IN V) ==> ((surrounded_node (V , (ESTD V))) v))) /\\ (!v. ((v IN V) ==> (((surrounded_node (V , (ECTC V))) v) \\/ ((vector_norm v) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))))))"}
{"_id": "1584", "title": "Calc_derivative.ratform", "text": "|- (!p. (!r. (!a. (!b. (((((ratform p) r) a) b) = (p ==> ((~ (b = (real_of_num (NUMERAL _0)))) /\\ (r = ((real_div a) b)))))))))"}
{"_id": "1585", "title": "EVENTUALLY_MONO", "text": "|- (!net. (!p. (!q. (((!x. ((p x) ==> (q x))) /\\ ((eventually p) net)) ==> ((eventually q) net)))))"}
{"_id": "1586", "title": "IN_CLOSURE_DELETE", "text": "|- (!s. (!x. ((x IN (closure (s DELETE x))) = ((limit_point_of x) s))))"}
{"_id": "1587", "title": "Trigonometry.UTNKIAC", "text": "|- (!x. (!y. ((((real_lt (real_neg ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) x) /\\ (((real_lt x) y) /\\ ((real_lt y) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) ==> ((real_lt (tan x)) (tan y)))))"}
{"_id": "1588", "title": "LEBESGUE_DENSITY_THEOREM_LIFT_CBALL", "text": "|- (!s. ((lebesgue_measurable s) ==> (?k. ((negligible k) /\\ (!x. ((~ (x IN k)) ==> (((--> (\\e. (lift ((real_div (measure (s INTER (cball (x , (drop e)))))) (measure (cball (x , (drop e)))))))) (((COND (x IN s)) (vec (NUMERAL (BIT1 _0)))) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\\GEN%PVAR%3726. (?t. (((SETSPEC GEN%PVAR%3726) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t))))))))))))"}
{"_id": "1589", "title": "Conforming.SUM_SOL_IN_FACE_SET_EQ_4PI", "text": "|- (!x. (!V. (!E. (((FAN (x , (V , E))) /\\ (conforming_fan (x , (V , E)))) ==> (((sum (face_set (hypermap1_of_fanx (x , (V , E))))) (\\f. ((sol x) ((((dartset_leads_into_fan x) V) E) f)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))"}
{"_id": "1590", "title": "VECTOR_SUB_REFL", "text": "|- (!x. (((vector_sub x) x) = (vec (NUMERAL _0))))"}
{"_id": "1591", "title": "IMAGE_LIFT_DROP", "text": "|- ((!s. (((IMAGE (lift o drop)) s) = s)) /\\ (!s. (((IMAGE (drop o lift)) s) = s)))"}
{"_id": "1592", "title": "IN_RELATIVE_INTERIOR", "text": "|- (!x. (!s. ((x IN (relative_interior s)) = ((x IN s) /\\ (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (((ball (x , e)) INTER ((hull affine) s)) SUBSET s)))))))"}
{"_id": "1593", "title": "REAL_CONTINUOUS_CONST", "text": "|- (!net. (!c. ((real_continuous (\\x. c)) net)))"}
{"_id": "1594", "title": "COMPONENTS_EQ_SING", "text": "|- (!s. (((components s) = (s INSERT EMPTY)) = ((connected s) /\\ (~ (s = EMPTY)))))"}
{"_id": "1595", "title": "MULT_ASSOC", "text": "|- (!m. (!n. (!p. ((m * (n * p)) = ((m * n) * p)))))"}
{"_id": "1596", "title": "RANK_CMUL", "text": "|- (!A. (!c. ((rank ((%% c) A)) = (((COND (c = (real_of_num (NUMERAL _0)))) (NUMERAL _0)) (rank A)))))"}
{"_id": "1597", "title": "Planarity.dartset_leads_into_is_topological_component_yfan", "text": "|- (!x. (!V. (!E. (!ds. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((fan80 (x , (V , E))) /\\ (ds IN (face_set (hypermap1_of_fanx (x , (V , E)))))))) ==> (((((dartset_leads_into_fan x) V) E) ds) IN (topological_component_yfan (x , (V , E)))))))))"}
{"_id": "1598", "title": "Dih2k_hypermap.INDEX_VECMAT", "text": "|- (((i < (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= i') /\\ (i' <= (dimindex UNIV)))) ==> (((NUMERAL (BIT1 _0)) <= ((i * (dimindex UNIV)) + i')) /\\ (((i * (dimindex UNIV)) + i') <= (dimindex UNIV))))"}
{"_id": "1599", "title": "LIMPT_TRANSLATION_EQ", "text": "|- (!a. (!s. (!x. (((limit_point_of ((vector_add a) x)) ((IMAGE (\\y. ((vector_add a) y))) s)) = ((limit_point_of x) s)))))"}
{"_id": "1600", "title": "minimal", "text": "|- (!P. ((minimal P) = (@n. ((P n) /\\ (!m. ((m < n) ==> (~ (P m))))))))"}
{"_id": "1601", "title": "HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS_IMAGE", "text": "|- (!f. (!s. (((FINITE s) /\\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ (~ (x = y)))) ==> (real_negligible ((f x) INTER (f y)))))))) ==> ((has_real_measure (UNIONS ((IMAGE f) s))) ((sum s) (\\x. (real_measure (f x))))))))"}
{"_id": "1602", "title": "LINEAR_INDEPENDENT_EXTEND", "text": "|- (!f. (!b. ((independent b) ==> (?g. ((linear g) /\\ (!x. ((x IN b) ==> ((g x) = (f x)))))))))"}
{"_id": "1603", "title": "GT", "text": "|- (!n. (!m. ((m > n) = (n < m))))"}
{"_id": "1604", "title": "IN_UNIONS", "text": "|- (!s. (!x. ((x IN (UNIONS s)) = (?t. ((t IN s) /\\ (x IN t))))))"}
{"_id": "1605", "title": "OPEN_CONVEX_HULL", "text": "|- (!s. ((open s) ==> (open ((hull convex) s))))"}
{"_id": "1606", "title": "MAT_CMUL", "text": "|- (!a. ((mat a) = ((%% (real_of_num a)) (mat (NUMERAL (BIT1 _0))))))"}
{"_id": "1607", "title": "REAL_ABS_SIGN2", "text": "|- (!x. (!y. (((real_lt (real_abs ((real_sub x) y))) (real_neg y)) ==> ((real_lt x) (real_of_num (NUMERAL _0))))))"}
{"_id": "1608", "title": "CONVEX_CONE_CONVEX_CONE_HULL", "text": "|- (!s. (convex_cone ((hull convex_cone) s)))"}
{"_id": "1609", "title": "LIM_WITHINREAL_WITHIN", "text": "|- ((((--> f) a) ((within (atreal x)) s)) = (((--> (f o drop)) a) ((within (at (lift x))) ((IMAGE lift) s))))"}
{"_id": "1610", "title": "COMPLEX_NEG_SUB", "text": "|- (!x. (!y. ((vector_neg ((vector_sub x) y)) = ((vector_sub y) x))))"}
{"_id": "1611", "title": "IMAGE_I", "text": "|- (!s. (((IMAGE I) s) = s))"}
{"_id": "1612", "title": "LINEAR_NEGATION", "text": "|- (linear vector_neg)"}
{"_id": "1613", "title": "CONTINUOUS_WITHIN_SQRT_COMPOSE", "text": "|- (!f. (!s. (!a. ((((continuous (\\x. (lift (f x)))) ((within (at a)) s)) /\\ (((real_lt (real_of_num (NUMERAL _0))) (f a)) \\/ (!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))))) ==> ((continuous (\\x. (lift (sqrt (f x))))) ((within (at a)) s))))))"}
{"_id": "1614", "title": "COMPACT_EQ_BOUNDED_CLOSED", "text": "|- (!s. ((compact s) = ((bounded s) /\\ (closed s))))"}
{"_id": "1615", "title": "Collect_geom.COPLANAR_3", "text": "|- (!a. (!b. (!c. (((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) ==> (coplanar_alt (a INSERT (b INSERT (c INSERT EMPTY))))))))"}
{"_id": "1616", "title": "SUBSET_IMAGE", "text": "|- (!f. (!s. (!t. ((s SUBSET ((IMAGE f) t)) = (?u. ((u SUBSET t) /\\ (s = ((IMAGE f) u))))))))"}
{"_id": "1617", "title": "Functional_equation.functional_taum_x", "text": "|- (taum_x = ((add6 rhazim_x) ((add6 rhazim2_x) ((sub6 rhazim3_x) (constant6 ((real_mul ((real_add (real_of_num (NUMERAL (BIT1 _0)))) const1)) pi))))))"}
{"_id": "1618", "title": "Collect_geom.RADV_FORMULAR", "text": "|- (!va. (!vb. (!vc. ((~ (collinear (va INSERT (vb INSERT (vc INSERT EMPTY))))) ==> ((radV (va INSERT (vb INSERT (vc INSERT EMPTY)))) = (((eta_y ((dist3 vb) vc)) ((dist3 va) vc)) ((dist3 va) vb)))))))"}
{"_id": "1619", "title": "Pack1.fcc_compatible", "text": "|- (!S. (!f. (((fcc_compatible f) S) = (!v. ((v IN S) ==> ((real_le (sqrt (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_add (measure ((voronoi_open S) v))) (f v))))))))"}
{"_id": "1620", "title": "REAL_OPEN_SET_RATIONAL", "text": "|- (!s. (((real_open s) /\\ (~ (s = EMPTY))) ==> (?x. ((rational x) /\\ (x IN s)))))"}
{"_id": "1621", "title": "FROM_INTER_NUMSEG_MAX", "text": "|- (!m. (!n. (!p. (((from p) INTER (m .. n)) = (((MAX p) m) .. n)))))"}
{"_id": "1622", "title": "Lp_ineqs_proofs.dart4_y_invf_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((inverse (f_fan_pair_ext (V , E))) d)) = ((y3_fan (V , E)) d)) /\\ (((y2_fan ((inverse (f_fan_pair_ext (V , E))) d)) = (y1_fan d)) /\\ ((((y3_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y7_fan (V , E)) d)) /\\ ((((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y8_fan (V , E)) d)) /\\ (((y6_fan ((inverse (f_fan_pair_ext (V , E))) d)) = ((y5_fan (V , E)) d)) /\\ ((((y7_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = (y2_fan d)) /\\ ((((y8_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y9_fan (V , E)) d)) /\\ (((y9_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = (y6_fan d)))))))))))))))))"}
{"_id": "1623", "title": "FINITE_INDUCT_STRONG", "text": "|- (!P. (((P EMPTY) /\\ (!x. (!s. (((P s) /\\ ((~ (x IN s)) /\\ (FINITE s))) ==> (P (x INSERT s)))))) ==> (!s. ((FINITE s) ==> (P s)))))"}
{"_id": "1624", "title": "Lp_details.quad_4240815464_a", "text": "|- (quad_nonlinear_v10 ==> (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. ((ineq (((((((((dart_std4 y1) y2) y3) y4) y5) y6) y7) y8) y9)) (((real_lt (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9)) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) \\/ (((real_lt ((real_add y2) y3)) y4) \\/ (((real_gt ((real_add (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9)) ((real_sub ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) \\/ ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))))))))))"}
{"_id": "1625", "title": "LTE_CASES", "text": "|- (!m. (!n. ((m < n) \\/ (n <= m))))"}
{"_id": "1626", "title": "POSITIVE_SEMIDEFINITE_SIMILAR", "text": "|- (!A. (!B. ((positive_semidefinite A) ==> (positive_semidefinite ((matrix_mul (transp B)) ((matrix_mul A) B))))))"}
{"_id": "1627", "title": "REAL_LE_NEGR", "text": "|- (!x. (((real_le x) (real_neg x)) = ((real_le x) (real_of_num (NUMERAL _0)))))"}
{"_id": "1628", "title": "POLYTOPE_PCROSS", "text": "|- (!s. (!t. (((polytope s) /\\ (polytope t)) ==> (polytope (s PCROSS t)))))"}
{"_id": "1629", "title": "INTEGRABLE_SUBINTERVAL", "text": "|- (!f. (!a. (!b. (!c. (!d. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((closed_interval ((CONS (c , d)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL)))) ==> ((integrable_on f) (closed_interval ((CONS (c , d)) NIL)))))))))"}
{"_id": "1630", "title": "Ssrbool.negb_or", "text": "|- (!a. (!b. ((~ (a \\/ b)) = ((~ a) /\\ (~ b)))))"}
{"_id": "1631", "title": "Miqmcsn.K_SCS_OPP_3T7", "text": "|- ((scs_k_v39 (scs_opp_v39 scs_3T7)) = (NUMERAL (BIT1 (BIT1 _0))))"}
{"_id": "1632", "title": "DROP_EQ", "text": "|- (!x. (!y. (((drop x) = (drop y)) = (x = y))))"}
{"_id": "1633", "title": "STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION_SUBSPACE", "text": "|- (!f. (!s. (!e. (!t. (((compact s) /\\ (((continuous_on f) s) /\\ (((real_lt (real_of_num (NUMERAL _0))) e) /\\ ((subspace t) /\\ (((IMAGE f) s) SUBSET t))))) ==> (?g. ((vector_polynomial_function g) /\\ ((((IMAGE g) s) SUBSET t) /\\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))"}
{"_id": "1634", "title": "pathfinish", "text": "|- (!g. ((pathfinish g) = (g (vec (NUMERAL (BIT1 _0))))))"}
{"_id": "1635", "title": "MAXIMAL_AFFINE_INDEPENDENT_SUBSET", "text": "|- (!s. (!b. (((b SUBSET s) /\\ ((~ (affine_dependent b)) /\\ (!b'. (((b SUBSET b') /\\ ((b' SUBSET s) /\\ (~ (affine_dependent b')))) ==> (b' = b))))) ==> (s SUBSET ((hull affine) b)))))"}
{"_id": "1636", "title": "Yrtafyh.YRTAFYH", "text": "|- (!s. (!i. (!j. (((is_scs_v39 s) /\\ ((scs_basic_v39 s) /\\ (((NUMERAL (BIT1 (BIT1 _0))) < (scs_k_v39 s)) /\\ ((((scs_diag (scs_k_v39 s)) i) j) /\\ ((real_le (((scs_a_v39 s) i) j)) cstab))))) ==> ((is_scs_v39 (((scs_stab_diag_v39 s) i) j)) /\\ (scs_basic_v39 (((scs_stab_diag_v39 s) i) j)))))))"}
{"_id": "1637", "title": "REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN", "text": "|- (!f. (!s. (((real_continuous_on f) s) = (!x. ((x IN s) ==> ((real_continuous f) ((within (atreal x)) s)))))))"}
{"_id": "1638", "title": "REAL_CONVEX_BOUND_LT", "text": "|- (!x. (!y. (!a. (!u. (!v. ((((real_lt x) a) /\\ (((real_lt y) a) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_lt ((real_add ((real_mul u) x)) ((real_mul v) y))) a)))))))"}
{"_id": "1639", "title": "Conforming.DART_FANADD_EQ_DART_FAN_ADD_2DART", "text": "|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((fan80 (x , (V , E))) /\\ ((ds IN (face_set (hypermap1_of_fanx (x , (V , E))))) /\\ (((CARD ds) > (NUMERAL (BIT1 (BIT1 _0)))) /\\ (((f1 INSERT (f2 INSERT (f3 INSERT EMPTY))) SUBSET ds) /\\ ((((((f1_fan x) V) E) f1) = f2) /\\ ((((((f1_fan x) V) E) f2) = f3) /\\ ((~ (((((f1_fan x) V) E) f3) = f1)) /\\ (((pr2 f1) = v) /\\ (((pr2 f2) = u) /\\ (((pr2 f3) = w) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((~ (v = w)) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1)))))))))))))))))) ==> (((IMAGE pr23) (dart (hypermap1_of_fanx (x , (V , E1))))) = (((IMAGE pr23) (dart (hypermap1_of_fanx (x , (V , E))))) UNION ((v , w) INSERT ((w , v) INSERT EMPTY))))))))))))))))"}
{"_id": "1640", "title": "vector_polynomial_function", "text": "|- (!f. ((vector_polynomial_function f) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (real_polynomial_function (\\x. ((f x) $ i)))))))"}
{"_id": "1641", "title": "INSERT_SUBSET", "text": "|- (!x. (!s. (!t. (((x INSERT s) SUBSET t) = ((x IN t) /\\ (s SUBSET t))))))"}
{"_id": "1642", "title": "SETDIST_UNIQUE", "text": "|- (!s. (!t. (!a. (!b. (!d. (((a IN s) /\\ ((b IN t) /\\ (((distance (a , b)) = d) /\\ (!x. (!y. (((x IN s) /\\ (y IN t)) ==> ((real_le (distance (a , b))) (distance (x , y))))))))) ==> ((setdist (s , t)) = d)))))))"}
{"_id": "1643", "title": "Ssrnat.addnK", "text": "|- (!n. (!x. (((x + n) - n) = x)))"}
{"_id": "1644", "title": "COMPLEMENT_CONNECTED_COMPONENT_UNIONS", "text": "|- (!s. (!x. ((s DIFF ((connected_component s) x)) = (UNIONS ((GSPEC (\\GEN%PVAR%900. (?y. (((SETSPEC GEN%PVAR%900) (y IN s)) ((connected_component s) y))))) DELETE ((connected_component s) x))))))"}
{"_id": "1645", "title": "CX_COS", "text": "|- (!x. ((Cx (cos x)) = (ccos (Cx x))))"}
{"_id": "1646", "title": "Nonlin_def.sub6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((sub6 f) g) x1) x2) x3) x4) x5) x6) = ((real_sub ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "1647", "title": "ZERO_AE_DERIVATIVE_IMP_CONSTANT_GEN", "text": "|- (!f. (!a. (!b. (!s. ((((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((negligible ((IMAGE f) s)) /\\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (vec (NUMERAL _0))) (at x)))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = (f a)))))))))"}
{"_id": "1648", "title": "CLOSED_SEGMENT_LINEAR_IMAGE", "text": "|- (!f. (!a. (!b. ((linear f) ==> ((closed_segment ((CONS ((f a) , (f b))) NIL)) = ((IMAGE f) (closed_segment ((CONS (a , b)) NIL))))))))"}
{"_id": "1649", "title": "REAL_CONTINUOUS_DIV_WITHINREAL", "text": "|- (!f. (!s. (!a. ((((real_continuous f) ((within (atreal a)) s)) /\\ (((real_continuous g) ((within (atreal a)) s)) /\\ (~ ((g a) = (real_of_num (NUMERAL _0)))))) ==> ((real_continuous (\\x. ((real_div (f x)) (g x)))) ((within (atreal a)) s))))))"}
{"_id": "1650", "title": "Sphere.rotate4", "text": "|- (!f. (!x4. (!x2. (!x6. (!x1. (!x5. (!x3. ((((((((rotate4 f) x1) x2) x3) x4) x5) x6) = ((((((f x4) x2) x6) x1) x5) x3)))))))))"}
{"_id": "1651", "title": "TAGGED_DIVISION_OF_ALT", "text": "|- (!p. (!s. (((tagged_division_of p) s) = (((tagged_partial_division_of p) s) /\\ (!x. ((x IN s) ==> (?t. (?k. (((t , k) IN p) /\\ (x IN k))))))))))"}
{"_id": "1652", "title": "SQRT_NEG", "text": "|- (!x. ((sqrt (real_neg x)) = (real_neg (sqrt x))))"}
{"_id": "1653", "title": "COMPLEX_EQ_ADD_LCANCEL", "text": "|- (!x. (!y. (!z. ((((vector_add x) y) = ((vector_add x) z)) = (y = z)))))"}
{"_id": "1654", "title": "CPRODUCT_EQ", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((cproduct s) f) = ((cproduct s) g))))))"}
{"_id": "1655", "title": "AFFINE_EMPTY", "text": "|- (affine EMPTY)"}
{"_id": "1656", "title": "UNBOUNDED_HALFSPACE_COMPONENT_GE", "text": "|- (!a. (!k. (~ (bounded (GSPEC (\\GEN%PVAR%837. (?x. (((SETSPEC GEN%PVAR%837) ((real_ge (x $ k)) a)) x))))))))"}
{"_id": "1657", "title": "SUMS_0", "text": "|- (!f. (!s. ((!n. ((n IN s) ==> ((f n) = (vec (NUMERAL _0))))) ==> (((sums f) (vec (NUMERAL _0))) s))))"}
{"_id": "1658", "title": "HOLOMORPHIC_ON_CONST", "text": "|- (!c. (!s. ((holomorphic_on (\\z. c)) s)))"}
{"_id": "1659", "title": "HAS_REAL_DERIVATIVE_EXP", "text": "|- (!x. (((has_real_derivative exp) (exp x)) (atreal x)))"}
{"_id": "1660", "title": "RE_SUB", "text": "|- (!x. (!y. ((Re ((vector_sub x) y)) = ((real_sub (Re x)) (Re y)))))"}
{"_id": "1661", "title": "ROOT_MONO_LE", "text": "|- (!n. (!x. (!y. (((real_le x) y) ==> ((real_le ((root n) x)) ((root n) y))))))"}
{"_id": "1662", "title": "VECTOR_VARIATION_COMPARISON", "text": "|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (distance ((g x) , (g y)))) (distance ((f x) , (f y)))))))) ==> ((real_le ((vector_variation s) g)) ((vector_variation s) f))))))"}
{"_id": "1663", "title": "HOMOTOPIC_NON_ANTIPODAL_SPHEREMAPS", "text": "|- (!f. (!g. (!s. (!r. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET (sphere ((vec (NUMERAL _0)) , r))) /\\ (((continuous_on g) s) /\\ ((((IMAGE g) s) SUBSET (sphere ((vec (NUMERAL _0)) , r))) /\\ (!x. ((x IN s) ==> (~ ((f x) = (vector_neg (g x)))))))))) ==> ((((homotopic_with (\\x. T)) (s , (sphere ((vec (NUMERAL _0)) , r)))) f) g))))))"}
{"_id": "1664", "title": "CONTINUOUS_ON_CASES_LOCAL", "text": "|- (!P. (!f. (!g. (!s. (!t. ((((closed_in ((subtopology euclidean) (s UNION t))) s) /\\ (((closed_in ((subtopology euclidean) (s UNION t))) t) /\\ (((continuous_on f) s) /\\ (((continuous_on g) t) /\\ (!x. ((((x IN s) /\\ (~ (P x))) \\/ ((x IN t) /\\ (P x))) ==> ((f x) = (g x)))))))) ==> ((continuous_on (\\x. (((COND (P x)) (f x)) (g x)))) (s UNION t))))))))"}
{"_id": "1665", "title": "SIMPLE_IMAGE_GEN", "text": "|- (!f. (!P. ((GSPEC (\\GEN%PVAR%12. (?x. (((SETSPEC GEN%PVAR%12) (P x)) (f x))))) = ((IMAGE f) (GSPEC (\\GEN%PVAR%13. (?x. (((SETSPEC GEN%PVAR%13) (P x)) x))))))))"}
{"_id": "1666", "title": "SUM_PERMUTE", "text": "|- (!f. (!p. (!s. (((permutes p) s) ==> (((sum s) f) = ((sum s) (f o p)))))))"}
{"_id": "1667", "title": "REAL_LE_MUL_EQ", "text": "|- ((!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) x) ==> (((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_le (real_of_num (NUMERAL _0))) y))))) /\\ (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) y) ==> (((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_le (real_of_num (NUMERAL _0))) x))))))"}
{"_id": "1668", "title": "HULL_UNION_RIGHT", "text": "|- (!P. (!s. (!t. (((hull P) (s UNION t)) = ((hull P) (s UNION ((hull P) t)))))))"}
{"_id": "1669", "title": "CARD_EQ_CLOSED_SETS", "text": "|- ((GSPEC (\\GEN%PVAR%1254. (?s. (((SETSPEC GEN%PVAR%1254) (closed s)) s)))) =_c UNIV)"}
{"_id": "1670", "title": "BOUNDED_SCALING", "text": "|- (!c. (!s. ((bounded s) ==> (bounded ((IMAGE (\\x. ((% c) x))) s)))))"}
{"_id": "1671", "title": "metric_tybij", "text": "|- ((!a. ((metric (dest_metric a)) = a)) /\\ (!r. ((is_metric_space r) = ((dest_metric (metric r)) = r))))"}
{"_id": "1672", "title": "CX_DEF", "text": "|- (!a. ((Cx a) = (complex (a , (real_of_num (NUMERAL _0))))))"}
{"_id": "1673", "title": "NORM_LE_0", "text": "|- (!x. (((real_le (vector_norm x)) (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))"}
{"_id": "1674", "title": "REAL_OF_NUM_MUL", "text": "|- (!m. (!n. (((real_mul (real_of_num m)) (real_of_num n)) = (real_of_num (m * n)))))"}
{"_id": "1675", "title": "CONVEX_AFFSIGN", "text": "|- (!sgn. ((!x. (!y. (!u. (((sgn x) /\\ ((sgn y) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ ((real_le u) (real_of_num (NUMERAL (BIT1 _0))))))) ==> (sgn ((real_add ((real_mul ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) x)) ((real_mul u) y))))))) ==> (!s. (!t. (convex (((affsign sgn) s) t))))))"}
{"_id": "1676", "title": "SUM_POS_EQ_0_NUMSEG", "text": "|- (!f. (!m. (!n. (((!p. (((m <= p) /\\ (p <= n)) ==> ((real_le (real_of_num (NUMERAL _0))) (f p)))) /\\ (((sum (m .. n)) f) = (real_of_num (NUMERAL _0)))) ==> (!p. (((m <= p) /\\ (p <= n)) ==> ((f p) = (real_of_num (NUMERAL _0)))))))))"}
{"_id": "1677", "title": "DOMINATED_CONVERGENCE_AE", "text": "|- (!f. (!g. (!h. (!s. (!t. (((!k. ((integrable_on (f k)) s)) /\\ (((integrable_on h) s) /\\ ((negligible t) /\\ ((!k. (!x. ((x IN (s DIFF t)) ==> ((real_le (vector_norm ((f k) x))) (drop (h x)))))) /\\ (!x. ((x IN (s DIFF t)) ==> (((--> (\\k. ((f k) x))) (g x)) sequentially))))))) ==> (((integrable_on g) s) /\\ (((--> (\\k. ((integral s) (f k)))) ((integral s) g)) sequentially))))))))"}
{"_id": "1678", "title": "REAL_SEQ_OFFSET", "text": "|- (!f. (!l. (!k. ((((---> f) l) sequentially) ==> (((---> (\\i. (f (i + k)))) l) sequentially)))))"}
{"_id": "1679", "title": "CONVEX_NEGATIONS", "text": "|- (!s. ((convex s) ==> (convex ((IMAGE vector_neg) s))))"}
{"_id": "1680", "title": "HAS_REAL_INTEGRAL_SPIKE_FINITE", "text": "|- (!f. (!g. (!s. (!t. (!y. (((FINITE s) /\\ ((!x. ((x IN (t DIFF s)) ==> ((g x) = (f x)))) /\\ (((has_real_integral f) y) t))) ==> (((has_real_integral g) y) t)))))))"}
{"_id": "1681", "title": "AFFINE_HULL_0_3_EXPLICIT", "text": "|- (!x. (!y. (!z. (((vec (NUMERAL _0)) IN ((hull affine) (x INSERT (y INSERT (z INSERT EMPTY))))) = (?a. (?b. (?c. ((((vector_add ((% a) x)) ((vector_add ((% b) y)) ((% c) z))) = (vec (NUMERAL _0))) /\\ (~ (((real_add a) ((real_add b) c)) = (real_of_num (NUMERAL _0))))))))))))"}
{"_id": "1682", "title": "DIVMOD_ELIM_THM", "text": "|- (((P (m DIV n)) (m MOD n)) = (!q. (!r. ((((n = (NUMERAL _0)) /\\ ((q = (NUMERAL _0)) /\\ (r = m))) \\/ ((m = ((q * n) + r)) /\\ (r < n))) ==> ((P q) r)))))"}
{"_id": "1683", "title": "Add_triangle.fan_hypermap_map1", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((face_map (hypermap_of_fan (V , E))) d) = ((f_fan_pair (V , E)) d)) /\\ ((((node_map (hypermap_of_fan (V , E))) d) = ((n_fan_pair (V , E)) d)) /\\ (((edge_map (hypermap_of_fan (V , E))) d) = ((e_fan_pair (V , E)) d)))))))))"}
{"_id": "1684", "title": "SQRT_LE_0", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) (sqrt x)) = ((real_le (real_of_num (NUMERAL _0))) x)))"}
{"_id": "1685", "title": "Functional_equation.functional_mud_135_x", "text": "|- (!y2. (!y4. (!y6. (((domain6 (\\x1. (\\x2. (\\x3. (\\x4. (\\x5. (\\x6. ((real_le (real_of_num (NUMERAL _0))) y2)))))))) (((mud_135_x_v1 y2) y4) y6)) ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y2) y2))) proj_x1) proj_x3) dummy6) dummy6) dummy6)) (uni (sqrt , (((delta_135_x ((real_mul y2) y2)) ((real_mul y4) y4)) ((real_mul y6) y6)))))))))"}
{"_id": "1686", "title": "Lp_ineqs_def.y4'_fan", "text": "|- (!V. (!E. (!x. (((y4'_fan (V , E)) x) = (distance ((SND x) , (FST ((inverse (f_fan_pair_ext (V , E))) x))))))))"}
{"_id": "1687", "title": "Packing3.lemma1", "text": "|- (!f. (!P. (((FINITE f) /\\ (P f)) ==> (?n. (?g. ((g SUBSET f) /\\ ((g HAS_SIZE n) /\\ (P g))))))))"}
{"_id": "1688", "title": "MEASURABLE_ON_SPIKE", "text": "|- (!f. (!g. (!s. (!t. (((negligible s) /\\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((measurable_on f) t) ==> ((measurable_on g) t)))))))"}
{"_id": "1689", "title": "CLOSURE_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((closure ((IMAGE f) s)) = ((IMAGE f) (closure s))))))"}
{"_id": "1690", "title": "AFFINE_HULL_EMPTY", "text": "|- (((hull affine) EMPTY) = EMPTY)"}
{"_id": "1691", "title": "UNIONS_2", "text": "|- ((UNIONS (s INSERT (t INSERT EMPTY))) = (s UNION t))"}
{"_id": "1692", "title": "Hypermap_iso.fan_hypermaps_iso_explicit2", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((hyp_iso (ext_dart (V , E))) ((hypermap_of_fan (V , E)) , (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E))))))))"}
{"_id": "1693", "title": "HAS_BOUNDED_VARIATION_ON_SUBSET", "text": "|- (!f. (!s. (!t. ((((has_bounded_variation_on f) s) /\\ (t SUBSET s)) ==> ((has_bounded_variation_on f) t)))))"}
{"_id": "1694", "title": "CEXP_NZ", "text": "|- (!z. (~ ((cexp z) = (Cx (real_of_num (NUMERAL _0))))))"}
{"_id": "1695", "title": "CLOSED_IN_DIFF", "text": "|- (!top. (!s. (!t. ((((closed_in top) s) /\\ ((open_in top) t)) ==> ((closed_in top) (s DIFF t))))))"}
{"_id": "1696", "title": "Ssrnat.addSn", "text": "|- (!m. (!n. (((SUC m) + n) = (SUC (m + n)))))"}
{"_id": "1697", "title": "Hypermap.FACE_FINITE", "text": "|- (!H. (!x. (FINITE ((face H) x))))"}
{"_id": "1698", "title": "Hypermap.lemma_face_representation", "text": "|- (!H. (!u. ((u IN (face_set H)) ==> (?x. ((x IN (dart H)) /\\ (u = ((face H) x)))))))"}
{"_id": "1699", "title": "INTERIOR_MAXIMAL_EQ", "text": "|- (!s. (!t. ((open s) ==> ((s SUBSET (interior t)) = (s SUBSET t)))))"}
{"_id": "1700", "title": "INT_OF_NUM_MAX", "text": "|- (!m. (!n. (((int_max (int_of_num m)) (int_of_num n)) = (int_of_num ((MAX m) n)))))"}
{"_id": "1701", "title": "REAL_INV_DIV", "text": "|- (!x. (!y. ((real_inv ((real_div x) y)) = ((real_div y) x))))"}
{"_id": "1702", "title": "Seq.has", "text": "|- (!a. (!x. (!s'. ((((has a) NIL) = F) /\\ (((has a) ((CONS x) s')) = ((a x) \\/ ((has a) s')))))))"}
{"_id": "1703", "title": "COMPLEX_NORM_II", "text": "|- ((vector_norm ii) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "1704", "title": "POW_2_SQRT", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((sqrt ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = x)))"}
{"_id": "1705", "title": "Hypermap.lemma_atom_sub_node", "text": "|- (!H. (!L. (!x. ((((atom H) L) x) SUBSET ((node H) x)))))"}
{"_id": "1706", "title": "OPEN_IN_SUBTOPOLOGY_REFL", "text": "|- (!top. (!u. (((open_in ((subtopology top) u)) u) = (u SUBSET (topspace top)))))"}
{"_id": "1707", "title": "COMPLEX_DIFFERENTIABLE_DIV_WITHIN", "text": "|- (!f. (!g. (!z. (!s. ((((complex_differentiable f) ((within (at z)) s)) /\\ (((complex_differentiable g) ((within (at z)) s)) /\\ (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((complex_differentiable (\\z. ((complex_div (f z)) (g z)))) ((within (at z)) s)))))))"}
{"_id": "1708", "title": "CTAN_ADD", "text": "|- (!w. (!z. (((~ ((ccos w) = (Cx (real_of_num (NUMERAL _0))))) /\\ ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\\ (~ ((ccos ((vector_add w) z)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((ctan ((vector_add w) z)) = ((complex_div ((vector_add (ctan w)) (ctan z))) ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_mul (ctan w)) (ctan z))))))))"}
{"_id": "1709", "title": "HAS_REAL_MEASURE_TRANSLATION_EQ", "text": "|- (!s. (!m. (((has_real_measure ((IMAGE (\\x. ((real_add a) x))) s)) m) = ((has_real_measure s) m))))"}
{"_id": "1710", "title": "INT_OF_NUM_ADD", "text": "|- (!m. (!n. (((int_add (int_of_num m)) (int_of_num n)) = (int_of_num (m + n)))))"}
{"_id": "1711", "title": "NOT_OUTSIDE_CONNECTED_COMPONENT_LE", "text": "|- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ (bounded s)) ==> ((UNIV DIFF (outside s)) = (GSPEC (\\GEN%PVAR%2014. (?x. (((SETSPEC GEN%PVAR%2014) (!B. (?y. (((real_le B) (vector_norm y)) /\\ (~ (((connected_component (UNIV DIFF s)) x) y)))))) x)))))))"}
{"_id": "1712", "title": "HAS_REAL_DERIVATIVE_MUL_ATREAL", "text": "|- (!f. (!f'. (!g. (!g'. (!x. (((((has_real_derivative f) f') (atreal x)) /\\ (((has_real_derivative g) g') (atreal x))) ==> (((has_real_derivative (\\x. ((real_mul (f x)) (g x)))) ((real_add ((real_mul (f x)) g')) ((real_mul f') (g x)))) (atreal x))))))))"}
{"_id": "1713", "title": "CONVEX_IMP_PATH_CONNECTED", "text": "|- (!s. ((convex s) ==> (path_connected s)))"}
{"_id": "1714", "title": "LT_EXISTS", "text": "|- (!m. (!n. ((m < n) = (?d. (n = (m + (SUC d)))))))"}
{"_id": "1715", "title": "REAL_DIFFERENTIABLE_TRANSFORM", "text": "|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((real_differentiable_on f) s)) ==> ((real_differentiable_on g) s)))))"}
{"_id": "1716", "title": "CONTINUOUS_CARD_LT_RANGE_CONSTANT_EQ", "text": "|- (!s. ((connected s) = (!f. ((((continuous_on f) s) /\\ (((IMAGE f) s) <_c UNIV)) ==> (?a. (!x. ((x IN s) ==> ((f x) = a))))))))"}
{"_id": "1717", "title": "subspace", "text": "|- (!s. ((subspace s) = (((vec (NUMERAL _0)) IN s) /\\ ((!x. (!y. (((x IN s) /\\ (y IN s)) ==> (((vector_add x) y) IN s)))) /\\ (!c. (!x. ((x IN s) ==> (((% c) x) IN s))))))))"}
{"_id": "1718", "title": "INTERIOR_SUBSET", "text": "|- (!s. ((interior s) SUBSET s))"}
{"_id": "1719", "title": "SUBSET_ANTISYM_EQ", "text": "|- (!s. (!t. (((s SUBSET t) /\\ (t SUBSET s)) = (s = t))))"}
{"_id": "1720", "title": "MEASURE_SCALING", "text": "|- (!s. (!c. ((measurable s) ==> ((measure ((IMAGE (\\x. ((% c) x))) s)) = ((real_mul ((real_pow (real_abs c)) (dimindex UNIV))) (measure s))))))"}
{"_id": "1721", "title": "COMPACT_TRANSLATION_EQ", "text": "|- (!a. (!s. ((compact ((IMAGE (\\x. ((vector_add a) x))) s)) = (compact s))))"}
{"_id": "1722", "title": "REAL_LE_MAX", "text": "|- (!x. (!y. (!z. (((real_le z) ((real_max x) y)) = (((real_le z) x) \\/ ((real_le z) y))))))"}
{"_id": "1723", "title": "Hypermap.card_orbit_le", "text": "|- (!f. (!n. (!x. (((~ (n = (NUMERAL _0))) /\\ ((((POWER f) n) x) = x)) ==> ((CARD ((orbit_map f) x)) <= n)))))"}
{"_id": "1724", "title": "INTEGRAL_SUB", "text": "|- (!f. (!g. (!k. (!l. (!s. ((((integrable_on f) s) /\\ ((integrable_on g) s)) ==> (((integral s) (\\x. ((vector_sub (f x)) (g x)))) = ((vector_sub ((integral s) f)) ((integral s) g)))))))))"}
{"_id": "1725", "title": "conic_cap", "text": "|- (!r. (!v0. (!v1. (!a. (((((conic_cap v0) v1) r) a) = (((normball v0) r) INTER (((rcone_gt v0) v1) a)))))))"}
{"_id": "1726", "title": "COMPACT_CLOSURE", "text": "|- (!s. ((compact (closure s)) = (bounded s)))"}
{"_id": "1727", "title": "INF_INSERT_FINITE", "text": "|- (!x. (!s. ((FINITE s) ==> ((inf (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_min x) (inf s)))))))"}
{"_id": "1728", "title": "L1_LE_NORM", "text": "|- (!x. ((real_le ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. (real_abs (x $ i))))) ((real_mul (sqrt (real_of_num (dimindex UNIV)))) (vector_norm x))))"}
{"_id": "1729", "title": "IN_SPHERE_0", "text": "|- (!x. (!e. ((x IN (sphere ((vec (NUMERAL _0)) , e))) = ((vector_norm x) = e))))"}
{"_id": "1730", "title": "HAS_COMPLEX_DERIVATIVE_CARATHEODORY_WITHIN", "text": "|- (!f. (!f'. (!z. (!s. ((((has_complex_derivative f) f') ((within (at z)) s)) = (?g. ((!w. (((vector_sub (f w)) (f z)) = ((complex_mul (g w)) ((vector_sub w) z)))) /\\ (((continuous g) ((within (at z)) s)) /\\ ((g z) = f')))))))))"}
{"_id": "1731", "title": "REAL_SEGMENT_INTERVAL", "text": "|- ((!a. (!b. ((closed_real_segment ((CONS (a , b)) NIL)) = (((COND ((real_le a) b)) (closed_real_interval ((CONS (a , b)) NIL))) (closed_real_interval ((CONS (b , a)) NIL)))))) /\\ (!a. (!b. ((open_real_segment (a , b)) = (((COND ((real_le a) b)) (open_real_interval (a , b))) (open_real_interval (b , a)))))))"}
{"_id": "1732", "title": "Lp_ineqs_proofs.y8_hi_std", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))"}
{"_id": "1733", "title": "LOCALLY_PATH_CONNECTED_PATH_IMAGE", "text": "|- (!p. ((path p) ==> ((locally path_connected) (path_image p))))"}
{"_id": "1734", "title": "List_hypermap.mem_list_of_elements", "text": "|- (!x. (!L. (((MEM x) (list_of_elements L)) = (?y. ((MEM (x , y)) (list_of_darts L))))))"}
{"_id": "1735", "title": "Appendix.scs_a_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_a_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = a))))))))))"}
{"_id": "1736", "title": "ssreflect_eq_def", "text": "|- (!v. (!P. (((ssreflect_eq v) P) = P)))"}
{"_id": "1737", "title": "BETWEEN_IMP_COLLINEAR", "text": "|- (!a. (!b. (!x. (((between x) (a , b)) ==> (collinear (a INSERT (x INSERT (b INSERT EMPTY))))))))"}
{"_id": "1738", "title": "complex_pow", "text": "|- ((((complex_pow x) (NUMERAL _0)) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) /\\ (!n. (((complex_pow x) (SUC n)) = ((complex_mul x) ((complex_pow x) n)))))"}
{"_id": "1739", "title": "SUM_EQ_NUMSEG", "text": "|- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\\ (i <= n)) ==> ((f i) = (g i)))) ==> (((sum (m .. n)) f) = ((sum (m .. n)) g)))))))"}
{"_id": "1740", "title": "INT_LE_RNEG", "text": "|- (!x. (!y. (((int_le x) (int_neg y)) = ((int_le ((int_add x) y)) (int_of_num (NUMERAL _0))))))"}
{"_id": "1741", "title": "INTEGER_CASES", "text": "|- ((integer x) = ((?n. (x = (real_of_num n))) \\/ (?n. (x = (real_neg (real_of_num n))))))"}
{"_id": "1742", "title": "IMAGE_UNIONS", "text": "|- (!f. (!s. (((IMAGE f) (UNIONS s)) = (UNIONS ((IMAGE (IMAGE f)) s)))))"}
{"_id": "1743", "title": "Seq.subseq_filter", "text": "|- (!s1. (!s2. (!a. (((subseq s1) ((filter a) s2)) = (((all a) s1) /\\ ((subseq s1) s2))))))"}
{"_id": "1744", "title": "HAS_BOUNDED_VARIATION_COMPOSE_INCREASING", "text": "|- (!f. (!g. (!a. (!b. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f x))) (drop (f y)))))) /\\ ((has_bounded_variation_on g) (closed_interval ((CONS ((f a) , (f b))) NIL)))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "1745", "title": "RE", "text": "|- ((Re (complex (x , y))) = x)"}
{"_id": "1746", "title": "rectifiable_path", "text": "|- (!g. ((rectifiable_path g) = ((path g) /\\ ((has_bounded_variation_on g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))))))"}
{"_id": "1747", "title": "Pack2.POLYHEDRON_VORONOI_CLOSED", "text": "|- (!s. (!v. (((packing s) /\\ ((saturated s) /\\ (v IN s))) ==> (polyhedron ((voronoi_closed s) v)))))"}
{"_id": "1748", "title": "REAL_LE_LNEG", "text": "|- (!x. (!y. (((real_le (real_neg x)) y) = ((real_le (real_of_num (NUMERAL _0))) ((real_add x) y)))))"}
{"_id": "1749", "title": "CEXP_NEG_RMUL", "text": "|- (!z. (((complex_mul (cexp z)) (cexp (vector_neg z))) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))"}
{"_id": "1750", "title": "vol_solid_triangle", "text": "|- (!r. (!v0. (!v1. (!v2. (!v3. ((((((vol_solid_triangle v0) v1) v2) v3) r) = ((LET (\\a123. (LET_END ((LET (\\a231. (LET_END ((LET (\\a312. (LET_END ((real_mul ((real_add a123) ((real_add a231) ((real_sub a312) pi)))) ((real_div ((real_pow r) (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))) ((((dihV v0) v3) v1) v2))))) ((((dihV v0) v2) v3) v1))))) ((((dihV v0) v1) v2) v3))))))))"}
{"_id": "1751", "title": "OPEN_IMP_GDELTA", "text": "|- (!s. ((open s) ==> (gdelta s)))"}
{"_id": "1752", "title": "Functional_equation.rhazim_x'", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\\ (((real_le (real_of_num (NUMERAL _0))) x2) /\\ (((real_le (real_of_num (NUMERAL _0))) x3) /\\ (((real_le (real_of_num (NUMERAL _0))) x4) /\\ (((real_le (real_of_num (NUMERAL _0))) x5) /\\ ((real_le (real_of_num (NUMERAL _0))) x6)))))) ==> (((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazim_x' x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "1753", "title": "REAL_LE_SUB_LADD", "text": "|- (!x. (!y. (!z. (((real_le x) ((real_sub y) z)) = ((real_le ((real_add x) z)) y)))))"}
{"_id": "1754", "title": "CLOSED_UNIONS", "text": "|- (!s. (((FINITE s) /\\ (!t. ((t IN s) ==> (closed t)))) ==> (closed (UNIONS s))))"}
{"_id": "1755", "title": "Ssrbool.contraR", "text": "|- (!c. (!b. (((~ c) ==> b) ==> ((~ b) ==> c))))"}
{"_id": "1756", "title": "ACS_MONO_LE", "text": "|- (!x. (!y. ((((real_le (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x) /\\ (((real_le x) y) /\\ ((real_le y) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((real_le (acs y)) (acs x)))))"}
{"_id": "1757", "title": "Sphere.sol_y", "text": "|- (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. (((((((sol_y y1) y2) y3) y4) y5) y6) = ((real_add ((((((dih_y y1) y2) y3) y4) y5) y6)) ((real_add ((((((dih_y y2) y3) y1) y5) y6) y4)) ((real_sub ((((((dih_y y3) y1) y2) y6) y4) y5)) pi))))))))))"}
{"_id": "1758", "title": "HAS_REAL_MEASURE_SCALING", "text": "|- (!s. (!m. (!y. (((has_real_measure s) y) ==> ((has_real_measure ((IMAGE (\\x. ((real_mul m) x))) s)) ((real_mul (real_abs m)) y))))))"}
{"_id": "1759", "title": "SERIES_0", "text": "|- (!s. (((sums (\\n. (vec (NUMERAL _0)))) (vec (NUMERAL _0))) s))"}
{"_id": "1760", "title": "REAL_LT_DIV", "text": "|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_div x) y)))))"}
{"_id": "1761", "title": "NADD_LE_WELLDEF", "text": "|- (!x. (!x'. (!y. (!y'. ((((nadd_eq x) x') /\\ ((nadd_eq y) y')) ==> (((nadd_le x) y) = ((nadd_le x') y')))))))"}
{"_id": "1762", "title": "IM_SUB", "text": "|- (!x. (!y. ((Im ((vector_sub x) y)) = ((real_sub (Im x)) (Im y)))))"}
{"_id": "1763", "title": "REAL_ADD_RID", "text": "|- (!x. (((real_add x) (real_of_num (NUMERAL _0))) = x))"}
{"_id": "1764", "title": "GDELTA_PCROSS", "text": "|- (!s. (!t. (((gdelta s) /\\ (gdelta t)) ==> (gdelta (s PCROSS t)))))"}
{"_id": "1765", "title": "FCONS", "text": "|- ((!a. (!f. ((((FCONS a) f) (NUMERAL _0)) = a))) /\\ (!a. (!f. (!n. ((((FCONS a) f) (SUC n)) = (f n))))))"}
{"_id": "1766", "title": "MAPROWS_COMPOSE", "text": "|- (!f. (!g. ((maprows (f o g)) = ((maprows f) o (maprows g)))))"}
{"_id": "1767", "title": "CBALL_EMPTY", "text": "|- (!x. (!e. (((real_lt e) (real_of_num (NUMERAL _0))) ==> ((cball (x , e)) = EMPTY))))"}
{"_id": "1768", "title": "COMPLEX_MVT_LINE", "text": "|- (!f. (!f'. (!w. (!z. ((!u. ((u IN (closed_segment ((CONS (w , z)) NIL))) ==> (((has_complex_derivative f) (f' u)) (at u)))) ==> (?u. ((u IN (closed_segment ((CONS (w , z)) NIL))) /\\ (((real_sub (Re (f z))) (Re (f w))) = (Re ((complex_mul (f' u)) ((vector_sub z) w)))))))))))"}
{"_id": "1769", "title": "CLOSED_CBALL", "text": "|- (!x. (!e. (closed (cball (x , e)))))"}
{"_id": "1770", "title": "BARYCENTRE_IN_RELATIVE_INTERIOR", "text": "|- (!s. (((FINITE s) /\\ (~ (s = EMPTY))) ==> ((barycentre s) IN (relative_interior ((hull convex) s)))))"}
{"_id": "1771", "title": "LIFT_EQ", "text": "|- (!x. (!y. (((lift x) = (lift y)) = (x = y))))"}
{"_id": "1772", "title": "Ssrnat.subKn", "text": "|- (!m. (!n. ((m <= n) ==> ((n - (n - m)) = m))))"}
{"_id": "1773", "title": "root", "text": "|- (!x. (!n. (((root n) x) = ((real_mul (real_sgn x)) (exp ((real_div (log (real_abs x))) (real_of_num n)))))))"}
{"_id": "1774", "title": "Functional_equation.rhazim_all_atn", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazimatn_x x1) x2) x3) x4) x5) x6)) /\\ ((((((((rhazim2_x x1) x2) x3) x4) x5) x6) = ((((((rhazim2atn_x x1) x2) x3) x4) x5) x6)) /\\ (((((((rhazim3_x x1) x2) x3) x4) x5) x6) = ((((((rhazim3atn_x x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "1775", "title": "CONTINUOUS_NEG", "text": "|- (!f. (!net. (((continuous f) net) ==> ((continuous (\\x. (vector_neg (f x)))) net))))"}
{"_id": "1776", "title": "EXTREME_POINT_OF_CONVEX_HULL_INSERT", "text": "|- (!s. (!a. (((FINITE s) /\\ (~ (a IN ((hull convex) s)))) ==> ((extreme_point_of a) ((hull convex) (a INSERT s))))))"}
{"_id": "1777", "title": "AFF_GE_EQ_AFFINE_HULL", "text": "|- (!s. (!t. (((aff_ge s) EMPTY) = ((hull affine) s))))"}
{"_id": "1778", "title": "SQRT_POW_2", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = x)))"}
{"_id": "1779", "title": "Lp_main_estimate.f_surr_in_e_prime", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!f. (!v. (!w. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((v INSERT (w INSERT EMPTY)) IN ((e_prime E) f)) = (((v , w) IN f) \\/ ((w , v) IN f)))))))))))"}
{"_id": "1780", "title": "COMPLEX_TRAD", "text": "|- (!x. (!y. ((complex (x , y)) = ((vector_add (Cx x)) ((complex_mul ii) (Cx y))))))"}
{"_id": "1781", "title": "Seq2.next_el_hd_cons2", "text": "|- (!h1. (!h2. (!t. (((next_el ((CONS h1) ((CONS h2) t))) h1) = h2))))"}
{"_id": "1782", "title": "VSUM_ADD_NUMSEG", "text": "|- (!f. (!g. (!m. (!n. (((vsum (m .. n)) (\\i. ((vector_add (f i)) (g i)))) = ((vector_add ((vsum (m .. n)) f)) ((vsum (m .. n)) g)))))))"}
{"_id": "1783", "title": "Lp_ineqs_proofs.tauq_cross_split", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((((((((tauq (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y9_fan (V , E)) d)) = ((real_add ((((((taum (y1_fan d)) ((y3_fan (V , E)) d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) ((y5_fan (V , E)) d))) ((((((taum (y1_fan d)) (y2_fan d)) ((y7_fan (V , E)) d)) ((y9_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) (y6_fan d))))))))))))"}
{"_id": "1784", "title": "COMPACT_CBALL", "text": "|- (!x. (!e. (compact (cball (x , e)))))"}
{"_id": "1785", "title": "Hypermap_and_fan.PAIR_IN_DART_OF_FAN", "text": "|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((v , w) IN (dart_of_fan (V , E)))) ==> ((v IN V) /\\ (w IN V)))))))"}
{"_id": "1786", "title": "Lp_ineqs_proofs.y8_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y8_fan (V , E)) d))))))))))"}
{"_id": "1787", "title": "FROM_0", "text": "|- ((from (NUMERAL _0)) = UNIV)"}
{"_id": "1788", "title": "FACET_OF_TRANSLATION_EQ", "text": "|- (!a. (!f. (!s. (((facet_of ((IMAGE (\\x. ((vector_add a) x))) f)) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((facet_of f) s)))))"}
{"_id": "1789", "title": "COPLANAR_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((coplanar ((IMAGE f) s)) = (coplanar s)))))"}
{"_id": "1790", "title": "REAL_EQ_RCANCEL_IMP", "text": "|- (!x. (!y. (!z. (((~ (z = (real_of_num (NUMERAL _0)))) /\\ (((real_mul x) z) = ((real_mul y) z))) ==> (x = y)))))"}
{"_id": "1791", "title": "joinpaths", "text": "|- (!g1. (!g2. (((++ g1) g2) = (\\x. (((COND ((real_le (drop x)) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) (g1 ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x))) (g2 ((vector_sub ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)) (vec (NUMERAL (BIT1 _0))))))))))"}
{"_id": "1792", "title": "HAS_BOUNDED_VARIATION_ON_MUL", "text": "|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\\ (((has_bounded_variation_on g) s) /\\ (is_interval s))) ==> ((has_bounded_variation_on (\\x. ((% (drop (f x))) (g x)))) s)))))"}
{"_id": "1793", "title": "REAL_POW_LT2", "text": "|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\\ (((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_lt x) y))) ==> ((real_lt ((real_pow x) n)) ((real_pow y) n))))))"}
{"_id": "1794", "title": "SPAN_MUL", "text": "|- (!x. (!c. (!s. ((x IN (span s)) ==> (((% c) x) IN (span s))))))"}
{"_id": "1795", "title": "COMPACT_IMP_CLOSED", "text": "|- (!s. ((compact s) ==> (closed s)))"}
{"_id": "1796", "title": "FACE_OF_TRANS", "text": "|- (!s. (!t. (!u. ((((face_of s) t) /\\ ((face_of t) u)) ==> ((face_of s) u)))))"}
{"_id": "1797", "title": "ge_c", "text": "|- (!t. (!s. ((s >=_c t) = (t <=_c s))))"}
{"_id": "1798", "title": "RSTC_SYM", "text": "|- (!R. (!x. (!y. ((((RSTC R) x) y) ==> (((RSTC R) y) x)))))"}
{"_id": "1799", "title": "Seq.count", "text": "|- (!a. (!x. (!s'. ((((count a) NIL) = (NUMERAL _0)) /\\ (((count a) ((CONS x) s')) = ((((COND (a x)) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + ((count a) s')))))))"}
{"_id": "1800", "title": "OPEN_NON_GENERAL_COMPONENT", "text": "|- (!c. (((!s. (!x. (!y. ((((c s) x) y) ==> ((x IN s) /\\ (y IN s)))))) /\\ ((!s. (!x. (!y. ((((c s) x) y) ==> (((c s) y) x))))) /\\ ((!s. (!x. (!y. (!z. (((((c s) x) y) /\\ (((c s) y) z)) ==> (((c s) x) z)))))) /\\ ((!s. (!t. (!x. (!y. (((s SUBSET t) /\\ (((c s) x) y)) ==> (((c t) x) y)))))) /\\ (!s. (!x. (!y. (!e. (((y IN (ball (x , e))) /\\ ((ball (x , e)) SUBSET s)) ==> (((c (ball (x , e))) x) y)))))))))) ==> (!s. (!x. ((open s) ==> (open (s DIFF ((c s) x))))))))"}
{"_id": "1801", "title": "Sphere.lmfun", "text": "|- (!h. ((lmfun h) = (((COND ((real_le h) h0)) ((real_div ((real_sub h0) h)) ((real_sub h0) (real_of_num (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL _0)))))"}
{"_id": "1802", "title": "num_WOP", "text": "|- (!P. ((?n. (P n)) = (?n. ((P n) /\\ (!m. ((m < n) ==> (~ (P m))))))))"}
{"_id": "1803", "title": "NPRODUCT_CLAUSES", "text": "|- ((!f. (((nproduct EMPTY) f) = (NUMERAL (BIT1 _0)))) /\\ (!x. (!f. (!s. ((FINITE s) ==> (((nproduct (x INSERT s)) f) = (((COND (x IN s)) ((nproduct s) f)) ((f x) * ((nproduct s) f)))))))))"}
{"_id": "1804", "title": "INT_LT_DISCRETE", "text": "|- (!x. (!y. (((int_lt x) y) = ((int_le ((int_add x) (int_of_num (NUMERAL (BIT1 _0))))) y))))"}
{"_id": "1805", "title": "SETDIST_LINEAR_IMAGE", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((setdist (((IMAGE f) s) , ((IMAGE f) t))) = (setdist (s , t)))))))"}
{"_id": "1806", "title": "SEGMENT_CONVEX_HULL", "text": "|- (!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = ((hull convex) (a INSERT (b INSERT EMPTY))))))"}
{"_id": "1807", "title": "SUM_LE", "text": "|- (!f. (!g. (!s. (((FINITE s) /\\ (!x. ((x IN s) ==> ((real_le (f x)) (g x))))) ==> ((real_le ((sum s) f)) ((sum s) g))))))"}
{"_id": "1808", "title": "AFF_GE_SUBSET_AFFINE_HULL", "text": "|- (!s. (!t. (((aff_ge s) t) SUBSET ((hull affine) (s UNION t)))))"}
{"_id": "1809", "title": "Seq.has_pred1", "text": "|- (!x. (!s. (((has (pred1 x)) s) = ((MEM x) s))))"}
{"_id": "1810", "title": "REAL_DIFFERENTIABLE_AT_RPOW_RIGHT", "text": "|- (!a. (!x. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((real_differentiable (\\x. ((rpow a) x))) (atreal x)))))"}
{"_id": "1811", "title": "Lp_main_estimate.packing_v_prime", "text": "|- (!V. (!f. ((packing V) ==> (packing ((v_prime V) f)))))"}
{"_id": "1812", "title": "Functional_equation.proj_x5", "text": "|- (((((((proj_x5 x1) x2) x3) x4) x5) x6) = x5)"}
{"_id": "1813", "title": "Nonlin_def.dummy6", "text": "|- (dummy6 = (constant6 (real_of_num (NUMERAL _0))))"}
{"_id": "1814", "title": "BOREL_UNIV", "text": "|- (borel UNIV)"}
{"_id": "1815", "title": "Planarity.fan80", "text": "|- (!x. (!V. (!E. ((fan80 (x , (V , E))) = (!v. (!u. (((v INSERT (u INSERT EMPTY)) IN E) ==> (((real_lt (real_of_num (NUMERAL _0))) ((((azim x) v) u) (((((sigma_fan x) V) E) v) u))) /\\ ((real_lt ((((azim x) v) u) (((((sigma_fan x) V) E) v) u))) pi)))))))))"}
{"_id": "1816", "title": "OPEN_IN_INTER", "text": "|- (!top. (!s. (!t. ((((open_in top) s) /\\ ((open_in top) t)) ==> ((open_in top) (s INTER t))))))"}
{"_id": "1817", "title": "Ssrnat.leq_mul2l", "text": "|- (!m. (!n1. (!n2. (((m * n1) <= (m * n2)) = ((m = (NUMERAL _0)) \\/ (n1 <= n2))))))"}
{"_id": "1818", "title": "REAL_ABS_SIN_BOUND_LT", "text": "|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (sin x))) (real_abs x))))"}
{"_id": "1819", "title": "VSUM_LMUL", "text": "|- (!f. (!c. (!s. (((vsum s) (\\x. ((% c) (f x)))) = ((% c) ((vsum s) f))))))"}
{"_id": "1820", "title": "Functional_equation.eta_all_atn", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) x1) /\\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\\ ((real_lt (real_of_num (NUMERAL _0))) x6)))))) ==> ((((((((eta2_126 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x2) x6)) (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((((((eta2_135 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((((((eta2_456 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x4) x5) x6)) (NUMERAL (BIT0 (BIT1 _0))))))))))))))"}
{"_id": "1821", "title": "Hypermap.lemma_node_identity", "text": "|- (!H. (!x. (!y. ((y IN ((node H) x)) ==> (((node H) x) = ((node H) y))))))"}
{"_id": "1822", "title": "UNIONS_SUBSET", "text": "|- (!f. (!t. (((UNIONS f) SUBSET t) = (!s. ((s IN f) ==> (s SUBSET t))))))"}
{"_id": "1823", "title": "IN_COMPONENTS_SUBSET", "text": "|- (!s. (!c. ((c IN (components s)) ==> (c SUBSET s))))"}
{"_id": "1824", "title": "Nonlinear_lemma.lmdih5_ldih5", "text": "|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le (real_of_num (NUMERAL _0))) y5) /\\ ((real_le y5) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> ((((((((y_of_x lmdih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6) = (((((((y_of_x ldih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6)))))))))"}
{"_id": "1825", "title": "Tame_inequalities.DELTA_X4_MONO_LT_4", "text": "|- (!x1. (!x2. (!x3. (!a. (!x5. (!x6. (!b. ((((real_lt a) b) /\\ ((real_lt (real_of_num (NUMERAL _0))) x1)) ==> ((real_lt ((((((delta_x4 x1) x2) x3) b) x5) x6)) ((((((delta_x4 x1) x2) x3) a) x5) x6))))))))))"}
{"_id": "1826", "title": "Tame_defs.tame_3", "text": "|- (!H. ((tame_3 H) = (is_edge_nondegenerate H)))"}
{"_id": "1827", "title": "VECTOR_VARIATION_CONTINUOUS_LEFT", "text": "|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (a , c)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (a , c)) NIL))))))))))"}
{"_id": "1828", "title": "Rnsyjxm.in_face_imp_in_dart", "text": "|- (!x. (!H. (!d. (((x IN (dart H)) /\\ (d IN ((face H) x))) ==> (d IN (dart H))))))"}
{"_id": "1829", "title": "INT_LE_ANTISYM", "text": "|- (!x. (!y. ((((int_le x) y) /\\ ((int_le y) x)) = (x = y))))"}
{"_id": "1830", "title": "BEPPO_LEVI_INCREASING", "text": "|- (!f. (!s. (((!k. ((integrable_on (f k)) s)) /\\ ((!k. (!x. ((x IN s) ==> ((real_le (drop ((f k) x))) (drop ((f (SUC k)) x)))))) /\\ (bounded (GSPEC (\\GEN%PVAR%3065. (?k. (((SETSPEC GEN%PVAR%3065) (k IN UNIV)) ((integral s) (f k))))))))) ==> (?g. (?k. ((negligible k) /\\ (!x. ((x IN (s DIFF k)) ==> (((--> (\\k. ((f k) x))) (g x)) sequentially)))))))))"}
{"_id": "1831", "title": "SUM_EQ_SUPERSET", "text": "|- (!f. (!s. (!t. (((FINITE t) /\\ ((t SUBSET s) /\\ ((!x. ((x IN t) ==> ((f x) = (g x)))) /\\ (!x. (((x IN s) /\\ (~ (x IN t))) ==> ((f x) = (real_of_num (NUMERAL _0)))))))) ==> (((sum s) f) = ((sum t) g))))))"}
{"_id": "1832", "title": "COLLINEAR_AFF_DIM", "text": "|- (!s. ((collinear s) = ((int_le (aff_dim s)) (int_of_num (NUMERAL (BIT1 _0))))))"}
{"_id": "1833", "title": "List_hypermap.uniq_list_of_elements", "text": "|- (!L. (uniq (list_of_elements L)))"}
{"_id": "1834", "title": "MOD_MULT", "text": "|- (!m. (!n. ((~ (m = (NUMERAL _0))) ==> (((m * n) MOD m) = (NUMERAL _0)))))"}
{"_id": "1835", "title": "HAS_COMPLEX_DERIVATIVE_DIV_AT", "text": "|- (!f. (!f'. (!g. (!g'. (!x. (((((has_complex_derivative f) f') (at x)) /\\ ((((has_complex_derivative g) g') (at x)) /\\ (~ ((g x) = (Cx (real_of_num (NUMERAL _0))))))) ==> (((has_complex_derivative (\\x. ((complex_div (f x)) (g x)))) ((complex_div ((vector_sub ((complex_mul f') (g x))) ((complex_mul (f x)) g'))) ((complex_pow (g x)) (NUMERAL (BIT0 (BIT1 _0)))))) (at x))))))))"}
{"_id": "1836", "title": "COMPLETE_TRANSLATION_EQ", "text": "|- (!a. (!s. ((complete ((IMAGE (\\x. ((vector_add a) x))) s)) = (complete s))))"}
{"_id": "1837", "title": "Add_triangle.inE_eq_in_dart1", "text": "|- (!V. (!E. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) = ((v , w) IN (dart1_of_fan (V , E))))))))"}
{"_id": "1838", "title": "REAL_POW_ONE", "text": "|- (!n. (((real_pow (real_of_num (NUMERAL (BIT1 _0)))) n) = (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "1839", "title": "Trigonometry.XHVXJVB", "text": "|- (!x. ((vector_norm x) = (sqrt ((dot x) x))))"}
{"_id": "1840", "title": "List_hypermap.e_list_ext", "text": "|- (!L. ((e_list_ext L) = ((res e_list) (darts_of_list L))))"}
{"_id": "1841", "title": "Appendix.scs_d_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_d_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = d))))))))))"}
{"_id": "1842", "title": "CLOG_CEXP", "text": "|- (!z. ((((real_lt (real_neg pi)) (Im z)) /\\ ((real_le (Im z)) pi)) ==> ((clog (cexp z)) = z)))"}
{"_id": "1843", "title": "COMPLEX_MUL_2", "text": "|- (!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x) = ((vector_add x) x)))"}
{"_id": "1844", "title": "Nonlin_def.mul6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((mul6 f) g) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "1845", "title": "HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homeomorphic ((IMAGE f) s)) t) = ((homeomorphic s) t))))))"}
{"_id": "1846", "title": "DEPENDENT_BIGGERSET_GENERAL", "text": "|- (!s. (((FINITE s) ==> ((CARD s) > (dim s))) ==> (dependent s)))"}
{"_id": "1847", "title": "REAL_SUMMABLE_RESTRICT", "text": "|- (!f. (!k. (((real_summable UNIV) (\\n. (((COND (n IN k)) (f n)) (real_of_num (NUMERAL _0))))) = ((real_summable k) f))))"}
{"_id": "1848", "title": "REAL_MEASURE_MEASURE", "text": "|- (!s. ((real_measure s) = (measure ((IMAGE lift) s))))"}
{"_id": "1849", "title": "MATRIX_VECTOR_MUL_COMPONENT", "text": "|- (!A. (!x. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV))) ==> ((((matrix_vector_mul A) x) $ k) = ((dot (A $ k)) x))))))"}
{"_id": "1850", "title": "LEBESGUE_MEASURABLE_COUNTABLE_UNIONS", "text": "|- (!f. (((COUNTABLE f) /\\ (!s. ((s IN f) ==> (lebesgue_measurable s)))) ==> (lebesgue_measurable (UNIONS f))))"}
{"_id": "1851", "title": "REAL_NEG_GE0", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) (real_neg x)) = ((real_le x) (real_of_num (NUMERAL _0)))))"}
{"_id": "1852", "title": "HAS_BOUNDED_VARIATION_ON_COMBINE", "text": "|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\\ ((real_le (drop c)) (drop b))) ==> (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = (((has_bounded_variation_on f) (closed_interval ((CONS (a , c)) NIL))) /\\ ((has_bounded_variation_on f) (closed_interval ((CONS (c , b)) NIL))))))))))"}
{"_id": "1853", "title": "LINEAR_COMPOSE_CMUL", "text": "|- (!f. (!c. ((linear f) ==> (linear (\\x. ((% c) (f x)))))))"}
{"_id": "1854", "title": "REAL_EXP_INJ", "text": "|- (!x. (!y. (((exp x) = (exp y)) = (x = y))))"}
{"_id": "1855", "title": "Lp_ineqs_proofs.dart4_tauVEF_eq_tauq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) d)))) = (((((((((tauq (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y9_fan (V , E)) d)))))))))))"}
{"_id": "1856", "title": "Float.REAL_LT", "text": "|- (!m. (!n. (((real_lt (real_of_num m)) (real_of_num n)) = (m < n))))"}
{"_id": "1857", "title": "ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG", "text": "|- (!v. (orthogonal_transformation (reflect_along v)))"}
{"_id": "1858", "title": "Seq.last", "text": "|- (!x. (!h. (!t. ((((last x) NIL) = x) /\\ (((last x) ((CONS h) t)) = ((last h) t))))))"}
{"_id": "1859", "title": "CLOSED_IN", "text": "|- (!s. ((closed s) = ((closed_in euclidean) s)))"}
{"_id": "1860", "title": "MATRIX_NEG", "text": "|- (!f. ((linear f) ==> ((matrix (\\x. (vector_neg (f x)))) = (matrix_neg (matrix f)))))"}
{"_id": "1861", "title": "FST", "text": "|- (!x. (!y. ((FST (x , y)) = x)))"}
{"_id": "1862", "title": "Tame_lemmas.sum_tauVEF_upper_bound", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (scriptL V)) ==> ((real_le ((sum (face_set (hypermap_of_fan (V , E)))) (\\f. (tauVEF (V , (E , f)))))) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))) sol0))))))))"}
{"_id": "1863", "title": "NORMAL_MATRIX_KERNEL_TRANSP_EXPLICIT", "text": "|- (!A. (!x. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> ((((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0))) = (((matrix_vector_mul A) x) = (vec (NUMERAL _0)))))))"}
{"_id": "1864", "title": "Ssrbool.XOR_DEF", "text": "|- (!p. (!q. (((XOR p) q) = (((COND p) (~ q)) q))))"}
{"_id": "1865", "title": "Functional_equation.proj_y6", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((proj_y6 x1) x2) x3) x4) x5) x6) = (sqrt x6))))))))"}
{"_id": "1866", "title": "CONNECTED_CONTINUOUS_IMAGE", "text": "|- (!f. (!s. ((((continuous_on f) s) /\\ (connected s)) ==> (connected ((IMAGE f) s)))))"}
{"_id": "1867", "title": "Planarity.aff_ge_eq_aff_gt_union_aff_ge", "text": "|- (!x. (!v. (!w. ((~ (collinear (x INSERT (v INSERT (w INSERT EMPTY))))) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) = (((aff_gt (x INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) UNION (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) UNION ((aff_ge (x INSERT EMPTY)) (w INSERT EMPTY)))))))))"}
{"_id": "1868", "title": "Misc_defs_and_lemmas.IN_ELIM_THM'", "text": "|- ((!P. (!x. ((x IN (GSPEC (\\v. (P (SETSPEC v))))) = (P (\\p. (\\t. (p /\\ (x = t)))))))) /\\ ((!p. (!x. ((x IN (GSPEC (\\v. (?y. (((SETSPEC v) (p y)) y))))) = (p x)))) /\\ ((!P. (!x. (((GSPEC (\\v. (P (SETSPEC v)))) x) = (P (\\p. (\\t. (p /\\ (x = t)))))))) /\\ ((!p. (!x. (((GSPEC (\\v. (?y. (((SETSPEC v) (p y)) y)))) x) = (p x)))) /\\ (!p. (!x. ((x IN (\\y. (p y))) = (p x))))))))"}
{"_id": "1869", "title": "LIM_UNIQUE", "text": "|- (!net. (!f. (!l. (!l'. (((~ (trivial_limit net)) /\\ ((((--> f) l) net) /\\ (((--> f) l') net))) ==> (l = l'))))))"}
{"_id": "1870", "title": "tagged_division_of", "text": "|- (!s. (!i. (((tagged_division_of s) i) = (((tagged_partial_division_of s) i) /\\ ((UNIONS (GSPEC (\\GEN%PVAR%2759. (?k. (((SETSPEC GEN%PVAR%2759) (?x. ((x , k) IN s))) k))))) = i)))))"}
{"_id": "1871", "title": "LIM_NULL_SUB", "text": "|- (!net. (!f. (!g. (((((--> f) (vec (NUMERAL _0))) net) /\\ (((--> g) (vec (NUMERAL _0))) net)) ==> (((--> (\\x. ((vector_sub (f x)) (g x)))) (vec (NUMERAL _0))) net)))))"}
{"_id": "1872", "title": "Fan_defs.set_of_edge", "text": "|- (!v. (!E. (!V. ((((set_of_edge v) V) E) = (GSPEC (\\GEN%PVAR%5439. (?w. (((SETSPEC GEN%PVAR%5439) (((v INSERT (w INSERT EMPTY)) IN E) /\\ (w IN V))) w))))))))"}
{"_id": "1873", "title": "CX_SUB", "text": "|- (!x. (!y. ((Cx ((real_sub x) y)) = ((vector_sub (Cx x)) (Cx y)))))"}
{"_id": "1874", "title": "UNIFORMLY_CONTINUOUS_ON_ADD", "text": "|- (!f. (!g. (!s. ((((uniformly_continuous_on f) s) /\\ ((uniformly_continuous_on g) s)) ==> ((uniformly_continuous_on (\\x. ((vector_add (f x)) (g x)))) s)))))"}
{"_id": "1875", "title": "HAS_COMPLEX_DERIVATIVE_MUL_AT", "text": "|- (!f. (!f'. (!g. (!g'. (!x. (((((has_complex_derivative f) f') (at x)) /\\ (((has_complex_derivative g) g') (at x))) ==> (((has_complex_derivative (\\x. ((complex_mul (f x)) (g x)))) ((vector_add ((complex_mul (f x)) g')) ((complex_mul f') (g x)))) (at x))))))))"}
{"_id": "1876", "title": "Localization.AZIM_CYCLE_EQ_SIGMA_FAN_ALT", "text": "|- (!V. (!E. (!u. (!v. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (u IN (((set_of_edge v) V) E))) ==> (((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) u) = (((((sigma_fan (vec (NUMERAL _0))) V) E) v) u)))))))"}
{"_id": "1877", "title": "manhattan", "text": "|- (manhattan = (metric (UNIV , (GABS (\\f. (!x. (!y. ((GEQ (f (x , y))) ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. (real_abs ((real_sub (x $ i)) (y $ i)))))))))))))"}
{"_id": "1878", "title": "ABSOLUTE_RETRACT_PATH_IMAGE_ARC", "text": "|- (!g. (!s. (((arc g) /\\ ((path_image g) SUBSET s)) ==> ((retract_of (path_image g)) s))))"}
{"_id": "1879", "title": "INTEGRABLE_BOUNDED_VARIATION_PRODUCT", "text": "|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((has_bounded_variation_on g) (closed_interval ((CONS (a , b)) NIL)))) ==> ((integrable_on (\\x. ((% (drop (g x))) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "1880", "title": "dist", "text": "|- (!x. (!y. ((distance (x , y)) = (vector_norm ((vector_sub x) y)))))"}
{"_id": "1881", "title": "OPEN_BIJECTIVE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> ((open ((IMAGE f) s)) = (open s)))))"}
{"_id": "1882", "title": "PERMUTES_INVERSES", "text": "|- (!p. (!s. (((permutes p) s) ==> ((!x. ((p ((inverse p) x)) = x)) /\\ (!x. (((inverse p) (p x)) = x))))))"}
{"_id": "1883", "title": "IN_COMPONENTS_NONEMPTY", "text": "|- (!s. (!c. ((c IN (components s)) ==> (~ (c = EMPTY)))))"}
{"_id": "1884", "title": "REAL_ARCH_INV", "text": "|- (!e. (((real_lt (real_of_num (NUMERAL _0))) e) = (?n. ((~ (n = (NUMERAL _0))) /\\ (((real_lt (real_of_num (NUMERAL _0))) (real_inv (real_of_num n))) /\\ ((real_lt (real_inv (real_of_num n))) e))))))"}
{"_id": "1885", "title": "Add_triangle.azim_split_fan_face_add", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face (hypermap_of_fan (V , E))) (v , w))))) ==> ((LET (\\w'. (LET_END ((LET (\\d2. (LET_END ((LET (\\F2. (LET_END ((((azim_dart (V , E)) d2) = ((real_add ((azim_dart F2) (w , w'))) ((azim_dart F2) d2))) /\\ (((azim_dart (V , E)) (w' , v)) = ((real_add ((azim_dart F2) (w' , w))) ((azim_dart F2) (w' , v)))))))) ((split_fan_face (V , E)) (v , w)))))) ((f_fan_pair (V , E)) (v , w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))"}
{"_id": "1886", "title": "positive_semidefinite", "text": "|- (!A. ((positive_semidefinite A) = (((transp A) = A) /\\ (!x. ((real_le (real_of_num (NUMERAL _0))) ((dot x) ((matrix_vector_mul A) x)))))))"}
{"_id": "1887", "title": "SET_PROVE_CASES", "text": "|- (!P. (((P EMPTY) /\\ (!a. (!s. ((~ (a IN s)) ==> (P (a INSERT s)))))) ==> (!s. (P s))))"}
{"_id": "1888", "title": "Add_triangle.n_split_fan_face", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face (hypermap_of_fan (V , E))) (v , w))))) ==> ((((LET (\\w1. (\\d2. (\\n. (LET_END (((n (w1 , w)) = ((n_fan_pair (V , E)) (w1 , v))) /\\ (((n (w1 , v)) = (w1 , w)) /\\ (((n d2) = (w , w1)) /\\ (((n (w , w1)) = (w , v)) /\\ (!d. (((d IN (dart_of_fan ((split_fan_face (V , E)) (v , w)))) /\\ (~ ((d = (w1 , w)) \\/ ((d = (w1 , v)) \\/ ((d = d2) \\/ (d = (w , w1))))))) ==> ((n d) = ((n_fan_pair (V , E)) d))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) ((f_fan_pair (V , E)) (v , w))) (n_fan_pair ((split_fan_face (V , E)) (v , w)))))))))))"}
{"_id": "1889", "title": "Dih2k_hypermap.F_SY", "text": "|- (!v. ((F_SY v) = (GSPEC (\\GEN%PVAR%7490. (?i. (((SETSPEC GEN%PVAR%7490) (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) (((row i) v) , ((row (SUC (i MOD (dimindex UNIV)))) v))))))))"}
{"_id": "1890", "title": "REAL_EQ_ADD_RCANCEL", "text": "|- (!x. (!y. (!z. ((((real_add x) z) = ((real_add y) z)) = (x = y)))))"}
{"_id": "1891", "title": "complex_neg", "text": "|- ((vector_neg z) = (complex ((real_neg (Re z)) , (real_neg (Im z)))))"}
{"_id": "1892", "title": "List_hypermap.elements_of_list", "text": "|- (!L. ((elements_of_list L) = (set_of_list (list_of_elements L))))"}
{"_id": "1893", "title": "Sphere.aff", "text": "|- (aff = (hull affine))"}
{"_id": "1894", "title": "Seq.flatten", "text": "|- (flatten = ((foldr cat) NIL))"}
{"_id": "1895", "title": "HAS_COMPLEX_DERIVATIVE_INV_AT", "text": "|- (!f. (!f'. (!x. (((((has_complex_derivative f) f') (at x)) /\\ (~ ((f x) = (Cx (real_of_num (NUMERAL _0)))))) ==> (((has_complex_derivative (\\x. (complex_inv (f x)))) ((complex_div (vector_neg f')) ((complex_pow (f x)) (NUMERAL (BIT0 (BIT1 _0)))))) (at x))))))"}
{"_id": "1896", "title": "Fnjlbxs.closed_ball_annulus", "text": "|- (closed ball_annulus)"}
{"_id": "1897", "title": "HOMOTOPIC_PATHS_IMP_SUBSET", "text": "|- (!s. (!p. (!q. ((((homotopic_paths s) p) q) ==> (((path_image p) SUBSET s) /\\ ((path_image q) SUBSET s))))))"}
{"_id": "1898", "title": "OPEN_SURJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (((linear f) /\\ (!y. (?x. ((f x) = y)))) ==> (!s. ((open s) ==> (open ((IMAGE f) s))))))"}
{"_id": "1899", "title": "CNJ_EQ_0", "text": "|- (!z. (((cnj z) = (Cx (real_of_num (NUMERAL _0)))) = (z = (Cx (real_of_num (NUMERAL _0))))))"}
{"_id": "1900", "title": "Sphere.dih3_y", "text": "|- (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. (((((((dih3_y y1) y2) y3) y4) y5) y6) = ((((((dih_y y3) y1) y2) y6) y4) y5))))))))"}
{"_id": "1901", "title": "Fan_defs.dart_of_fan", "text": "|- (!V. (!E. ((dart_of_fan (V , E)) = ((GSPEC (\\GEN%PVAR%5441. (?v. (((SETSPEC GEN%PVAR%5441) ((v IN V) /\\ ((((set_of_edge v) V) E) = EMPTY))) (v , v))))) UNION (GSPEC (\\GEN%PVAR%5442. (?v. (?w. (((SETSPEC GEN%PVAR%5442) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w))))))))))"}
{"_id": "1902", "title": "ANR_SPHERE", "text": "|- (!a. (!r. (ANR (sphere (a , r)))))"}
{"_id": "1903", "title": "REAL_LE_REFL", "text": "|- (!x. ((real_le x) x))"}
{"_id": "1904", "title": "CONVEX_UNIV", "text": "|- (convex UNIV)"}
{"_id": "1905", "title": "MEASURE_UNIQUE", "text": "|- (!s. (!m. (((has_measure s) m) ==> ((measure s) = m))))"}
{"_id": "1906", "title": "Hypermap_and_fan.SURROUNDED_IMP_CARD_SET_OF_EDGE_GE_3", "text": "|- (!V. (!E. (!v. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((v IN V) /\\ ((surrounded_node (V , E)) v))) ==> ((CARD (((set_of_edge v) V) E)) >= (NUMERAL (BIT1 (BIT1 _0))))))))"}
{"_id": "1907", "title": "UNIFORMLY_CONTINUOUS_ON_VSUM", "text": "|- (!t. (!f. (!s. (((FINITE s) /\\ (!a. ((a IN s) ==> ((uniformly_continuous_on (f a)) t)))) ==> ((uniformly_continuous_on (\\x. ((vsum s) (\\a. ((f a) x))))) t)))))"}
{"_id": "1908", "title": "REAL_INTEGRABLE_LMUL_EQ", "text": "|- (!f. (!s. (!c. (((real_integrable_on (\\x. ((real_mul c) (f x)))) s) = ((c = (real_of_num (NUMERAL _0))) \\/ ((real_integrable_on f) s))))))"}
{"_id": "1909", "title": "CLOSEST_POINT_SELF", "text": "|- (!s. (!x. ((x IN s) ==> (((closest_point s) x) = x))))"}
{"_id": "1910", "title": "Add_triangle.sigma_fan_inV", "text": "|- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) IN V)))))))"}
{"_id": "1911", "title": "int_add_th", "text": "|- (!x. (!y. ((real_of_int ((int_add x) y)) = ((real_add (real_of_int x)) (real_of_int y)))))"}
{"_id": "1912", "title": "Ssrbool.orbC", "text": "|- (!b. (!c. ((b \\/ c) = (c \\/ b))))"}
{"_id": "1913", "title": "ARG_EQ_0_PI", "text": "|- (!z. ((((Arg z) = (real_of_num (NUMERAL _0))) \\/ ((Arg z) = pi)) = (real z)))"}
{"_id": "1914", "title": "INSIDE_NO_OVERLAP", "text": "|- (!s. (((inside s) INTER s) = EMPTY))"}
{"_id": "1915", "title": "LIPSCHITZ_IMP_CONTINUOUS_ON", "text": "|- (!f. (!s. ((?B. (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))))) ==> ((continuous_on f) s))))"}
{"_id": "1916", "title": "Collect_geom2.ETA_Y_POS_LE", "text": "|- ((real_le (real_of_num (NUMERAL _0))) (((eta_y ((dist3 v1) v2)) ((dist3 v1) v3)) ((dist3 v2) v3)))"}
{"_id": "1917", "title": "Topology.AFF_GE_1_1", "text": "|- (!x. (!v. ((~ (x = v)) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) = (GSPEC (\\GEN%PVAR%5522. (?y. (((SETSPEC GEN%PVAR%5522) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t2) /\\ ((((real_add t1) t2) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (y = ((vector_add ((% t1) x)) ((% t2) v)))))))) y))))))))"}
{"_id": "1918", "title": "REAL_CONTINUOUS_INV_WITHINREAL", "text": "|- (!f. (!s. (!a. ((((real_continuous f) ((within (atreal a)) s)) /\\ (~ ((f a) = (real_of_num (NUMERAL _0))))) ==> ((real_continuous (\\x. (real_inv (f x)))) ((within (atreal a)) s))))))"}
{"_id": "1919", "title": "HAS_REAL_MEASURE_REAL_NEGLIGIBLE_SYMDIFF", "text": "|- (!s. (!t. (!m. ((((has_real_measure s) m) /\\ (real_negligible ((s DIFF t) UNION (t DIFF s)))) ==> ((has_real_measure t) m)))))"}
{"_id": "1920", "title": "CLOSED_TRANSLATION_EQ", "text": "|- (!a. (!s. ((closed ((IMAGE (\\x. ((vector_add a) x))) s)) = (closed s))))"}
{"_id": "1921", "title": "HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE", "text": "|- (!a. (!b. (!k. (!c. (((~ (a = (vec (NUMERAL _0)))) /\\ (((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV)))) ==> ((homeomorphic (GSPEC (\\GEN%PVAR%1058. (?x. (((SETSPEC GEN%PVAR%1058) (((dot a) x) = b)) x))))) (GSPEC (\\GEN%PVAR%1059. (?x. (((SETSPEC GEN%PVAR%1059) ((x $ k) = c)) x))))))))))"}
{"_id": "1922", "title": "VECTOR_SUB", "text": "|- (((vector_sub x) y) = ((vector_add x) (vector_neg y)))"}
{"_id": "1923", "title": "ARG", "text": "|- (!z. (((real_le (real_of_num (NUMERAL _0))) (Arg z)) /\\ (((real_lt (Arg z)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\\ (z = ((complex_mul (Cx (vector_norm z))) (cexp ((complex_mul ii) (Cx (Arg z)))))))))"}
{"_id": "1924", "title": "Hypermap_iso.hyp_iso_ext", "text": "|- (!f. (!H. (!G. (((hyp_iso f) (H , G)) ==> (!g. ((!d. ((d IN (dart H)) ==> ((f d) = (g d)))) ==> ((hyp_iso g) (H , G))))))))"}
{"_id": "1925", "title": "BETWEEN_DIST_LE", "text": "|- (!r. (!a. (!b. (!c. ((((real_le (distance (c , a))) r) /\\ (((real_le (distance (c , b))) r) /\\ ((between x) (a , b)))) ==> ((real_le (distance (c , x))) r))))))"}
{"_id": "1926", "title": "SEGMENT_CLOSED_OPEN", "text": "|- (!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = ((open_segment (a , b)) UNION (a INSERT (b INSERT EMPTY))))))"}
{"_id": "1927", "title": "Seq.allP", "text": "|- (!a. (!s. ((!x. (((MEM x) s) ==> (a x))) = ((all a) s))))"}
{"_id": "1928", "title": "Sphere.pathL", "text": "|- (!a. (!b. ((pathL (a , b)) = (a , ((real_div ((real_add a) b)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "1929", "title": "HAS_COMPLEX_DERIVATIVE_SUB", "text": "|- (!f. (!f'. (!g. (!g'. (!net. (((((has_complex_derivative f) f') net) /\\ (((has_complex_derivative g) g') net)) ==> (((has_complex_derivative (\\x. ((vector_sub (f x)) (g x)))) ((vector_sub f') g')) net)))))))"}
{"_id": "1930", "title": "NORM_PASTECART_LE", "text": "|- (!x. (!y. ((real_le (vector_norm ((pastecart x) y))) ((real_add (vector_norm x)) (vector_norm y)))))"}
{"_id": "1931", "title": "INT_NOT_LT", "text": "|- (!x. (!y. ((~ ((int_lt x) y)) = ((int_le y) x))))"}
{"_id": "1932", "title": "Fan_defs.f_fan_pair_ext", "text": "|- (!V. (!E. (!x. (((f_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((f_fan_pair (V , E)) x)) x)))))"}
{"_id": "1933", "title": "Hypermap_iso.IN_TRANS", "text": "|- (!x. (!s. (!t. (((t SUBSET s) /\\ (x IN t)) ==> (x IN s)))))"}
{"_id": "1934", "title": "Ssrnat.minnC", "text": "|- (!m. (!n. (((minn m) n) = ((minn n) m))))"}
{"_id": "1935", "title": "REAL_EQ_ADD_LCANCEL_0", "text": "|- (!x. (!y. ((((real_add x) y) = x) = (y = (real_of_num (NUMERAL _0))))))"}
{"_id": "1936", "title": "Sphere.periodic", "text": "|- (!n. (!f. (((periodic f) n) = (!i. ((f (i + n)) = (f i))))))"}
{"_id": "1937", "title": "Seq.reshape", "text": "|- ((((reshape ((CONS n) sh)) s) = ((CONS ((take n) s)) ((reshape sh) ((dropl n) s)))) /\\ (((reshape NIL) s) = NIL))"}
{"_id": "1938", "title": "FACET_OF_HALFSPACE_LE", "text": "|- (!f. (!a. (!b. (((facet_of f) (GSPEC (\\GEN%PVAR%2216. (?x. (((SETSPEC GEN%PVAR%2216) ((real_le ((dot a) x)) b)) x))))) = ((~ (a = (vec (NUMERAL _0)))) /\\ (f = (GSPEC (\\GEN%PVAR%2217. (?x. (((SETSPEC GEN%PVAR%2217) (((dot a) x) = b)) x))))))))))"}
{"_id": "1939", "title": "FINITE_RECURSION", "text": "|- (!f. (!b. ((!x. (!y. (!s. ((~ (x = y)) ==> (((f x) ((f y) s)) = ((f y) ((f x) s))))))) ==> (((((ITSET f) EMPTY) b) = b) /\\ (!x. (!s. ((FINITE s) ==> ((((ITSET f) (x INSERT s)) b) = (((COND (x IN s)) (((ITSET f) s) b)) ((f x) (((ITSET f) s) b)))))))))))"}
{"_id": "1940", "title": "REAL_CONVEX_ON_SECOND_DERIVATIVE", "text": "|- (!f. (!f'. (!f''. (!s. (((is_realinterval s) /\\ ((~ (?a. (s = (a INSERT EMPTY)))) /\\ ((!x. ((x IN s) ==> (((has_real_derivative f) (f' x)) ((within (atreal x)) s)))) /\\ (!x. ((x IN s) ==> (((has_real_derivative f') (f'' x)) ((within (atreal x)) s))))))) ==> (((real_convex_on f) s) = (!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f'' x))))))))))"}
{"_id": "1941", "title": "Hypermap.lemma_dart_invariant", "text": "|- (!H. (!x. ((x IN (dart H)) ==> ((((edge_map H) x) IN (dart H)) /\\ ((((node_map H) x) IN (dart H)) /\\ (((face_map H) x) IN (dart H)))))))"}
{"_id": "1942", "title": "COUNTABLE_RATIONAL", "text": "|- (COUNTABLE rational)"}
{"_id": "1943", "title": "Seq.has_predU", "text": "|- (!a1. (!a2. (!s. (((has ((predU a1) a2)) s) = (((has a1) s) \\/ ((has a2) s))))))"}
{"_id": "1944", "title": "AFF_GT_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((aff_gt ((IMAGE f) s)) ((IMAGE f) t)) = ((IMAGE f) ((aff_gt s) t)))))))"}
{"_id": "1945", "title": "REAL_LT_IMP_NZ", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> (~ (x = (real_of_num (NUMERAL _0))))))"}
{"_id": "1946", "title": "log_convex_on", "text": "|- (!s. (!f. (((log_convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\\ ((y IN s) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((real_le (real_of_num (NUMERAL _0))) (f ((vector_add ((% u) x)) ((% v) y)))) /\\ ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_mul ((rpow (f x)) u)) ((rpow (f y)) v))))))))))))"}
{"_id": "1947", "title": "Tame_defs.tame_9a", "text": "|- (!H. ((tame_9a H) = (!x. ((x IN (dart H)) ==> (((CARD ((face H) x)) >= (NUMERAL (BIT1 (BIT1 _0)))) /\\ ((CARD ((face H) x)) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))))))"}
{"_id": "1948", "title": "CLOSED_IN_CLOSED_TRANS", "text": "|- (!s. (!t. ((((closed_in ((subtopology euclidean) t)) s) /\\ (closed t)) ==> (closed s))))"}
{"_id": "1949", "title": "COMPLEX_MUL_RZERO", "text": "|- (!x. (((complex_mul x) (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL _0)))))"}
{"_id": "1950", "title": "RE_CCOS", "text": "|- (!z. ((Re (ccos z)) = ((real_mul ((real_div ((real_add (exp (Im z))) (exp (real_neg (Im z))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (cos (Re z)))))"}
{"_id": "1951", "title": "HAS_SIZE_FACES_OF_SIMPLEX", "text": "|- (!n. (!s. (((simplex n) s) ==> ((GSPEC (\\GEN%PVAR%2479. (?f. (((SETSPEC GEN%PVAR%2479) ((face_of f) s)) f)))) HAS_SIZE ((NUMERAL (BIT0 (BIT1 _0))) EXP (num_of_int ((int_add n) (int_of_num (NUMERAL (BIT1 _0))))))))))"}
{"_id": "1952", "title": "segment", "text": "|- (((closed_segment ((CONS (a , b)) NIL)) = (GSPEC (\\GEN%PVAR%559. (?u. (((SETSPEC GEN%PVAR%559) (((real_le (real_of_num (NUMERAL _0))) u) /\\ ((real_le u) (real_of_num (NUMERAL (BIT1 _0)))))) ((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) a)) ((% u) b))))))) /\\ ((open_segment (a , b)) = ((closed_segment ((CONS (a , b)) NIL)) DIFF (a INSERT (b INSERT EMPTY)))))"}
{"_id": "1953", "title": "REAL_CX", "text": "|- (!x. (real (Cx x)))"}
{"_id": "1954", "title": "bilinear", "text": "|- (!f. ((bilinear f) = ((!x. (linear (\\y. ((f x) y)))) /\\ (!y. (linear (\\x. ((f x) y)))))))"}
{"_id": "1955", "title": "LIPSCHITZ_IMP_HAS_BOUNDED_VARIATION", "text": "|- (!f. (!s. (!B. (((bounded s) /\\ (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))))) ==> ((has_bounded_variation_on f) s)))))"}
{"_id": "1956", "title": "WITHINREAL_UNIV", "text": "|- (!x. (((within (atreal x)) UNIV) = (atreal x)))"}
{"_id": "1957", "title": "IN_COMPONENTS_CONNECTED", "text": "|- (!s. (!c. ((c IN (components s)) ==> (connected c))))"}
{"_id": "1958", "title": "NUM_OF_INT", "text": "|- (!x. (((int_le (int_of_num (NUMERAL _0))) x) = ((int_of_num (num_of_int x)) = x)))"}
{"_id": "1959", "title": "HOMOTOPIC_WITH_IMP_SUBSET", "text": "|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> ((((IMAGE f) X) SUBSET Y) /\\ (((IMAGE g) X) SUBSET Y))))))))"}
{"_id": "1960", "title": "Hypermap_and_fan.N_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((permutes (n_fan_pair_ext (V , E))) (dart_of_fan (V , E))))))"}
{"_id": "1961", "title": "ORTHOGONAL_TRANSFORMATION_ID", "text": "|- (orthogonal_transformation (\\x. x))"}
{"_id": "1962", "title": "BASIS_EXISTS", "text": "|- (!v. (?b. ((b SUBSET v) /\\ ((independent b) /\\ ((v SUBSET (span b)) /\\ (b HAS_SIZE (dim v)))))))"}
{"_id": "1963", "title": "RE_DEF", "text": "|- (!z. ((Re z) = (z $ (NUMERAL (BIT1 _0)))))"}
{"_id": "1964", "title": "ORTHOGONAL_SUBSPACE_DECOMP_UNIQUE", "text": "|- (!s. (!t. (!x. (!y. (!x'. (!y'. (((!a. (!b. (((a IN s) /\\ (b IN t)) ==> ((orthogonal a) b)))) /\\ ((x IN (span s)) /\\ ((x' IN (span s)) /\\ ((y IN (span t)) /\\ ((y' IN (span t)) /\\ (((vector_add x) y) = ((vector_add x') y'))))))) ==> ((x = x') /\\ (y = y')))))))))"}
{"_id": "1965", "title": "FACE_OF_HALFSPACE_LE", "text": "|- (!f. (!a. (!b. (((face_of f) (GSPEC (\\GEN%PVAR%2153. (?x. (((SETSPEC GEN%PVAR%2153) ((real_le ((dot a) x)) b)) x))))) = ((f = EMPTY) \\/ ((f = (GSPEC (\\GEN%PVAR%2154. (?x. (((SETSPEC GEN%PVAR%2154) (((dot a) x) = b)) x))))) \\/ (f = (GSPEC (\\GEN%PVAR%2155. (?x. (((SETSPEC GEN%PVAR%2155) ((real_le ((dot a) x)) b)) x)))))))))))"}
{"_id": "1966", "title": "DIAMETER_SUBSET", "text": "|- (!s. (!t. (((s SUBSET t) /\\ (bounded t)) ==> ((real_le (diameter s)) (diameter t)))))"}
{"_id": "1967", "title": "HAS_MEASURE_0", "text": "|- (!s. (((has_measure s) (real_of_num (NUMERAL _0))) = (negligible s)))"}
{"_id": "1968", "title": "INTEGRAL_COMPONENT", "text": "|- (!f. (((integrable_on f) s) ==> ((((integral s) f) $ k) = (drop ((integral s) (\\x. (lift ((f x) $ k))))))))"}
{"_id": "1969", "title": "LIFT_NEG", "text": "|- (!x. ((lift (real_neg x)) = (vector_neg (lift x))))"}
{"_id": "1970", "title": "SETDIST_EMPTY", "text": "|- ((!t. ((setdist (EMPTY , t)) = (real_of_num (NUMERAL _0)))) /\\ (!s. ((setdist (s , EMPTY)) = (real_of_num (NUMERAL _0)))))"}
{"_id": "1971", "title": "REAL_INV_POW", "text": "|- (!x. (!n. ((real_inv ((real_pow x) n)) = ((real_pow (real_inv x)) n))))"}
{"_id": "1972", "title": "real_infsum", "text": "|- (!f. (!s. (((real_infsum s) f) = (@l. (((real_sums f) l) s)))))"}
{"_id": "1973", "title": "CLOSED_IN_EMPTY", "text": "|- (!top. ((closed_in top) EMPTY))"}
{"_id": "1974", "title": "CONTINUOUS_ON_ADD", "text": "|- (!f. (!g. (!s. ((((continuous_on f) s) /\\ ((continuous_on g) s)) ==> ((continuous_on (\\x. ((vector_add (f x)) (g x)))) s)))))"}
{"_id": "1975", "title": "SUM_SUB", "text": "|- (!f. (!g. (!s. ((FINITE s) ==> (((sum s) (\\x. ((real_sub (f x)) (g x)))) = ((real_sub ((sum s) f)) ((sum s) g)))))))"}
{"_id": "1976", "title": "NOT_LE", "text": "|- (!m. (!n. ((~ (m <= n)) = (n < m))))"}
{"_id": "1977", "title": "Planarity.exists_dartset_leads_into_fan", "text": "|- (!x. (!V. (!E. (!ds. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((fan80 (x , (V , E))) /\\ (ds IN (face_set (hypermap1_of_fanx (x , (V , E)))))))) ==> (?s. (!y. ((y IN ds) ==> (s = (((((dart_leads_into x) V) E) (pr2 y)) (pr3 y)))))))))))"}
{"_id": "1978", "title": "real_uniformly_continuous_on", "text": "|- (!s. (!f. (((real_uniformly_continuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (!x'. (((x IN s) /\\ ((x' IN s) /\\ ((real_lt (real_abs ((real_sub x') x))) d))) ==> ((real_lt (real_abs ((real_sub (f x')) (f x)))) e)))))))))))"}
{"_id": "1979", "title": "SERIES_BOUND", "text": "|- (!f. (!g. (!s. (!a. (!b. (((((sums f) a) s) /\\ ((((sums (lift o g)) (lift b)) s) /\\ (!i. ((i IN s) ==> ((real_le (vector_norm (f i))) (g i)))))) ==> ((real_le (vector_norm a)) b)))))))"}
{"_id": "1980", "title": "DIST_IN_CLOSED_SEGMENT", "text": "|- (!a. (!b. (!x. ((x IN (closed_segment ((CONS (a , b)) NIL))) ==> (((real_le (distance (x , a))) (distance (a , b))) /\\ ((real_le (distance (x , b))) (distance (a , b))))))))"}
{"_id": "1981", "title": "POLYHEDRON_NEGATIONS", "text": "|- (!s. ((polyhedron s) ==> (polyhedron ((IMAGE vector_neg) s))))"}
{"_id": "1982", "title": "HAS_COMPLEX_DERIVATIVE_CSIN", "text": "|- (!z. (((has_complex_derivative csin) (ccos z)) (at z)))"}
{"_id": "1983", "title": "SKOLEM_THM", "text": "|- (!P. ((!x. (?y. ((P x) y))) = (?y. (!x. ((P x) (y x))))))"}
{"_id": "1984", "title": "COMPLEX_POLY_CLAUSES", "text": "|- ((!x. (!y. (!z. (((vector_add x) ((vector_add y) z)) = ((vector_add ((vector_add x) y)) z))))) /\\ ((!x. (!y. (((vector_add x) y) = ((vector_add y) x)))) /\\ ((!x. (((vector_add (Cx (real_of_num (NUMERAL _0)))) x) = x)) /\\ ((!x. (!y. (!z. (((complex_mul x) ((complex_mul y) z)) = ((complex_mul ((complex_mul x) y)) z))))) /\\ ((!x. (!y. (((complex_mul x) y) = ((complex_mul y) x)))) /\\ ((!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT1 _0))))) x) = x)) /\\ ((!x. (((complex_mul (Cx (real_of_num (NUMERAL _0)))) x) = (Cx (real_of_num (NUMERAL _0))))) /\\ ((!x. (!y. (!z. (((complex_mul x) ((vector_add y) z)) = ((vector_add ((complex_mul x) y)) ((complex_mul x) z)))))) /\\ ((!x. (((complex_pow x) (NUMERAL _0)) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))) /\\ (!x. (!n. (((complex_pow x) (SUC n)) = ((complex_mul x) ((complex_pow x) n))))))))))))))"}
{"_id": "1985", "title": "CONVEX_ON_LEFT_SECANT", "text": "|- (!f. (!s. (((convex_on f) s) = (!a. (!b. (!x. (((a IN s) /\\ ((b IN s) /\\ (x IN (open_segment (a , b))))) ==> ((real_le ((real_div ((real_sub (f x)) (f a))) (vector_norm ((vector_sub x) a)))) ((real_div ((real_sub (f b)) (f a))) (vector_norm ((vector_sub b) a)))))))))))"}
{"_id": "1986", "title": "Tame_defs2.list_update", "text": "|- (((((list_update NIL) i) v) = NIL) /\\ ((((list_update ((CONS x) xs)) i) v) = (((COND (i = (NUMERAL _0))) ((CONS v) xs)) ((CONS x) (((list_update xs) (i - (NUMERAL (BIT1 _0)))) v)))))"}
{"_id": "1987", "title": "Trigonometry2.TWO_PI_POS", "text": "|- ((real_lt (real_of_num (NUMERAL _0))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))"}
{"_id": "1988", "title": "HAS_COMPLEX_DERIVATIVE_AT_WITHIN", "text": "|- (!f. (!f'. (!x. (!s. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative f) f') ((within (at x)) s)))))))"}
{"_id": "1989", "title": "RETRACT_OF_TRANSLATION_EQ", "text": "|- (!a. (!t. (!s. (((retract_of ((IMAGE (\\x. ((vector_add a) x))) t)) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((retract_of t) s)))))"}
{"_id": "1990", "title": "CONTRACTIBLE_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((contractible ((IMAGE f) s)) = (contractible s)))))"}
{"_id": "1991", "title": "BASIS_EXPANSION", "text": "|- (!x. (((vsum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. ((% (x $ i)) (basis i)))) = x))"}
{"_id": "1992", "title": "MEASURE_INTERVAL_2", "text": "|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ ((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))) /\\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ ((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))))"}
{"_id": "1993", "title": "HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_IMP", "text": "|- (!f. (!g. (((orthogonal_transformation f) /\\ ((orthogonal_transformation g) /\\ ((((homotopic_with (\\x. T)) ((sphere ((vec (NUMERAL _0)) , (real_of_num (NUMERAL (BIT1 _0))))) , (sphere ((vec (NUMERAL _0)) , (real_of_num (NUMERAL (BIT1 _0))))))) f) g))) ==> ((det (matrix f)) = (det (matrix g))))))"}
{"_id": "1994", "title": "int_eq", "text": "|- (!x. (!y. ((x = y) = ((real_of_int x) = (real_of_int y)))))"}
{"_id": "1995", "title": "connected_component", "text": "|- (!s. (!x. (!y. ((((connected_component s) x) y) = (?t. ((connected t) /\\ ((t SUBSET s) /\\ ((x IN t) /\\ (y IN t)))))))))"}
{"_id": "1996", "title": "HAS_REAL_INTEGRAL_SPIKE_INTERIOR", "text": "|- (!f. (!g. (!a. (!b. (!y. (((!x. ((x IN (open_real_interval (a , b))) ==> ((g x) = (f x)))) /\\ (((has_real_integral f) y) (closed_real_interval ((CONS (a , b)) NIL)))) ==> (((has_real_integral g) y) (closed_real_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "1997", "title": "TOPSPACE_EUCLIDEAN", "text": "|- ((topspace euclidean) = UNIV)"}
{"_id": "1998", "title": "Hypermap.number_of_faces", "text": "|- (!H. ((number_of_faces H) = (CARD (face_set H))))"}
{"_id": "1999", "title": "OPEN_UNIONS", "text": "|- ((!s. ((s IN f) ==> (open s))) ==> (open (UNIONS f)))"}
{"_id": "2000", "title": "Lp_ineqs_proofs.add_diag_list", "text": "|- (!V. (!g. (!E. (!diags. (((add_diag_list (g , (V , E))) diags) = (((foldr (\\d. (\\E'. ((add_diag (V , E')) (g d))))) E) diags))))))"}
{"_id": "2001", "title": "HULL_INC", "text": "|- (!P. (!s. (!x. ((x IN s) ==> (x IN ((hull P) s))))))"}
{"_id": "2002", "title": "FACE_OF_CONVEX_HULLS", "text": "|- (!f. (!s. (((FINITE s) /\\ ((f SUBSET s) /\\ ((DISJOINT ((hull affine) f)) ((hull convex) (s DIFF f))))) ==> ((face_of ((hull convex) f)) ((hull convex) s)))))"}
{"_id": "2003", "title": "REAL_LT_REFL", "text": "|- (!x. (~ ((real_lt x) x)))"}
{"_id": "2004", "title": "REALLIM_SEQUENTIALLY", "text": "|- (!s. (!l. ((((---> s) l) sequentially) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs ((real_sub (s n)) l))) e)))))))))"}
{"_id": "2005", "title": "COUNTABLE_ALT", "text": "|- (!s. ((COUNTABLE s) = (s <=_c UNIV)))"}
{"_id": "2006", "title": "Lp_ineqs_proofs.y1_hi", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (y1_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))"}
{"_id": "2007", "title": "lt_c", "text": "|- (!t. (!s. ((s <_c t) = ((s <=_c t) /\\ (~ (t <=_c s))))))"}
{"_id": "2008", "title": "WLOG_LE", "text": "|- (((!m. (!n. (((P m) n) = ((P n) m)))) /\\ (!m. (!n. ((m <= n) ==> ((P m) n))))) ==> (!m. (!n. ((P m) n))))"}
{"_id": "2009", "title": "CBALL_INTERVAL", "text": "|- (!x. (!e. ((cball (x , e)) = (closed_interval ((CONS (((vector_sub x) (lift e)) , ((vector_add x) (lift e)))) NIL)))))"}
{"_id": "2010", "title": "Real_ext.REAL_PROP_POS_MUL2", "text": "|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_mul x) y)))))"}
{"_id": "2011", "title": "Collect_geom.C_COEF_FORMULA", "text": "|- (!x12. (!x13. (!x14. (!x23. (!x24. ((((((c_coef x12) x13) x14) x23) x24) = ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24))))))))))))))))))"}
{"_id": "2012", "title": "CASN_CSIN", "text": "|- (!z. ((((real_lt (real_abs (Re z))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) \\/ (((real_abs (Re z)) = ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\\ ((Im z) = (real_of_num (NUMERAL _0))))) ==> ((casn (csin z)) = z)))"}
{"_id": "2013", "title": "Dih2k_hypermap.SUC_NOT", "text": "|- (((NUMERAL (BIT1 _0)) < k) ==> (((NUMERAL (BIT1 _0)) <= (SUC (i MOD k))) /\\ (((SUC (i MOD k)) <= k) /\\ (~ (i = (SUC (i MOD k)))))))"}
{"_id": "2014", "title": "LEBESGUE_DENSITY_THEOREM_INTEGRAL_CBALL", "text": "|- (!f. (!s. ((!a. (!b. ((absolutely_integrable_on f) (s INTER (closed_interval ((CONS (a , b)) NIL)))))) ==> (?k. ((negligible k) /\\ (!x. ((~ (x IN k)) ==> (((--> (\\e. ((% (real_inv (measure (cball (x , (drop e)))))) ((integral (s INTER (cball (x , (drop e))))) f)))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\\GEN%PVAR%3724. (?t. (((SETSPEC GEN%PVAR%3724) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))))))))))))"}
{"_id": "2015", "title": "Nonlinear_lemma.rho_alt", "text": "|- (!y. ((rho y) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_div ((real_sub y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0)))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))"}
{"_id": "2016", "title": "Ssrnat.subn_gt0", "text": "|- (!m. (!n. (((NUMERAL _0) < (n - m)) = (m < n))))"}
{"_id": "2017", "title": "Fan.f1_fan", "text": "|- (!x. (!V. (!E. ((((f1_fan x) V) E) = (GABS (\\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (w , ((((((inverse1_sigma_fan x) V) E) w) v) , v))))))))))))))"}
{"_id": "2018", "title": "Lp_ineqs_def.y3_fan", "text": "|- (!V. (!E. (!x. (((y3_fan (V , E)) x) = (vector_norm (FST ((inverse (f_fan_pair_ext (V , E))) x)))))))"}
{"_id": "2019", "title": "ANR_BALL", "text": "|- (!a. (!r. (ANR (ball (a , r)))))"}
{"_id": "2020", "title": "CLOSED_IN_TRANS", "text": "|- (!s. (!t. (!u. ((((closed_in ((subtopology euclidean) t)) s) /\\ ((closed_in ((subtopology euclidean) u)) t)) ==> ((closed_in ((subtopology euclidean) u)) s)))))"}
{"_id": "2021", "title": "Nonlin_def.scalar6", "text": "|- (!f. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!r. (((((((((scalar6 f) r) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) r))))))))))"}
{"_id": "2022", "title": "vertex_image", "text": "|- (!f. (!s. (((vertex_image f) s) = ((hull convex) ((IMAGE f) (GSPEC (\\GEN%PVAR%2552. (?v. (((SETSPEC GEN%PVAR%2552) ((extreme_point_of v) s)) v)))))))))"}
{"_id": "2023", "title": "WF_INSEG_WOSET", "text": "|- (WF (\\x. (\\y. ((woset x) /\\ ((woset y) /\\ (((inseg x) y) /\\ (~ (x = y))))))))"}
{"_id": "2024", "title": "List_hypermap.list_of_elements", "text": "|- (!L. ((list_of_elements L) = (undup (flatten L))))"}
{"_id": "2025", "title": "CHOICE_DEF", "text": "|- (!s. ((~ (s = EMPTY)) ==> ((CHOICE s) IN s)))"}
{"_id": "2026", "title": "SUM_SUPPORT", "text": "|- (!f. (!s. (((sum (((support real_add) f) s)) f) = ((sum s) f))))"}
{"_id": "2027", "title": "Fnjlbxs.infnorm_lt", "text": "|- (!e. (!v. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_lt (real_abs (v $ i))) e))) ==> ((real_lt (infnorm v)) e))))"}
{"_id": "2028", "title": "I_O_ID", "text": "|- (!f. (((I o f) = f) /\\ ((f o I) = f)))"}
{"_id": "2029", "title": "Fan.collinear_imp_azim_is_rezo_fan", "text": "|- (!x. (!v. (!u. (!y. (!h1. (!h2. (((~ (v = x)) /\\ ((~ (u = x)) /\\ ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) /\\ (((real_le (real_of_num (NUMERAL _0))) y) /\\ (((real_lt y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\\ (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\\ (((% (distance (v , x))) e3) = ((vector_sub v) x))) ==> (?psi. (?r1. (?r2. ((((vector_sub u) x) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub v) x))))) /\\ ((((vector_sub u) x) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) y)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) y)))) e2)) ((% h2) ((vector_sub v) x))))) /\\ (((real_lt (real_of_num (NUMERAL _0))) r1) /\\ ((real_lt (real_of_num (NUMERAL _0))) r2)))))))))))))))) ==> (y = (real_of_num (NUMERAL _0))))))))))"}
{"_id": "2030", "title": "Collect_geom.ETA_Y_SYYM", "text": "|- (!x. (!y. (!z. (((((eta_y x) y) z) = (((eta_y y) x) z)) /\\ ((((eta_y x) y) z) = (((eta_y z) y) x))))))"}
{"_id": "2031", "title": "rcone_ge", "text": "|- (rcone_ge = (rconesgn real_ge))"}
{"_id": "2032", "title": "Hypermap.face_set", "text": "|- (!H. ((face_set H) = ((set_of_orbits (dart H)) (face_map H))))"}
{"_id": "2033", "title": "STC", "text": "|- (!R. ((STC R) = (TC (SC R))))"}
{"_id": "2034", "title": "IN_DELETE", "text": "|- (!s. (!x. (!y. ((x IN (s DELETE y)) = ((x IN s) /\\ (~ (x = y)))))))"}
{"_id": "2035", "title": "TRACE_COVARIANCE_CAUCHY_SCHWARZ", "text": "|- (!A. (!B. ((real_le (trace ((matrix_mul (transp A)) B))) ((real_mul (sqrt (trace ((matrix_mul (transp A)) A)))) (sqrt (trace ((matrix_mul (transp B)) B)))))))"}
{"_id": "2036", "title": "Planarity.AFFINE_HULL_1", "text": "|- (!a. (((hull affine) (a INSERT EMPTY)) = (GSPEC (\\GEN%PVAR%5568. (?u. (((SETSPEC GEN%PVAR%5568) (u = (real_of_num (NUMERAL (BIT1 _0))))) ((% u) a)))))))"}
{"_id": "2037", "title": "Ssrnat.subn_sub", "text": "|- (!m. (!n. (!p. (((n - m) - p) = (n - (m + p))))))"}
{"_id": "2038", "title": "EMPTY_INTERIOR_LOWDIM", "text": "|- (!s. (((dim s) < (dimindex UNIV)) ==> ((interior s) = EMPTY)))"}
{"_id": "2039", "title": "INTERVAL_TRANSLATION", "text": "|- ((!c. (!a. (!b. ((closed_interval ((CONS (((vector_add c) a) , ((vector_add c) b))) NIL)) = ((IMAGE (\\x. ((vector_add c) x))) (closed_interval ((CONS (a , b)) NIL))))))) /\\ (!c. (!a. (!b. ((open_interval (((vector_add c) a) , ((vector_add c) b))) = ((IMAGE (\\x. ((vector_add c) x))) (open_interval (a , b))))))))"}
{"_id": "2040", "title": "PATH_COMPONENT_EQ_EMPTY", "text": "|- (!s. (!x. ((((path_component s) x) = EMPTY) = (~ (x IN s)))))"}
{"_id": "2041", "title": "NORM_POS_LE", "text": "|- (!x. ((real_le (real_of_num (NUMERAL _0))) (vector_norm x)))"}
{"_id": "2042", "title": "SEPARATING_HYPERPLANE_AFFINE_AFFINE", "text": "|- (!s. (!t. (((affine s) /\\ ((affine t) /\\ ((~ (s = EMPTY)) /\\ ((~ (t = EMPTY)) /\\ ((DISJOINT s) t))))) ==> (?a. (?b. (?c. ((~ (a = (vec (NUMERAL _0)))) /\\ (((real_lt b) c) /\\ ((!x. ((x IN s) ==> (((dot a) x) = b))) /\\ (!x. ((x IN t) ==> (((dot a) x) = c))))))))))))"}
{"_id": "2043", "title": "AFFINE_HULL_SING", "text": "|- (!a. (((hull affine) (a INSERT EMPTY)) = (a INSERT EMPTY)))"}
{"_id": "2044", "title": "LIM_COMPLEX_REAL", "text": "|- (!f. (!g. (!l. (!m. ((((eventually (\\n. ((Re (g n)) = (f n)))) sequentially) /\\ (((Re m) = l) /\\ (((--> g) m) sequentially))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs ((real_sub (f n)) l))) e)))))))))))"}
{"_id": "2045", "title": "UNIONS_INSERT", "text": "|- ((UNIONS (s INSERT u)) = (s UNION (UNIONS u)))"}
{"_id": "2046", "title": "CONNECTED_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((connected ((IMAGE f) s)) = (connected s)))))"}
{"_id": "2047", "title": "CLOSED_IN_COMPONENT", "text": "|- (!s. (!c. ((c IN (components s)) ==> ((closed_in ((subtopology euclidean) s)) c))))"}
{"_id": "2048", "title": "REAL_ADD_LDISTRIB", "text": "|- (!x. (!y. (!z. (((real_mul x) ((real_add y) z)) = ((real_add ((real_mul x) y)) ((real_mul x) z))))))"}
{"_id": "2049", "title": "RANK_DIM_IM", "text": "|- (!A. ((rank A) = (dim ((IMAGE (\\x. ((matrix_vector_mul A) x))) UNIV))))"}
{"_id": "2050", "title": "Trigonometry2.SIMPLIZE_COS_IF_OTHOR", "text": "|- (!v0. (!v1. (!w. (!p. (((~ (v0 = w)) /\\ ((~ (v0 = v1)) /\\ ((((vector_sub p) v0) = ((% k) ((vector_sub v1) v0))) /\\ (((dot ((vector_sub w) p)) ((vector_sub v1) v0)) = (real_of_num (NUMERAL _0)))))) ==> ((cos (((arcV v0) v1) w)) = ((real_mul k) ((real_div (vector_norm ((vector_sub v1) v0))) (vector_norm ((vector_sub w) v0))))))))))"}
{"_id": "2051", "title": "Hypermap.lemma_dart_invariant_power_face", "text": "|- (!H. (!x. (!n. ((x IN (dart H)) ==> ((((POWER (face_map H)) n) x) IN (dart H))))))"}
{"_id": "2052", "title": "INTERVAL_UPPERBOUND_NONEMPTY", "text": "|- (!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) ==> ((interval_upperbound (closed_interval ((CONS (a , b)) NIL))) = b))))"}
{"_id": "2053", "title": "REAL_COMPACT_INTERVAL", "text": "|- (!a. (!b. (real_compact (closed_real_interval ((CONS (a , b)) NIL)))))"}
{"_id": "2054", "title": "LIM_WITHINREAL", "text": "|- (!f. (!l. (!a. (!s. ((((--> f) l) ((within (atreal a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (((x IN s) /\\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\\ ((real_lt (real_abs ((real_sub x) a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))"}
{"_id": "2055", "title": "LINEAR_FROM_1", "text": "|- (!f. ((linear f) = (?c. (f = (\\x. ((% (drop x)) c))))))"}
{"_id": "2056", "title": "Sphere.eventually_radial", "text": "|- (!C. (!x. (((eventually_radial x) C) = (?r. (((real_gt r) (real_of_num (NUMERAL _0))) /\\ (((radial r) x) (C INTER (ball (x , r)))))))))"}
{"_id": "2057", "title": "ITER_POINTLESS", "text": "|- ((!f. (((ITER (NUMERAL _0)) f) = I)) /\\ (!f. (!n. (((ITER (SUC n)) f) = (f o ((ITER n) f))))))"}
{"_id": "2058", "title": "LINEAR_CONTINUOUS_ON", "text": "|- (!f. (!s. ((linear f) ==> ((continuous_on f) s))))"}
{"_id": "2059", "title": "LIM_CONST", "text": "|- (!net. (!a. (((--> (\\x. a)) a) net)))"}
{"_id": "2060", "title": "REAL_MEASURABLE_IMP_REAL_LEBESGUE_MEASURABLE", "text": "|- (!s. ((real_measurable s) ==> (real_lebesgue_measurable s)))"}
{"_id": "2061", "title": "INT_LE_REFL", "text": "|- (!x. ((int_le x) x))"}
{"_id": "2062", "title": "RE_DIV_CX", "text": "|- (!z. (!x. ((Re ((complex_div z) (Cx x))) = ((real_div (Re z)) x))))"}
{"_id": "2063", "title": "HAS_DERIVATIVE_ID", "text": "|- (!net. (((has_derivative (\\x. x)) (\\h. h)) net))"}
{"_id": "2064", "title": "Hypermap_and_fan.F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((permutes (f_fan_pair_ext (V , E))) (dart_of_fan (V , E))))))"}
{"_id": "2065", "title": "REAL_POS", "text": "|- (!n. ((real_le (real_of_num (NUMERAL _0))) (real_of_num n)))"}
{"_id": "2066", "title": "DROP_DIFFERENTIAL_NEG_AT_MAXIMUM", "text": "|- (!f. (!f'. (!x. (!s. (!e. (((x IN s) /\\ ((convex s) /\\ ((((has_derivative f) f') ((within (at x)) s)) /\\ (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (!w. ((w IN (s INTER (ball (x , e)))) ==> ((real_le (drop (f w))) (drop (f x))))))))) ==> (!y. ((y IN s) ==> ((real_le (drop (f' ((vector_sub y) x)))) (real_of_num (NUMERAL _0)))))))))))"}
{"_id": "2067", "title": "path_length", "text": "|- (!g. ((path_length g) = ((vector_variation (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) g)))"}
{"_id": "2068", "title": "PERMUTES_IN_IMAGE", "text": "|- (!p. (!s. (!x. (((permutes p) s) ==> (((p x) IN s) = (x IN s))))))"}
{"_id": "2069", "title": "Appendix.FZIOTEF_TRANS", "text": "|- (!S1. (!S2. (!S3. ((((scs_arrow_v39 S1) S2) /\\ ((scs_arrow_v39 S2) S3)) ==> ((scs_arrow_v39 S1) S3)))))"}
{"_id": "2070", "title": "NORM_CPOW_REAL", "text": "|- (!w. (!z. (((real w) /\\ ((real_lt (real_of_num (NUMERAL _0))) (Re w))) ==> ((vector_norm ((cpow w) z)) = (exp ((real_mul (Re z)) (log (Re w))))))))"}
{"_id": "2071", "title": "REAL_INTEGRABLE_ON", "text": "|- (((real_integrable_on f) s) = ((integrable_on (lift o (f o drop))) ((IMAGE lift) s)))"}
{"_id": "2072", "title": "LIM_NULL_COMPARISON_COMPLEX", "text": "|- (!net. (!f. (!g. ((((eventually (\\x. ((real_le (vector_norm (f x))) (vector_norm (g x))))) net) /\\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> f) (Cx (real_of_num (NUMERAL _0)))) net)))))"}
{"_id": "2073", "title": "EXTREME_POINT_OF_CONVEX_HULL", "text": "|- (!x. (!s. (((extreme_point_of x) ((hull convex) s)) ==> (x IN s))))"}
{"_id": "2074", "title": "OPEN_MAP_CLOSED_SUPERSET_PREIMAGE", "text": "|- (!f. (!s. (!t. (!u. (!w. (((!k. (((open_in ((subtopology euclidean) s)) k) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) k)))) /\\ (((closed_in ((subtopology euclidean) s)) u) /\\ ((w SUBSET t) /\\ ((GSPEC (\\GEN%PVAR%697. (?x. (((SETSPEC GEN%PVAR%697) ((x IN s) /\\ ((f x) IN w))) x)))) SUBSET u)))) ==> (?v. (((closed_in ((subtopology euclidean) t)) v) /\\ ((w SUBSET v) /\\ ((GSPEC (\\GEN%PVAR%698. (?x. (((SETSPEC GEN%PVAR%698) ((x IN s) /\\ ((f x) IN v))) x)))) SUBSET u))))))))))"}
{"_id": "2075", "title": "Tame_lemmas.reflection_eq", "text": "|- (!n. (!x. ((((dot x) n) = (real_of_num (NUMERAL _0))) ==> (((reflection n) x) = x))))"}
{"_id": "2076", "title": "differentiable", "text": "|- (!f. (!net. (((differentiable f) net) = (?f'. (((has_derivative f) f') net)))))"}
{"_id": "2077", "title": "HOLOMORPHIC_ON_MUL", "text": "|- (!f. (!g. (!s. ((((holomorphic_on f) s) /\\ ((holomorphic_on g) s)) ==> ((holomorphic_on (\\z. ((complex_mul (f z)) (g z)))) s)))))"}
{"_id": "2078", "title": "REAL_ADD_ASSOC", "text": "|- (!x. (!y. (!z. (((real_add x) ((real_add y) z)) = ((real_add ((real_add x) y)) z)))))"}
{"_id": "2079", "title": "HAS_COMPLEX_DERIVATIVE_CDIV_AT", "text": "|- (!f. (!f'. (!c. (!x. (!s. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative (\\x. ((complex_div (f x)) c))) ((complex_div f') c)) (at x))))))))"}
{"_id": "2080", "title": "CONVEX_HULL_FINITE", "text": "|- (!s. (((hull convex) s) = (GSPEC (\\GEN%PVAR%1491. (?y. (((SETSPEC GEN%PVAR%1491) (?u. ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((vsum s) (\\x. ((% (u x)) x))) = y))))) y))))))"}
{"_id": "2081", "title": "cross", "text": "|- (!a. (!b. (((cross a) b) = (vector ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 _0))))) ((real_mul (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 (BIT1 _0))))))) ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT1 _0)))))) NIL)))))))"}
{"_id": "2082", "title": "REAL_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET", "text": "|- (!f. (!s. (!t. (((s SUBSET t) /\\ (((real_measurable_on f) t) /\\ (real_lebesgue_measurable s))) ==> ((real_measurable_on f) s)))))"}
{"_id": "2083", "title": "ADMISSIBLE_COND", "text": "|- (!<<. (!p. (!P. (!s. (!h. (!k. ((((((admissible <<) p) s) P) /\\ (((((admissible <<) (\\f. (\\x. (((p f) x) /\\ ((P f) x))))) s) h) /\\ ((((admissible <<) (\\f. (\\x. (((p f) x) /\\ (~ ((P f) x)))))) s) k))) ==> ((((admissible <<) p) s) (\\f. (\\x. (((COND ((P f) x)) ((h f) x)) ((k f) x))))))))))))"}
{"_id": "2084", "title": "Lp_ineqs_proofs.dart4_azim_cross_split", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((azim_dart (V , E)) d) = ((real_add ((((((dih_y (y1_fan d)) ((y3_fan (V , E)) d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) ((y5_fan (V , E)) d))) ((((((dih_y (y1_fan d)) (y2_fan d)) ((y7_fan (V , E)) d)) ((y9_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) (y6_fan d))))))))))))"}
{"_id": "2085", "title": "REAL_LE_LCANCEL_IMP", "text": "|- (!x. (!y. (!z. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_le ((real_mul x) y)) ((real_mul x) z))) ==> ((real_le y) z)))))"}
{"_id": "2086", "title": "INTEGRAL_UNIQUE", "text": "|- (!f. (!y. (!k. ((((has_integral f) y) k) ==> (((integral k) f) = y)))))"}
{"_id": "2087", "title": "INTEGRAL_EQ_0", "text": "|- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (vec (NUMERAL _0))))) ==> (((integral s) f) = (vec (NUMERAL _0))))))"}
{"_id": "2088", "title": "CONNECTED_COMPONENT_EQ", "text": "|- (!s. (!x. (!y. ((y IN ((connected_component s) x)) ==> (((connected_component s) y) = ((connected_component s) x))))))"}
{"_id": "2089", "title": "REAL_LE_LADD", "text": "|- (!x. (!y. (!z. (((real_le ((real_add x) y)) ((real_add x) z)) = ((real_le y) z)))))"}
{"_id": "2090", "title": "REAL_CONTINUOUS_COMPLEX_COMPONENTS_AT", "text": "|- (!z. (((real_continuous Re) (at z)) /\\ ((real_continuous Im) (at z))))"}
{"_id": "2091", "title": "REAL_MEASURABLE_SCALING", "text": "|- (!s. (!c. ((real_measurable s) ==> (real_measurable ((IMAGE (\\x. ((real_mul c) x))) s)))))"}
{"_id": "2092", "title": "OPEN_UNION", "text": "|- (!s. (!t. (((open s) /\\ (open t)) ==> (open (s UNION t)))))"}
{"_id": "2093", "title": "NUMSEG_LREC", "text": "|- (!m. (!n. ((m <= n) ==> ((m INSERT ((m + (NUMERAL (BIT1 _0))) .. n)) = (m .. n)))))"}
{"_id": "2094", "title": "EQ_EXT", "text": "|- (!f. (!g. ((!x. ((f x) = (g x))) ==> (f = g))))"}
{"_id": "2095", "title": "REAL_ABS_BETWEEN1", "text": "|- (!x. (!y. (!z. ((((real_lt x) z) /\\ ((real_lt (real_abs ((real_sub y) x))) ((real_sub z) x))) ==> ((real_lt y) z)))))"}
{"_id": "2096", "title": "ITERATE_BIJECTION", "text": "|- (!op. ((monoidal op) ==> (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\\ ((p x) = y)))))) ==> ((((iterate op) s) f) = (((iterate op) s) (f o p)))))))))"}
{"_id": "2097", "title": "Ssrbool.andFb", "text": "|- (!b. ((F /\\ b) = F))"}
{"_id": "2098", "title": "REAL_CONTINUOUS_ON_RPOW", "text": "|- (!s. (!y. ((((real_of_num (NUMERAL _0)) IN s) ==> ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_continuous_on (\\x. ((rpow x) y))) s))))"}
{"_id": "2099", "title": "NSUM_CLAUSES_LEFT", "text": "|- (!f. (!m. (!n. ((m <= n) ==> (((nsum (m .. n)) f) = ((f m) + ((nsum ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))"}
{"_id": "2100", "title": "DIM_EQ_CARD", "text": "|- (!s. ((independent s) ==> ((dim s) = (CARD s))))"}
{"_id": "2101", "title": "DIFFERENTIABLE_IMP_CONTINUOUS_AT", "text": "|- (!f. (!x. (((differentiable f) (at x)) ==> ((continuous f) (at x)))))"}
{"_id": "2102", "title": "SUM_INJECTION", "text": "|- (!f. (!p. (!s. (((FINITE s) /\\ ((!x. ((x IN s) ==> ((p x) IN s))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((p x) = (p y)))) ==> (x = y)))))) ==> (((sum s) (f o p)) = ((sum s) f))))))"}
{"_id": "2103", "title": "Hypermap_and_fan.CARD_FACE_GT_1", "text": "|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((CARD ((face (hypermap_of_fan (V , E))) x)) > (NUMERAL (BIT1 _0)))) ==> (x IN (dart1_of_fan (V , E)))))))"}
{"_id": "2104", "title": "COND_RAND", "text": "|- (!b. (!f. (!x. (!y. ((f (((COND b) x) y)) = (((COND b) (f x)) (f y)))))))"}
{"_id": "2105", "title": "Nonlin_def.promote3_to_6", "text": "|- (!x4. (!x5. (!x6. (!f. (!x1. (!x2. (!x3. ((((((((promote3_to_6 f) x1) x2) x3) x4) x5) x6) = (((f x1) x2) x3)))))))))"}
{"_id": "2106", "title": "HAS_DERIVATIVE_NEG_EQ", "text": "|- (!f. (!f'. (!net. ((((has_derivative (\\x. (vector_neg (f x)))) (\\h. (vector_neg (f' h)))) net) = (((has_derivative f) f') net)))))"}
{"_id": "2107", "title": "Hypermap_iso.bij_disjoint_union", "text": "|- (!f. (!g. (!s. (!t. (!s'. (!t'. ((((DISJOINT s) t) /\\ (((DISJOINT s') t') /\\ ((((BIJ f) s) s') /\\ (((BIJ g) t) t')))) ==> (((BIJ (\\x. (((COND (x IN s)) (f x)) (g x)))) (s UNION t)) (s' UNION t')))))))))"}
{"_id": "2108", "title": "ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV", "text": "|- (!f. (!s. (((absolutely_integrable_on (\\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) UNIV) = ((absolutely_integrable_on f) s))))"}
{"_id": "2109", "title": "Fan_misc.INVERSE_SIGMA_FAN_EQ_INVERSE1_SIGMA_FAN", "text": "|- (!x. (!V. (!E. (!v. (!w. (((FAN (x , (V , E))) /\\ ((v INSERT (w INSERT EMPTY)) IN E)) ==> ((((((inverse1_sigma_fan x) V) E) v) w) = (((((inverse_sigma_fan x) V) E) v) w))))))))"}
{"_id": "2110", "title": "DROP_VEC", "text": "|- (!n. ((drop (vec n)) = (real_of_num n)))"}
{"_id": "2111", "title": "Functional_equation.gchi_all_atn", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((gchi1_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x1))) ((((((dihatn_x x1) x2) x3) x4) x5) x6))) /\\ ((((((((gchi2_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x2))) ((((((dih2atn_x x1) x2) x3) x4) x5) x6))) /\\ ((((((((gchi3_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x3))) ((((((dih3atn_x x1) x2) x3) x4) x5) x6))) /\\ ((((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x4))) ((((((dih4atn_x x1) x2) x3) x4) x5) x6))) /\\ ((((((((gchi5_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x5))) ((((((dih5atn_x x1) x2) x3) x4) x5) x6))) /\\ (((((((gchi6_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x6))) ((((((dih6atn_x x1) x2) x3) x4) x5) x6)))))))))))))))"}
{"_id": "2112", "title": "Tame_defs.contracted_dart", "text": "|- (!x. (!w1. (!v. (!w. ((contracted_dart (x , (v , (w , w1)))) = (v , w))))))"}
{"_id": "2113", "title": "CONDENSATION_POINTS_EQ_EMPTY", "text": "|- (!s. (((GSPEC (\\GEN%PVAR%1861. (?x. (((SETSPEC GEN%PVAR%1861) ((condensation_point_of x) s)) x)))) = EMPTY) = (COUNTABLE s)))"}
{"_id": "2114", "title": "SPAN_SPAN", "text": "|- (!s. ((span (span s)) = (span s)))"}
{"_id": "2115", "title": "Planarity.properties_inside_collinear_fan", "text": "|- (!x. (!u. (!w. (!a. ((((real_lt (real_of_num (NUMERAL _0))) a) /\\ (((real_lt a) (real_of_num (NUMERAL (BIT1 _0)))) /\\ (~ (collinear (x INSERT (u INSERT (w INSERT EMPTY))))))) ==> ((~ (collinear (x INSERT (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) INSERT (u INSERT EMPTY))))) /\\ (~ (collinear (x INSERT (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) INSERT (w INSERT EMPTY)))))))))))"}
{"_id": "2116", "title": "Fan.pr3", "text": "|- (pr3 = (GABS (\\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) w))))))))"}
{"_id": "2117", "title": "RELATIVE_INTERIOR_EMPTY", "text": "|- ((relative_interior EMPTY) = EMPTY)"}
{"_id": "2118", "title": "Miqmcsn.K_SCS_3T4_prime2", "text": "|- ((scs_k_v39 scs_3T4_prime2) = (NUMERAL (BIT1 (BIT1 _0))))"}
{"_id": "2119", "title": "Lp_ineqs_proofs.y3_hi", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y3_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))"}
{"_id": "2120", "title": "SIMPLICIAL_COMPLEX_IMP_TRIANGULATION", "text": "|- (!tr. ((simplicial_complex tr) ==> (triangulation tr)))"}
{"_id": "2121", "title": "UNIFORM_LIM_COMPLEX_MUL", "text": "|- (!net. (!P. (!f. (!g. (!l. (!m. (!b1. (!b2. ((((eventually (\\x. (!n. ((P n) ==> ((real_le (vector_norm (l n))) b1))))) net) /\\ (((eventually (\\x. (!n. ((P n) ==> ((real_le (vector_norm (m n))) b2))))) net) /\\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) (l n)))) e))))) net))) /\\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((g n) x)) (m n)))) e))))) net)))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((complex_mul ((f n) x)) ((g n) x))) ((complex_mul (l n)) (m n))))) e))))) net))))))))))))"}
{"_id": "2122", "title": "ISO", "text": "|- (!g. (!f. (((ISO f) g) = ((!x. ((f (g x)) = x)) /\\ (!y. ((g (f y)) = y))))))"}
{"_id": "2123", "title": "MEM_EXISTS_EL", "text": "|- (!l. (!x. (((MEM x) l) = (?i. ((i < (LENGTH l)) /\\ (x = ((EL i) l)))))))"}
{"_id": "2124", "title": "CONVEX_SIMPLEX", "text": "|- (!n. (!s. (((simplex n) s) ==> (convex s))))"}
{"_id": "2125", "title": "Hypermap.orbit_map", "text": "|- (!f. (!x. (((orbit_map f) x) = (GSPEC (\\GEN%PVAR%5390. (?n. (((SETSPEC GEN%PVAR%5390) (n >= (NUMERAL _0))) (((POWER f) n) x))))))))"}
{"_id": "2126", "title": "VECTOR_MUL_RCANCEL", "text": "|- (!a. (!b. (!x. ((((% a) x) = ((% b) x)) = ((a = b) \\/ (x = (vec (NUMERAL _0))))))))"}
{"_id": "2127", "title": "Real_ext.REAL_PROP_POS_LMUL", "text": "|- ((!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) x) ==> (((real_lt (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_lt (real_of_num (NUMERAL _0))) y))))) /\\ (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) y) ==> (((real_lt (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_lt (real_of_num (NUMERAL _0))) x))))))"}
{"_id": "2128", "title": "COMPLEX_ADD_LINV", "text": "|- (!x. (((vector_add (vector_neg x)) x) = (Cx (real_of_num (NUMERAL _0)))))"}
{"_id": "2129", "title": "TRIANGULATION_TRANSLATION", "text": "|- (!a. (!tr. ((triangulation ((IMAGE (IMAGE (\\x. ((vector_add a) x)))) tr)) = (triangulation tr))))"}
{"_id": "2130", "title": "COMPLEX_ADD_CSIN", "text": "|- (!w. (!z. (((vector_add (csin w)) (csin z)) = ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_mul (csin ((complex_div ((vector_add w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) (ccos ((complex_div ((vector_sub w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))"}
{"_id": "2131", "title": "REAL_LE_SETDIST_EQ", "text": "|- (!d. (!s. (!t. (((real_le d) (setdist (s , t))) = ((!x. (!y. (((x IN s) /\\ (y IN t)) ==> ((real_le d) (distance (x , y)))))) /\\ (((s = EMPTY) \\/ (t = EMPTY)) ==> ((real_le d) (real_of_num (NUMERAL _0)))))))))"}
{"_id": "2132", "title": "Ssrnat.leqSS", "text": "|- (!n. (!m. (((SUC n) <= (SUC m)) = (n <= m))))"}
{"_id": "2133", "title": "FINITE_SING", "text": "|- (!a. (FINITE (a INSERT EMPTY)))"}
{"_id": "2134", "title": "Lp_ineqs_quads.invf_in_dart", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((inverse (f_fan_pair_ext (V , E))) d) IN (dart_of_fan (V , E))))))))"}
{"_id": "2135", "title": "SUM_SUPERSET", "text": "|- (!f. (!u. (!v. (((u SUBSET v) /\\ (!x. (((x IN v) /\\ (~ (x IN u))) ==> ((f x) = (real_of_num (NUMERAL _0)))))) ==> (((sum v) f) = ((sum u) f))))))"}
{"_id": "2136", "title": "homotopic_with", "text": "|- (!X. (!Y. (!p. (!q. (!P. (((((homotopic_with P) (X , Y)) p) q) = (?h. (((continuous_on h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) /\\ ((((IMAGE h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) SUBSET Y) /\\ ((!x. ((h ((pastecart (vec (NUMERAL _0))) x)) = (p x))) /\\ ((!x. ((h ((pastecart (vec (NUMERAL (BIT1 _0)))) x)) = (q x))) /\\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> (P (\\x. (h ((pastecart t) x)))))))))))))))))"}
{"_id": "2137", "title": "Seq.zip", "text": "|- ((((zip ((CONS x) s)) ((CONS y) t)) = ((CONS (x , y)) ((zip s) t))) /\\ ((((zip NIL) t) = NIL) /\\ (((zip s) NIL) = NIL)))"}
{"_id": "2138", "title": "HAS_BOUNDED_VARIATION_ON_EMPTY", "text": "|- (!f. ((has_bounded_variation_on f) EMPTY))"}
{"_id": "2139", "title": "COLLINEAR_DIST_BETWEEN", "text": "|- (!a. (!b. (!x. (((collinear (x INSERT (a INSERT (b INSERT EMPTY)))) /\\ (((real_le (distance (x , a))) (distance (a , b))) /\\ ((real_le (distance (x , b))) (distance (a , b))))) ==> ((between x) (a , b))))))"}
{"_id": "2140", "title": "FORALL_BOOL_THM", "text": "|- ((!b. (P b)) = ((P T) /\\ (P F)))"}
{"_id": "2141", "title": "Ssrbool.contraL", "text": "|- (!c. (!b. ((c ==> (~ b)) ==> (b ==> (~ c)))))"}
{"_id": "2142", "title": "numseg", "text": "|- (!m. (!n. ((m .. n) = (GSPEC (\\GEN%PVAR%150. (?x. (((SETSPEC GEN%PVAR%150) ((m <= x) /\\ (x <= n))) x)))))))"}
{"_id": "2143", "title": "BROUWER_REDUCTION_THEOREM", "text": "|- (!P. (!s. (((!f. (((!n. ((compact (f n)) /\\ ((~ ((f n) = EMPTY)) /\\ (P (f n))))) /\\ (!n. ((f (SUC n)) SUBSET (f n)))) ==> (P (INTERS (GSPEC (\\GEN%PVAR%1277. (?n. (((SETSPEC GEN%PVAR%1277) (n IN UNIV)) (f n))))))))) /\\ ((compact s) /\\ ((~ (s = EMPTY)) /\\ (P s)))) ==> (?t. ((t SUBSET s) /\\ ((compact t) /\\ ((~ (t = EMPTY)) /\\ ((P t) /\\ (!u. (((u SUBSET s) /\\ ((closed u) /\\ ((~ (u = EMPTY)) /\\ (P u)))) ==> (~ (u PSUBSET t))))))))))))"}
{"_id": "2144", "title": "DIAMETER_TRANSLATION", "text": "|- (!a. (!s. ((diameter ((IMAGE (\\x. ((vector_add a) x))) s)) = (diameter s))))"}
{"_id": "2145", "title": "Lp_main_estimate.f_surr_tau_fun_eq_tauVEF", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((((tau_fun V) E) f) = (tauVEF (V , (E , f)))))))))))"}
{"_id": "2146", "title": "REAL_EQ_RDIV_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> ((x = ((real_div y) z)) = (((real_mul x) z) = y))))))"}
{"_id": "2147", "title": "SUMMABLE_IMP_BOUNDED", "text": "|- (!f. (!k. (((summable k) f) ==> (bounded ((IMAGE f) k)))))"}
{"_id": "2148", "title": "NEGLIGIBLE_CONVEX_HULL", "text": "|- (!s. (((FINITE s) /\\ ((CARD s) <= (dimindex UNIV))) ==> (negligible ((hull convex) s))))"}
{"_id": "2149", "title": "HAS_BOUNDED_REAL_VARIATION_TRANSLATION_EQ", "text": "|- (!a. (!f. (!s. (((has_bounded_real_variation_on (\\x. (f ((real_add a) x)))) s) = ((has_bounded_real_variation_on f) ((IMAGE (\\x. ((real_add a) x))) s))))))"}
{"_id": "2150", "title": "ROWS_MAPROWS", "text": "|- (!f. (!A. ((rows ((maprows f) A)) = ((IMAGE f) (rows A)))))"}
{"_id": "2151", "title": "HAS_DERIVATIVE_SEQUENCE", "text": "|- (!s. (!f. (!f'. (!g'. (((convex s) /\\ ((!n. (!x. ((x IN s) ==> (((has_derivative (f n)) ((f' n) x)) ((within (at x)) s))))) /\\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (!h. (((n >= N) /\\ (x IN s)) ==> ((real_le (vector_norm ((vector_sub (((f' n) x) h)) ((g' x) h)))) ((real_mul e) (vector_norm h)))))))))) /\\ (?x. (?l. ((x IN s) /\\ (((--> (\\n. ((f n) x))) l) sequentially))))))) ==> (?g. (!x. ((x IN s) ==> ((((--> (\\n. ((f n) x))) (g x)) sequentially) /\\ (((has_derivative g) (g' x)) ((within (at x)) s)))))))))))"}
{"_id": "2152", "title": "POSITIVE_SEMIDEFINITE_ADD", "text": "|- (!A. (!B. (((positive_semidefinite A) /\\ (positive_semidefinite B)) ==> (positive_semidefinite ((matrix_add A) B)))))"}
{"_id": "2153", "title": "REAL_CONTINUOUS_AT_ACS", "text": "|- (!x. (((real_lt (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))) ==> ((real_continuous acs) (atreal x))))"}
{"_id": "2154", "title": "lim", "text": "|- (!f. (!net. (((lim net) f) = (@l. (((--> f) l) net)))))"}
{"_id": "2155", "title": "MOD_LT", "text": "|- (!m. (!n. ((m < n) ==> ((m MOD n) = m))))"}
{"_id": "2156", "title": "STC_TRANS", "text": "|- (!R. (!x. (!y. (!z. (((((STC R) x) y) /\\ (((STC R) y) z)) ==> (((STC R) x) z))))))"}
{"_id": "2157", "title": "AFFINE_INDEPENDENT_IMP_FINITE", "text": "|- (!s. ((~ (affine_dependent s)) ==> (FINITE s)))"}
{"_id": "2158", "title": "Miqmcsn.K_SCS_4M8_13", "text": "|- ((scs_k_v39 scs_4M8_13) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))"}
{"_id": "2159", "title": "POLYTOPE_EMPTY", "text": "|- (polytope EMPTY)"}
{"_id": "2160", "title": "SEGMENT_TO_FRONTIER_SIMPLE", "text": "|- (!s. (!x. (((bounded s) /\\ (x IN s)) ==> (?a. (?b. ((a IN (frontier s)) /\\ ((b IN (frontier s)) /\\ (x IN (closed_segment ((CONS (a , b)) NIL))))))))))"}
{"_id": "2161", "title": "RPOW_ONE", "text": "|- (!z. (((rpow (real_of_num (NUMERAL (BIT1 _0)))) z) = (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "2162", "title": "Hypermap_and_fan.PAIR_IN_DART1_OF_FAN", "text": "|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((v , w) IN (dart1_of_fan (V , E)))) ==> ((v IN V) /\\ ((w IN V) /\\ ((w IN (((set_of_edge v) V) E)) /\\ (v IN (((set_of_edge w) V) E))))))))))"}
{"_id": "2163", "title": "SEGMENT_TO_RELATIVE_FRONTIER_SIMPLE", "text": "|- (!s. (!x. (((bounded s) /\\ ((x IN s) /\\ (~ (s = (x INSERT EMPTY))))) ==> (?a. (?b. ((a IN (relative_frontier s)) /\\ ((b IN (relative_frontier s)) /\\ (x IN (closed_segment ((CONS (a , b)) NIL))))))))))"}
{"_id": "2164", "title": "HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS", "text": "|- (!m. (!f. (((FINITE f) /\\ ((!s. ((s IN f) ==> ((has_real_measure s) (m s)))) /\\ (!s. (!t. (((s IN f) /\\ ((t IN f) /\\ (~ (s = t)))) ==> (real_negligible (s INTER t))))))) ==> ((has_real_measure (UNIONS f)) ((sum f) m)))))"}
{"_id": "2165", "title": "LE_MULT_RCANCEL", "text": "|- (!m. (!n. (!p. (((m * p) <= (n * p)) = ((m <= n) \\/ (p = (NUMERAL _0)))))))"}
{"_id": "2166", "title": "NPRODUCT_POS_LT", "text": "|- (!f. (!s. (((FINITE s) /\\ (!x. ((x IN s) ==> ((NUMERAL _0) < (f x))))) ==> ((NUMERAL _0) < ((nproduct s) f)))))"}
{"_id": "2167", "title": "CONVEX_HULL_SUBSET_AFFINE_HULL", "text": "|- (!s. (((hull convex) s) SUBSET ((hull affine) s)))"}
{"_id": "2168", "title": "Nonlin_def.sol_euler156_x_div_sqrtdelta", "text": "|- (sol_euler156_x_div_sqrtdelta = (rotate6 sol_euler_x_div_sqrtdelta))"}
{"_id": "2169", "title": "IM_CNJ", "text": "|- (!z. ((Im (cnj z)) = (real_neg (Im z))))"}
{"_id": "2170", "title": "Seq2.list_sum_eq0", "text": "|- (!s. (!f. ((!x. (((MEM x) s) ==> ((f x) = (real_of_num (NUMERAL _0))))) ==> (((list_sum s) f) = (real_of_num (NUMERAL _0))))))"}
{"_id": "2171", "title": "IMAGE_FSTCART_PCROSS", "text": "|- (!s. (!t. (((IMAGE fstcart) (s PCROSS t)) = (((COND (t = EMPTY)) EMPTY) s))))"}
{"_id": "2172", "title": "LTE_TRANS", "text": "|- (!m. (!n. (!p. (((m < n) /\\ (n <= p)) ==> (m < p)))))"}
{"_id": "2173", "title": "has_bounded_real_variation_on", "text": "|- (!f. (!s. (((has_bounded_real_variation_on f) s) = ((has_bounded_variation_on (lift o (f o drop))) ((IMAGE lift) s)))))"}
{"_id": "2174", "title": "INSIDE_OUTSIDE_INTERSECT_CONNECTED", "text": "|- (!s. (!t. (((connected t) /\\ ((~ (((inside s) INTER t) = EMPTY)) /\\ (~ (((outside s) INTER t) = EMPTY)))) ==> (~ ((s INTER t) = EMPTY)))))"}
{"_id": "2175", "title": "OPEN_IN_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((open_in ((subtopology euclidean) ((IMAGE f) t))) ((IMAGE f) s)) = ((open_in ((subtopology euclidean) t)) s))))))"}
{"_id": "2176", "title": "LINEAR_SNDCART", "text": "|- (linear sndcart)"}
{"_id": "2177", "title": "UNBOUNDED_COMPLEMENT_COMPONENT_CONVEX", "text": "|- (!s. (!c. (((convex s) /\\ (c IN (components (UNIV DIFF s)))) ==> (~ (bounded c)))))"}
{"_id": "2178", "title": "REAL_NEGLIGIBLE_COUNTABLE_UNIONS", "text": "|- (!s. ((!n. (real_negligible (s n))) ==> (real_negligible (UNIONS (GSPEC (\\GEN%PVAR%4943. (?n. (((SETSPEC GEN%PVAR%4943) (n IN UNIV)) (s n)))))))))"}
{"_id": "2179", "title": "sgn_ge", "text": "|- (sgn_ge = (\\t. ((real_le (real_of_num (NUMERAL _0))) t)))"}
{"_id": "2180", "title": "Merge_ineq.delta_y_sym", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x2) x1) x3) x5) x4) x6)) /\\ ((((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x1) x3) x2) x4) x6) x5)) /\\ (((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x5) x4) x3) x2) x1) x6))))))))))"}
{"_id": "2181", "title": "CLOSURE_HALFSPACE_COMPONENT_GT", "text": "|- (!a. (!k. ((closure (GSPEC (\\GEN%PVAR%812. (?x. (((SETSPEC GEN%PVAR%812) ((real_gt (x $ k)) a)) x))))) = (GSPEC (\\GEN%PVAR%813. (?x. (((SETSPEC GEN%PVAR%813) ((real_ge (x $ k)) a)) x)))))))"}
{"_id": "2182", "title": "FINITE_PERMUTATIONS", "text": "|- (!s. ((FINITE s) ==> (FINITE (GSPEC (\\GEN%PVAR%249. (?p. (((SETSPEC GEN%PVAR%249) ((permutes p) s)) p)))))))"}
{"_id": "2183", "title": "MEASURE_SING", "text": "|- (!a. ((measure (a INSERT EMPTY)) = (real_of_num (NUMERAL _0))))"}
{"_id": "2184", "title": "POLYTOPE_IMP_POLYHEDRON", "text": "|- (!p. ((polytope p) ==> (polyhedron p)))"}
{"_id": "2185", "title": "Hypermap.orbit_cyclic", "text": "|- (!f. (!m. (!x. (((~ (m = (NUMERAL _0))) /\\ ((((POWER f) m) x) = x)) ==> (((orbit_map f) x) = (GSPEC (\\GEN%PVAR%5396. (?k. (((SETSPEC GEN%PVAR%5396) (k < m)) (((POWER f) k) x))))))))))"}
{"_id": "2186", "title": "REALLIM_NULL_COMPARISON", "text": "|- (!net. (!f. (!g. ((((eventually (\\x. ((real_le (real_abs (f x))) (g x)))) net) /\\ (((---> g) (real_of_num (NUMERAL _0))) net)) ==> (((---> f) (real_of_num (NUMERAL _0))) net)))))"}
{"_id": "2187", "title": "Real_ext.REAL_PROP_NN_INV", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) (real_inv x)) = ((real_le (real_of_num (NUMERAL _0))) x)))"}
{"_id": "2188", "title": "QUANTIFY_SURJECTION_HIGHER_THM", "text": "|- (!f. ((!y. (?x. ((f x) = y))) ==> (((!P. ((!x. (P x)) = (!x. (P (f x))))) /\\ ((!P. ((?x. (P x)) = (?x. (P (f x))))) /\\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE f) s))))) /\\ ((!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE f) s))))) /\\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE (IMAGE f)) s))))) /\\ ((!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE (IMAGE f)) s))))) /\\ ((!P. ((!g. (P g)) = (!g. (P (f o g))))) /\\ ((!P. ((?g. (P g)) = (?g. (P (f o g))))) /\\ ((!P. ((!g. (P g)) = (!g. (P (f o g))))) /\\ ((!P. ((?g. (P g)) = (?g. (P (f o g))))) /\\ ((!Q. ((!l. (Q l)) = (!l. (Q ((MAP f) l))))) /\\ (!Q. ((?l. (Q l)) = (?l. (Q ((MAP f) l)))))))))))))))) /\\ ((!P. ((GSPEC (\\GEN%PVAR%238. (?x. (((SETSPEC GEN%PVAR%238) (P x)) x)))) = ((IMAGE f) (GSPEC (\\GEN%PVAR%239. (?x. (((SETSPEC GEN%PVAR%239) (P (f x))) x))))))) /\\ ((!Q. ((GSPEC (\\GEN%PVAR%240. (?s. (((SETSPEC GEN%PVAR%240) (Q s)) s)))) = ((IMAGE (IMAGE f)) (GSPEC (\\GEN%PVAR%241. (?s. (((SETSPEC GEN%PVAR%241) (Q ((IMAGE f) s))) s))))))) /\\ (!R. ((GSPEC (\\GEN%PVAR%242. (?l. (((SETSPEC GEN%PVAR%242) (R l)) l)))) = ((IMAGE (MAP f)) (GSPEC (\\GEN%PVAR%243. (?l. (((SETSPEC GEN%PVAR%243) (R ((MAP f) l))) l))))))))))))"}
{"_id": "2189", "title": "Ssrnat.maxn", "text": "|- (!n. (!m. (((maxn m) n) = (((COND (m < n)) n) m))))"}
{"_id": "2190", "title": "SIMPLEX_ORDERING_EXISTS", "text": "|- (!s. (((simplex ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))) s) ==> (?A. ((rows A) = (GSPEC (\\GEN%PVAR%2546. (?v. (((SETSPEC GEN%PVAR%2546) ((extreme_point_of v) s)) v))))))))"}
{"_id": "2191", "title": "IS_INTERVAL_IMP_ENR", "text": "|- (!s. ((is_interval s) ==> (ENR s)))"}
{"_id": "2192", "title": "REAL_MUL_ASSOC", "text": "|- (!x. (!y. (!z. (((real_mul x) ((real_mul y) z)) = ((real_mul ((real_mul x) y)) z)))))"}
{"_id": "2193", "title": "VECTOR_VARIATION_REFLECT2", "text": "|- (!f. (!s. (((vector_variation ((IMAGE vector_neg) s)) (\\x. (f (vector_neg x)))) = ((vector_variation s) f))))"}
{"_id": "2194", "title": "BALL_EQ_EMPTY", "text": "|- (!x. (!e. (((ball (x , e)) = EMPTY) = ((real_le e) (real_of_num (NUMERAL _0))))))"}
{"_id": "2195", "title": "IN_SING", "text": "|- (!x. (!y. ((x IN (y INSERT EMPTY)) = (x = y))))"}
{"_id": "2196", "title": "AFF_DIM_UNIV", "text": "|- ((aff_dim UNIV) = (int_of_num (dimindex UNIV)))"}
{"_id": "2197", "title": "AFFINE_DEPENDENT_IMP_DEPENDENT", "text": "|- (!s. ((affine_dependent s) ==> (dependent s)))"}
{"_id": "2198", "title": "BOUNDED_HALFSPACE_LT", "text": "|- (!a. (!b. ((bounded (GSPEC (\\GEN%PVAR%844. (?x. (((SETSPEC GEN%PVAR%844) ((real_lt ((dot a) x)) b)) x))))) = ((a = (vec (NUMERAL _0))) /\\ ((real_le b) (real_of_num (NUMERAL _0)))))))"}
{"_id": "2199", "title": "ONE", "text": "|- ((NUMERAL (BIT1 _0)) = (SUC (NUMERAL _0)))"}
{"_id": "2200", "title": "BARYCENTRE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((barycentre ((IMAGE f) s)) = (f (barycentre s))))))"}
{"_id": "2201", "title": "CLOSURE_SEGMENT", "text": "|- ((!a. (!b. ((closure (closed_segment ((CONS (a , b)) NIL))) = (closed_segment ((CONS (a , b)) NIL))))) /\\ (!a. (!b. ((closure (open_segment (a , b))) = (((COND (a = b)) EMPTY) (closed_segment ((CONS (a , b)) NIL)))))))"}
{"_id": "2202", "title": "SECOND_MEAN_VALUE_THEOREM_BONNET_FULL", "text": "|- (!f. (!g. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_interval ((CONS (a , b)) NIL))) /\\ (((has_integral (\\x. ((% (g x)) (f x)))) ((% (g b)) ((integral (closed_interval ((CONS (c , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL))))))))))"}
{"_id": "2203", "title": "CLOSURE_EMPTY", "text": "|- ((closure EMPTY) = EMPTY)"}
{"_id": "2204", "title": "SERIES_LINEAR", "text": "|- (!f. (!h. (!l. (!s. (((((sums f) l) s) /\\ (linear h)) ==> (((sums (\\n. (h (f n)))) (h l)) s))))))"}
{"_id": "2205", "title": "Tame_lemmas.fully_surrounded_sol_eq_sol_y", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0))))) ==> ((LET (\\w'. (LET_END ((LET (\\y1. (LET_END ((LET (\\y2. (LET_END ((LET (\\y3. (LET_END ((LET (\\y4. (LET_END ((LET (\\y5. (LET_END ((LET (\\y6. (LET_END (((sol (vec (NUMERAL _0))) (((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w)) = ((((((sol_y y1) y2) y3) y4) y5) y6))))) (distance (v , w)))))) (distance (v , w')))))) (distance (w , w')))))) (vector_norm w'))))) (vector_norm w))))) (vector_norm v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))"}
{"_id": "2206", "title": "COMPACT_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((compact ((IMAGE f) s)) = (compact s)))))"}
{"_id": "2207", "title": "NOT_IN_EMPTY", "text": "|- (!x. (~ (x IN EMPTY)))"}
{"_id": "2208", "title": "Functional_equation.promote1_to_6", "text": "|- (!x2. (!x3. (!x4. (!x5. (!x6. (!f. (!x1. ((((((((promote1_to_6 f) x1) x2) x3) x4) x5) x6) = (f x1)))))))))"}
{"_id": "2209", "title": "Hypermap_and_fan.FULLY_SURROUNDED_IMP_CARD_FACE_GE_3", "text": "|- (!V. (!E. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (fully_surrounded (V , E))) ==> (!x. ((x IN (dart_of_fan (V , E))) ==> ((CARD ((face (hypermap_of_fan (V , E))) x)) >= (NUMERAL (BIT1 (BIT1 _0)))))))))"}
{"_id": "2210", "title": "real_sub", "text": "|- (!x. (!y. (((real_sub x) y) = ((real_add x) (real_neg y)))))"}
{"_id": "2211", "title": "NOT_LT", "text": "|- (!m. (!n. ((~ (m < n)) = (n <= m))))"}
{"_id": "2212", "title": "ARITH_ZERO", "text": "|- (((NUMERAL (NUMERAL _0)) = (NUMERAL _0)) /\\ ((BIT0 _0) = _0))"}
{"_id": "2213", "title": "COMPLEX_DIV_1", "text": "|- (!z. (((complex_div z) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = z))"}
{"_id": "2214", "title": "Appendix.scs_str_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_str_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = str))))))))))"}
{"_id": "2215", "title": "Yxionxl.PRO_EQU_ID1", "text": "|- ((((scs_k_v39 s) = k) /\\ (is_scs_v39 s)) ==> (s = ((scs_prop_equ_v39 ((scs_prop_equ_v39 s) i)) (k - (i MOD k)))))"}
{"_id": "2216", "title": "list_INDUCT", "text": "|- (!P. (((P NIL) /\\ (!a0. (!a1. ((P a1) ==> (P ((CONS a0) a1)))))) ==> (!x. (P x))))"}
{"_id": "2217", "title": "MEM_TRANSLATION", "text": "|- (!a. (!x. (!l. (((MEM ((vector_add a) x)) ((MAP (\\x. ((vector_add a) x))) l)) = ((MEM x) l)))))"}
{"_id": "2218", "title": "LINEAR_CONTINUOUS_AT", "text": "|- (!f. (!a. ((linear f) ==> ((continuous f) (at a)))))"}
{"_id": "2219", "title": "HAS_REAL_MEASURE", "text": "|- (!s. (!m. (((has_real_measure s) m) = (((has_real_integral (\\x. (((COND (x IN s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0))))) m) UNIV))))"}
{"_id": "2220", "title": "MATRIX_NEG_MINUS1", "text": "|- (!A. ((matrix_neg A) = ((%% (real_neg (real_of_num (NUMERAL (BIT1 _0))))) A)))"}
{"_id": "2221", "title": "Seq.rotr1_rcons", "text": "|- (!x. (!s. (((rotr (NUMERAL (BIT1 _0))) ((rcons s) x)) = ((CONS x) s))))"}
{"_id": "2222", "title": "RELATIVE_INTERIOR_SUBSET", "text": "|- (!s. ((relative_interior s) SUBSET s))"}
{"_id": "2223", "title": "Kepler_spec.BIJ_REAL3_TRIPLE", "text": "|- (((triple_of_real3 o real3_of_triple) = I) /\\ ((real3_of_triple o triple_of_real3) = I))"}
{"_id": "2224", "title": "SQRT_MONO_LT", "text": "|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_lt x) y)) ==> ((real_lt (sqrt x)) (sqrt y)))))"}
{"_id": "2225", "title": "FACE_OF_TRANSLATION_EQ", "text": "|- (!a. (!f. (!s. (((face_of ((IMAGE (\\x. ((vector_add a) x))) f)) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((face_of f) s)))))"}
{"_id": "2226", "title": "components", "text": "|- (!s. ((components s) = (GSPEC (\\GEN%PVAR%904. (?x. (((SETSPEC GEN%PVAR%904) (x IN s)) ((connected_component s) x)))))))"}
{"_id": "2227", "title": "Trigonometry2.EXISTS_PROJECTING_POINT2", "text": "|- (!p. (!u. (!v0. (?pp. ((pp IN (aff (u INSERT (v0 INSERT EMPTY)))) /\\ (((dot ((vector_sub p) pp)) ((vector_sub u) v0)) = (real_of_num (NUMERAL _0))))))))"}
{"_id": "2228", "title": "REAL_POW_ROOT", "text": "|- (!n. (!x. (((ODD n) \\/ ((~ (n = (NUMERAL _0))) /\\ ((real_le (real_of_num (NUMERAL _0))) x))) ==> (((real_pow ((root n) x)) n) = x))))"}
{"_id": "2229", "title": "COND_ID", "text": "|- (!b. (!t. ((((COND b) t) t) = t)))"}
{"_id": "2230", "title": "REAL_LTE_ANTISYM", "text": "|- (!x. (!y. (~ (((real_lt x) y) /\\ ((real_le y) x)))))"}
{"_id": "2231", "title": "EMPTY_AS_INTERVAL", "text": "|- (EMPTY = (closed_interval ((CONS ((vec (NUMERAL (BIT1 _0))) , (vec (NUMERAL _0)))) NIL)))"}
{"_id": "2232", "title": "NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE", "text": "|- (!f. (!s. (!t. ((((absolutely_continuous_on f) s) /\\ ((is_interval s) /\\ ((negligible t) /\\ (t SUBSET s)))) ==> (negligible ((IMAGE f) t))))))"}
{"_id": "2233", "title": "Trigonometry.AVWKGNB", "text": "|- (!x3. (!x1. (!x2. (!x4. (!x5. (!x6. (((((((delta_x x1) x2) x3) x4) x5) x6) = ((real_add ((real_mul x1) ((real_mul x4) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6))))))) ((real_add ((real_mul x2) ((real_mul x5) ((real_add ((real_sub x1) x2)) ((real_add x3) ((real_add ((real_sub x4) x5)) x6)))))) ((real_sub ((real_sub ((real_sub ((real_sub ((real_mul x3) ((real_mul x6) ((real_add x1) ((real_add ((real_sub x2) x3)) ((real_add x4) ((real_sub x5) x6))))))) ((real_mul x2) ((real_mul x3) x4)))) ((real_mul x1) ((real_mul x3) x5)))) ((real_mul x1) ((real_mul x2) x6)))) ((real_mul x4) ((real_mul x5) x6))))))))))))"}
{"_id": "2234", "title": "Lp_ineqs_def.y2_fan", "text": "|- (!x. ((y2_fan x) = (vector_norm (SND x))))"}
{"_id": "2235", "title": "List_hypermap_iso.fan_list_nodes_iso", "text": "|- (!f. (!x. (!V. (!E. (((fan_list_nodes_iso (f , (V , E))) x) = (FST (f (x , (((COND ((((set_of_edge x) V) E) = EMPTY)) x) (CHOICE (((set_of_edge x) V) E)))))))))))"}
{"_id": "2236", "title": "PATH_IMAGE_LINEAR_IMAGE", "text": "|- (!f. (!g. ((linear f) ==> ((path_image (f o g)) = ((IMAGE f) (path_image g))))))"}
{"_id": "2237", "title": "REAL_LT_LDIV_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_lt ((real_div x) z)) y) = ((real_lt x) ((real_mul y) z)))))))"}
{"_id": "2238", "title": "MEASURABLE_ON_PREIMAGE_BOREL", "text": "|- (!f. (((measurable_on f) UNIV) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3549. (?x. (((SETSPEC GEN%PVAR%3549) ((f x) IN t)) x)))))))))"}
{"_id": "2239", "title": "PATH_CONNECTED_IFF_PATH_COMPONENT", "text": "|- (!s. ((path_connected s) = (!x. (!y. (((x IN s) /\\ (y IN s)) ==> (((path_component s) x) y))))))"}
{"_id": "2240", "title": "Seq.filter", "text": "|- (!a. (!x. (!s'. ((((filter a) NIL) = NIL) /\\ (((filter a) ((CONS x) s')) = (((COND (a x)) ((CONS x) ((filter a) s'))) ((filter a) s')))))))"}
{"_id": "2241", "title": "HAS_COMPLEX_DERIVATIVE_CASN", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative casn) (complex_inv (ccos (casn z)))) (at z))))"}
{"_id": "2242", "title": "EVEN_MOD", "text": "|- (!n. ((EVEN n) = ((n MOD (NUMERAL (BIT0 (BIT1 _0)))) = (NUMERAL _0))))"}
{"_id": "2243", "title": "AFF_DIM_HYPERPLANE", "text": "|- (!a. (!b. ((~ (a = (vec (NUMERAL _0)))) ==> ((aff_dim (GSPEC (\\GEN%PVAR%1576. (?x. (((SETSPEC GEN%PVAR%1576) (((dot a) x) = b)) x))))) = ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "2244", "title": "LIPSCHITZ_LIM", "text": "|- (!f. (!g. (!s. (!b. (((!n. (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) ((real_mul b) (vector_norm ((vector_sub x) y)))))))) /\\ (!x. ((x IN s) ==> (((--> (\\n. ((f n) x))) (g x)) sequentially)))) ==> (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))))))))"}
{"_id": "2245", "title": "Appendix.tgt", "text": "|- (tgt = ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))"}
{"_id": "2246", "title": "CCOS_ADD", "text": "|- (!w. (!z. ((ccos ((vector_add w) z)) = ((vector_sub ((complex_mul (ccos w)) (ccos z))) ((complex_mul (csin w)) (csin z))))))"}
{"_id": "2247", "title": "REAL_MUL_AC", "text": "|- ((((real_mul m) n) = ((real_mul n) m)) /\\ ((((real_mul ((real_mul m) n)) p) = ((real_mul m) ((real_mul n) p))) /\\ (((real_mul m) ((real_mul n) p)) = ((real_mul n) ((real_mul m) p)))))"}
{"_id": "2248", "title": "Hypermap_iso.hyp_iso_inj", "text": "|- (!G. (!H. (!f. (((hyp_iso f) (H , G)) ==> (!x. (!y. (((x IN (dart H)) /\\ ((y IN (dart H)) /\\ ((f x) = (f y)))) ==> (x = y))))))))"}
{"_id": "2249", "title": "OPEN_IN_IMP_SUBSET", "text": "|- (!top. (!s. (!t. (((open_in ((subtopology top) s)) t) ==> (t SUBSET s)))))"}
{"_id": "2250", "title": "Hypermap_iso.hyp_iso_surj", "text": "|- (!G. (!H. (!f. (((hyp_iso f) (H , G)) ==> (!y. ((y IN (dart G)) ==> (?x. ((x IN (dart H)) /\\ ((f x) = y)))))))))"}
{"_id": "2251", "title": "Wrgcvdr_cizmrrh.wedge_in_fan_ge", "text": "|- (!E. (!v. (!w. (((wedge_in_fan_ge (v , w)) E) = (((COND ((CARD ((EE v) E)) > (NUMERAL (BIT1 _0)))) ((((wedge_ge (vec (NUMERAL _0))) v) w) ((((azim_cycle ((EE v) E)) (vec (NUMERAL _0))) v) w))) (GSPEC (\\GEN%PVAR%6280. (?x. (((SETSPEC GEN%PVAR%6280) T) x)))))))))"}
{"_id": "2252", "title": "Ssrnat.ltE", "text": "|- (!n. (!m. ((n < m) = ((SUC n) <= m))))"}
{"_id": "2253", "title": "HOMOTOPIC_TRIVIALITY", "text": "|- (!s. (!t. ((!f. (!g. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((continuous_on g) s) /\\ (((IMAGE g) s) SUBSET t)))) ==> ((((homotopic_with (\\x. T)) (s , t)) f) g)))) = (((s = EMPTY) \\/ (path_connected t)) /\\ (!f. ((((continuous_on f) s) /\\ (((IMAGE f) s) SUBSET t)) ==> (?c. ((((homotopic_with (\\x. T)) (s , t)) f) (\\x. c)))))))))"}
{"_id": "2254", "title": "complex_differentiable", "text": "|- (!f. (!net. (((complex_differentiable f) net) = (?f'. (((has_complex_derivative f) f') net)))))"}
{"_id": "2255", "title": "MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE", "text": "|- (!f. (!g. (!s. ((((measurable_on f) s) /\\ (((integrable_on g) s) /\\ (!x. ((x IN s) ==> ((real_le (vector_norm (f x))) (drop (g x))))))) ==> ((absolutely_integrable_on f) s)))))"}
{"_id": "2256", "title": "QUANTIFY_SURJECTION_THM", "text": "|- (!f. ((!y. (?x. ((f x) = y))) ==> (((!P. ((!x. (P x)) = (!x. (P (f x))))) /\\ ((!P. ((?x. (P x)) = (?x. (P (f x))))) /\\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE f) s))))) /\\ (!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE f) s)))))))) /\\ (!P. ((GSPEC (\\GEN%PVAR%236. (?x. (((SETSPEC GEN%PVAR%236) (P x)) x)))) = ((IMAGE f) (GSPEC (\\GEN%PVAR%237. (?x. (((SETSPEC GEN%PVAR%237) (P (f x))) x))))))))))"}
{"_id": "2257", "title": "Lp_main_estimate.pent_diags_imp", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (!a. (!b. ((LET (\\f. (LET_END ((LET (\\w'. (LET_END ((LET (\\u. (LET_END ((LET (\\z. (LET_END ((((v , w) IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) /\\ ((~ (a = b)) /\\ ((a IN ((v_prime V) f)) /\\ ((b IN ((v_prime V) f)) /\\ (~ ((a INSERT (b INSERT EMPTY)) IN ((e_prime E) f))))))) ==> ((((a , b) = (w , w')) \\/ ((a , b) = (w' , w))) \\/ ((((a , b) = (z , v)) \\/ ((a , b) = (v , z))) \\/ ((((a , b) = (u , w)) \\/ ((a , b) = (w , u))) \\/ ((((a , b) = (w' , z)) \\/ ((a , b) = (z , w'))) \\/ (((a , b) = (v , u)) \\/ ((a , b) = (u , v))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) u) w'))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))"}
{"_id": "2258", "title": "COUNTABLE_COMPACT_OPEN_IN", "text": "|- (!u. (COUNTABLE (GSPEC (\\GEN%PVAR%1245. (?c. (((SETSPEC GEN%PVAR%1245) ((compact c) /\\ ((open_in ((subtopology euclidean) u)) c))) c))))))"}
{"_id": "2259", "title": "PCROSS_EQ_EMPTY", "text": "|- (!s. (!t. (((s PCROSS t) = EMPTY) = ((s = EMPTY) \\/ (t = EMPTY)))))"}
{"_id": "2260", "title": "int_neg_th", "text": "|- (!x. ((real_of_int (int_neg x)) = (real_neg (real_of_int x))))"}
{"_id": "2261", "title": "NORM_FSTCART", "text": "|- (!x. ((real_le (vector_norm (fstcart x))) (vector_norm x)))"}
{"_id": "2262", "title": "OPEN_IN", "text": "|- (!s. ((open s) = ((open_in euclidean) s)))"}
{"_id": "2263", "title": "VECTOR_SUB_LDISTRIB", "text": "|- (((% c) ((vector_sub x) y)) = ((vector_sub ((% c) x)) ((% c) y)))"}
{"_id": "2264", "title": "Ssrnat.fact_gt0", "text": "|- (!n. ((NUMERAL _0) < (FACT n)))"}
{"_id": "2265", "title": "HULL_MINIMAL", "text": "|- (!P. (!s. (!t. (((s SUBSET t) /\\ (P t)) ==> (((hull P) s) SUBSET t)))))"}
{"_id": "2266", "title": "LEFT_ADD_DISTRIB", "text": "|- (!m. (!n. (!p. ((m * (n + p)) = ((m * n) + (m * p))))))"}
{"_id": "2267", "title": "HOMEOMORPHIC_OPEN_INTERVALS_1", "text": "|- (!a. (!b. (!c. (!d. ((((real_lt (drop a)) (drop b)) /\\ ((real_lt (drop c)) (drop d))) ==> ((homeomorphic (open_interval (a , b))) (open_interval (c , d))))))))"}
{"_id": "2268", "title": "OPEN_IN_DELETE", "text": "|- (!u. (!s. (!a. (((open_in ((subtopology euclidean) u)) s) ==> ((open_in ((subtopology euclidean) u)) (s DELETE a))))))"}
{"_id": "2269", "title": "Seq.unzip2", "text": "|- (unzip2 = (map SND))"}
{"_id": "2270", "title": "has_derivative_at", "text": "|- (!f. (!f'. (!x. ((((has_derivative f) f') (at x)) = ((linear f') /\\ (((--> (\\y. ((% (real_inv (vector_norm ((vector_sub y) x)))) ((vector_sub (f y)) ((vector_add (f x)) (f' ((vector_sub y) x))))))) (vec (NUMERAL _0))) (at x)))))))"}
{"_id": "2271", "title": "RELATIVE_INTERIOR_TRANSLATION", "text": "|- (!a. (!s. ((relative_interior ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) (relative_interior s)))))"}
{"_id": "2272", "title": "IN_SPHERE", "text": "|- (!x. (!y. (!e. ((y IN (sphere (x , e))) = ((distance (x , y)) = e)))))"}
{"_id": "2273", "title": "Ssrnat.mulSnr", "text": "|- (!m. (!n. (((SUC m) * n) = ((m * n) + n))))"}
{"_id": "2274", "title": "complex_div", "text": "|- (!w. (!z. (((complex_div w) z) = ((complex_mul w) (complex_inv z)))))"}
{"_id": "2275", "title": "MIDPOINT_LINEAR_IMAGE", "text": "|- (!f. (!a. (!b. ((linear f) ==> ((midpoint ((f a) , (f b))) = (f (midpoint (a , b))))))))"}
{"_id": "2276", "title": "BOUNDED_SUBSET_BALL", "text": "|- (!s. (!x. ((bounded s) ==> (?r. (((real_lt (real_of_num (NUMERAL _0))) r) /\\ (s SUBSET (ball (x , r))))))))"}
{"_id": "2277", "title": "PATH_APPROX_VECTOR_POLYNOMIAL_FUNCTION", "text": "|- (!g. (!e. (((path g) /\\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?p. ((vector_polynomial_function p) /\\ (((pathstart p) = (pathstart g)) /\\ (((pathfinish p) = (pathfinish g)) /\\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> ((real_lt (vector_norm ((vector_sub (p t)) (g t)))) e))))))))))"}
{"_id": "2278", "title": "VEC_COMPONENT", "text": "|- (!k. (!i. (((vec k) $ i) = (real_of_num k))))"}
{"_id": "2279", "title": "INT_LT_ANTISYM", "text": "|- (!x. (!y. (~ (((int_lt x) y) /\\ ((int_lt y) x)))))"}
{"_id": "2280", "title": "VECTOR_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL", "text": "|- (!f. (!a. (!b. (!x. (!f'. (!f''. ((((real_lt (drop a)) (drop b)) /\\ ((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((((has_vector_derivative f) f') ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) /\\ (((has_vector_derivative f) f'') ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))) ==> (f' = f''))))))))"}
{"_id": "2281", "title": "REAL_CONTINUOUS_FRAC", "text": "|- (!x. ((~ (integer x)) ==> ((real_continuous frac) (atreal x))))"}
{"_id": "2282", "title": "LOWDIM_EQ_INTER_HYPERPLANE", "text": "|- (!s. (!t. (((subspace s) /\\ ((subspace t) /\\ ((t SUBSET s) /\\ (((dim t) + (NUMERAL (BIT1 _0))) = (dim s))))) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\\ (((GSPEC (\\GEN%PVAR%191. (?x. (((SETSPEC GEN%PVAR%191) (((dot a) x) = (real_of_num (NUMERAL _0)))) x)))) INTER s) = t))))))"}
{"_id": "2283", "title": "AFFINE_IMP_SUBSPACE", "text": "|- (!s. (((affine s) /\\ ((vec (NUMERAL _0)) IN s)) ==> (subspace s)))"}
{"_id": "2284", "title": "NORM_EQ_SQUARE", "text": "|- (!x. (((vector_norm x) = a) = (((real_le (real_of_num (NUMERAL _0))) a) /\\ (((dot x) x) = ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "2285", "title": "columnvector", "text": "|- (!v. ((columnvector v) = (lambdai. (lambdaj. (v $ i)))))"}
{"_id": "2286", "title": "HAS_REAL_DERIVATIVE_CARATHEODORY_WITHINREAL", "text": "|- (!f. (!f'. (!z. (!s. ((((has_real_derivative f) f') ((within (atreal z)) s)) = (?g. ((!w. (((real_sub (f w)) (f z)) = ((real_mul (g w)) ((real_sub w) z)))) /\\ (((real_continuous g) ((within (atreal z)) s)) /\\ ((g z) = f')))))))))"}
{"_id": "2287", "title": "Ssrnat.leq_sub_add", "text": "|- (!m. (!n. (!p. (((m - n) <= p) = (m <= (n + p))))))"}
{"_id": "2288", "title": "Ssrnat.muln_addr", "text": "|- (!m. (!n1. (!n2. ((m * (n1 + n2)) = ((m * n1) + (m * n2))))))"}
{"_id": "2289", "title": "FACET_OF_LINEAR_IMAGE", "text": "|- (!f. (!c. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((facet_of ((IMAGE f) c)) ((IMAGE f) s)) = ((facet_of c) s))))))"}
{"_id": "2290", "title": "Lp_ineqs_def.y5_fan", "text": "|- (!V. (!E. (!x. (((y5_fan (V , E)) x) = (distance ((inverse (f_fan_pair_ext (V , E))) x))))))"}
{"_id": "2291", "title": "Ssrnat.minn_maxl", "text": "|- (!m1. (!m2. (!n. (((minn ((maxn m1) m2)) n) = ((maxn ((minn m1) n)) ((minn m2) n))))))"}
{"_id": "2292", "title": "Conforming.properties_of_elements_in_node_fully_surroundedfan", "text": "|- (!x. (!V. (!E. (!ds. (!y. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((ds IN (node_set (hypermap1_of_fanx (x , (V , E))))) /\\ (y IN ds)))) ==> (((pr2 y) INSERT ((pr3 y) INSERT EMPTY)) IN E)))))))"}
{"_id": "2293", "title": "POSITIVE_SEMIDEFINITE_CMUL", "text": "|- (!c. (!A. (((positive_semidefinite A) /\\ ((real_le (real_of_num (NUMERAL _0))) c)) ==> (positive_semidefinite ((%% c) A)))))"}
{"_id": "2294", "title": "Tame_general.CONTRAVENING_IMP_FULLY_SURROUNDED", "text": "|- (!V. (((FAN ((vec (NUMERAL _0)) , (V , (ESTD V)))) /\\ (contravening V)) ==> (fully_surrounded (V , (ESTD V)))))"}
{"_id": "2295", "title": "RE_CMUL", "text": "|- (!a. (!z. ((Re ((% a) z)) = ((real_mul a) (Re z)))))"}
{"_id": "2296", "title": "EDGE_OF_LINEAR_IMAGE", "text": "|- (!f. (!c. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((edge_of ((IMAGE f) c)) ((IMAGE f) s)) = ((edge_of c) s))))))"}
{"_id": "2297", "title": "FORALL_VECTOR_3", "text": "|- ((!v. (P v)) = (!x. (!y. (!z. (P (vector ((CONS x) ((CONS y) ((CONS z) NIL)))))))))"}
{"_id": "2298", "title": "Real_ext.REAL_PROP_LT_LMUL", "text": "|- (!x. (!y. (!z. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt y) z)) ==> ((real_lt ((real_mul x) y)) ((real_mul x) z))))))"}
{"_id": "2299", "title": "FSIGMA_LOCALLY_COMPACT", "text": "|- (!s. (((locally compact) s) ==> (fsigma s)))"}
{"_id": "2300", "title": "REAL_EXP_N", "text": "|- (!n. (!x. ((exp ((real_mul (real_of_num n)) x)) = ((real_pow (exp x)) n))))"}
{"_id": "2301", "title": "Lp_ineqs_proofs.lp_cond_trans1", "text": "|- (!L. (!V. (!E. (!g. (!h. (!d. (((lp_cond (L , (g , h))) (V , E)) ==> ((lp_cond (((split_list_hyp L) d) , (g , h))) (V , ((add_diag (V , E)) (g d)))))))))))"}
{"_id": "2302", "title": "BILINEAR_DROP_MUL", "text": "|- (bilinear (\\x. (\\y. ((% (drop x)) y))))"}
{"_id": "2303", "title": "LOCALLY_CONNECTED_IM_KLEINEN", "text": "|- (!s. (((locally connected) s) = (!v. (!x. ((((open_in ((subtopology euclidean) s)) v) /\\ (x IN v)) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\\ ((x IN u) /\\ ((u SUBSET v) /\\ (!y. ((y IN u) ==> (?c. ((connected c) /\\ ((c SUBSET v) /\\ ((x IN c) /\\ (y IN c))))))))))))))))"}
{"_id": "2304", "title": "real_abs", "text": "|- (!x. ((real_abs x) = (((COND ((real_le (real_of_num (NUMERAL _0))) x)) x) (real_neg x))))"}
{"_id": "2305", "title": "Hypermap_iso.inE", "text": "|- ((!p. (!x. ((x IN (GSPEC (\\v. (?y. (((SETSPEC v) (p y)) y))))) = (p x)))) /\\ ((!P. (!x. (((GSPEC (\\v. (P (SETSPEC v)))) x) = (P (\\p. (\\t. (p /\\ (x = t)))))))) /\\ ((!p. (!x. (((GSPEC (\\v. (?y. (((SETSPEC v) (p y)) y)))) x) = (p x)))) /\\ (!p. (!x. ((x IN (\\y. (p y))) = (p x)))))))"}
{"_id": "2306", "title": "RC_INC", "text": "|- (!R. (!x. (!y. (((R x) y) ==> (((RC R) x) y)))))"}
{"_id": "2307", "title": "Functional_equation.functional_mud_234_x", "text": "|- (!y1. (!y5. (!y6. (((domain6 (\\x1. (\\x2. (\\x3. (\\x4. (\\x5. (\\x6. ((real_le (real_of_num (NUMERAL _0))) y1)))))))) (((mud_234_x_v1 y1) y5) y6)) ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y1) y1))) proj_x2) proj_x3) dummy6) dummy6) dummy6)) (uni (sqrt , (((delta_234_x ((real_mul y1) y1)) ((real_mul y5) y5)) ((real_mul y6) y6)))))))))"}
{"_id": "2308", "title": "REAL_LE_TRANS", "text": "|- (!x. (!y. (!z. ((((real_le x) y) /\\ ((real_le y) z)) ==> ((real_le x) z)))))"}
{"_id": "2309", "title": "Hypermap.lemma_face_identity", "text": "|- (!H. (!x. (!y. ((y IN ((face H) x)) ==> (((face H) x) = ((face H) y))))))"}
{"_id": "2310", "title": "Tame_lemmas.reflection_at0", "text": "|- (!x. (((reflection (vec (NUMERAL _0))) x) = x))"}
{"_id": "2311", "title": "HAS_REAL_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG", "text": "|- (!f. (!s. (((FINITE (GSPEC (\\GEN%PVAR%4933. (?x. (((SETSPEC GEN%PVAR%4933) ((x IN s) /\\ (~ ((f x) = EMPTY)))) x))))) /\\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ (~ (x = y)))) ==> ((DISJOINT (f x)) (f y))))))) ==> ((has_real_measure (UNIONS ((IMAGE f) s))) ((sum s) (\\x. (real_measure (f x))))))))"}
{"_id": "2312", "title": "Trigonometry.KGLLRQT1", "text": "|- (!x. ((cos (real_neg x)) = (cos x)))"}
{"_id": "2313", "title": "DIVISION_OF_TAGGED_DIVISION", "text": "|- (!s. (!i. (((tagged_division_of s) i) ==> ((division_of ((IMAGE SND) s)) i))))"}
{"_id": "2314", "title": "LIM_AT_POSINFINITY", "text": "|- (!f. (!l. ((((--> f) l) at_posinfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge x) b) ==> ((real_lt (distance ((f x) , l))) e)))))))))"}
{"_id": "2315", "title": "ATN_ABS_LE_X", "text": "|- (!x. ((real_le (real_abs (atn x))) (real_abs x)))"}
{"_id": "2316", "title": "Functional_equation.cos797", "text": "|- (cos797 = (cos ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))"}
{"_id": "2317", "title": "NORM_EQ_0", "text": "|- (!x. (((vector_norm x) = (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))"}
{"_id": "2318", "title": "REAL_SUMS_INFSUM", "text": "|- (!f. (!s. ((((real_sums f) ((real_infsum s) f)) s) = ((real_summable s) f))))"}
{"_id": "2319", "title": "CX_ABS", "text": "|- (!x. ((Cx (real_abs x)) = (Cx (vector_norm (Cx x)))))"}
{"_id": "2320", "title": "Seq2.card_set_of_list_uniq", "text": "|- (!l. ((uniq l) ==> ((CARD (set_of_list l)) = (sizel l))))"}
{"_id": "2321", "title": "CONVEX_INTER", "text": "|- (!s. (!t. (((convex s) /\\ (convex t)) ==> (convex (s INTER t)))))"}
{"_id": "2322", "title": "Ssrnat.eqn_leq", "text": "|- (!m. (!n. ((m = n) = ((m <= n) /\\ (n <= m)))))"}
{"_id": "2323", "title": "MIDPOINT_IN_CONVEX", "text": "|- (!s. (!x. (!y. (((convex s) /\\ ((x IN s) /\\ (y IN s))) ==> ((midpoint (x , y)) IN s)))))"}
{"_id": "2324", "title": "DIM_SUBSET_UNIV", "text": "|- (!s. ((dim s) <= (dimindex UNIV)))"}
{"_id": "2325", "title": "REALLIM_WITHIN_SUBSET", "text": "|- (!f. (!l. (!a. (!s. (!t. (((((---> f) l) ((within (at a)) s)) /\\ (t SUBSET s)) ==> (((---> f) l) ((within (at a)) t))))))))"}
{"_id": "2326", "title": "BOUNDED_DIFFS", "text": "|- (!s. (!t. (((bounded s) /\\ (bounded t)) ==> (bounded (GSPEC (\\GEN%PVAR%586. (?x. (?y. (((SETSPEC GEN%PVAR%586) ((x IN s) /\\ (y IN t))) ((vector_sub x) y))))))))))"}
{"_id": "2327", "title": "LIMIT_POINT_FINITE", "text": "|- (!s. (!a. ((FINITE s) ==> (~ ((limit_point_of a) s)))))"}
{"_id": "2328", "title": "RPOW_POS_LE", "text": "|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (real_of_num (NUMERAL _0))) ((rpow x) y)))))"}
{"_id": "2329", "title": "sgn_le", "text": "|- (sgn_le = (\\t. ((real_le t) (real_of_num (NUMERAL _0)))))"}
{"_id": "2330", "title": "Fan.fan7", "text": "|- (!E. (!V. (!x. ((fan7 (x , (V , E))) = (!e1. (!e2. (((e1 IN (E UNION (GSPEC (\\GEN%PVAR%5449. (?v. (((SETSPEC GEN%PVAR%5449) (v IN V)) (v INSERT EMPTY))))))) /\\ (e2 IN (E UNION (GSPEC (\\GEN%PVAR%5450. (?v. (((SETSPEC GEN%PVAR%5450) (v IN V)) (v INSERT EMPTY)))))))) ==> ((((aff_ge (x INSERT EMPTY)) e1) INTER ((aff_ge (x INSERT EMPTY)) e2)) = ((aff_ge (x INSERT EMPTY)) (e1 INTER e2))))))))))"}
{"_id": "2331", "title": "SURJ", "text": "|- (!t. (!s. (!f. ((((SURJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\\ (!x. ((x IN t) ==> (?y. ((y IN s) /\\ ((f y) = x))))))))))"}
{"_id": "2332", "title": "CX_DIV", "text": "|- (!x. (!y. ((Cx ((real_div x) y)) = ((complex_div (Cx x)) (Cx y)))))"}
{"_id": "2333", "title": "MEASURE_INTERVAL_1", "text": "|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND ((real_le (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))) /\\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND ((real_le (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))))"}
{"_id": "2334", "title": "ROWS_TRANSP", "text": "|- (!A. ((rows (transp A)) = (columns A)))"}
{"_id": "2335", "title": "REAL_SUB_LDISTRIB", "text": "|- (!x. (!y. (!z. (((real_mul x) ((real_sub y) z)) = ((real_sub ((real_mul x) y)) ((real_mul x) z))))))"}
{"_id": "2336", "title": "DIST_1", "text": "|- (!x. (!y. ((distance (x , y)) = (real_abs ((real_sub (drop x)) (drop y))))))"}
{"_id": "2337", "title": "SIMPLY_CONNECTED_FUNDAMENTAL_GROUP", "text": "|- (!s. ((simply_connected s) = ((path_connected s) /\\ (!a. ((a IN s) ==> ((fundamental_group (s , a)) = (((homotopic_paths s) (linepath (a , a))) INSERT EMPTY)))))))"}
{"_id": "2338", "title": "HAS_INTEGRAL_LIM_AT_POSINFINITY", "text": "|- (!f. (!l. ((((has_integral f) l) (GSPEC (\\GEN%PVAR%2953. (?t. (((SETSPEC GEN%PVAR%2953) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t))))) = ((!a. ((integrable_on f) (closed_interval ((CONS ((vec (NUMERAL _0)) , a)) NIL)))) /\\ (((--> (\\a. ((integral (closed_interval ((CONS ((vec (NUMERAL _0)) , (lift a))) NIL))) f))) l) at_posinfinity)))))"}
{"_id": "2339", "title": "Local_lemmas1.localization", "text": "|- (!V. (!E. (!FF. (((localization (V , E)) FF) = (((v_prime V) FF) , ((e_prime E) FF))))))"}
{"_id": "2340", "title": "Hypermap_and_fan.PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ", "text": "|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((v , w) IN (dart1_of_fan (V , E)))) ==> (~ (v = w)))))))"}
{"_id": "2341", "title": "INTEGRABLE_BOUNDED_VARIATION_COMPLEX_LMUL", "text": "|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((has_bounded_variation_on g) (closed_interval ((CONS (a , b)) NIL)))) ==> ((integrable_on (\\x. ((complex_mul (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "2342", "title": "CONVEX_HULL_INSERT_REDUNDANT_POINT", "text": "|- (!s. (!a. (!b. (!c. (((a IN ((hull convex) (c INSERT s))) /\\ ((b IN ((hull convex) (c INSERT s))) /\\ (c IN (open_segment (a , b))))) ==> (((hull convex) (c INSERT s)) = ((hull convex) s)))))))"}
{"_id": "2343", "title": "ORTHOGONAL_BASIS_BASIS", "text": "|- (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ (j <= (dimindex UNIV))))) ==> (((orthogonal (basis i)) (basis j)) = (~ (i = j))))))"}
{"_id": "2344", "title": "AFFINE_TRANSLATION_SUBSPACE", "text": "|- (!t. (((affine t) /\\ (~ (t = EMPTY))) = (?a. (?s. ((subspace s) /\\ (t = ((IMAGE (\\x. ((vector_add a) x))) s)))))))"}
{"_id": "2345", "title": "LOCALLY_CONNECTED", "text": "|- (!s. (((locally connected) s) = (!v. (!x. ((((open_in ((subtopology euclidean) s)) v) /\\ (x IN v)) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\\ ((connected u) /\\ ((x IN u) /\\ (u SUBSET v))))))))))"}
{"_id": "2346", "title": "MATRIX_INV_INV", "text": "|- (!A. ((matrix_inv (matrix_inv A)) = A))"}
{"_id": "2347", "title": "INVERSE_I", "text": "|- ((inverse I) = I)"}
{"_id": "2348", "title": "INDEPENDENT_STDBASIS", "text": "|- (independent (GSPEC (\\GEN%PVAR%152. (?i. (((SETSPEC GEN%PVAR%152) (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) (basis i))))))"}
{"_id": "2349", "title": "mat", "text": "|- (!k. ((mat k) = (lambdai. (lambdaj. (((COND (i = j)) (real_of_num k)) (real_of_num (NUMERAL _0)))))))"}
{"_id": "2350", "title": "Local_lemmas.PRESERABLE_AFF_GE_SUBSET", "text": "|- ((a IN ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))) ==> (((aff_ge (x INSERT EMPTY)) (a INSERT EMPTY)) SUBSET ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))"}
{"_id": "2351", "title": "CONTINUOUS_ON", "text": "|- (!f. (!s. (((continuous_on f) s) = (!x. ((x IN s) ==> (((--> f) (f x)) ((within (at x)) s)))))))"}
{"_id": "2352", "title": "SUBPATH_TRANSLATION", "text": "|- (!a. (!g. (!u. (!v. ((((subpath u) v) ((\\x. ((vector_add a) x)) o g)) = ((\\x. ((vector_add a) x)) o (((subpath u) v) g)))))))"}
{"_id": "2353", "title": "Real_ext.REAL_PROP_LE_RMUL", "text": "|- (!x. (!y. (!z. ((((real_le x) y) /\\ ((real_le (real_of_num (NUMERAL _0))) z)) ==> ((real_le ((real_mul x) z)) ((real_mul y) z))))))"}
{"_id": "2354", "title": "FORALL_PAIR_THM", "text": "|- (!P. ((!p. (P p)) = (!p1. (!p2. (P (p1 , p2))))))"}
{"_id": "2355", "title": "CNJ_DIV", "text": "|- (!w. (!z. ((cnj ((complex_div w) z)) = ((complex_div (cnj w)) (cnj z)))))"}
{"_id": "2356", "title": "Ssrbool.andbN", "text": "|- (!b. ((b /\\ (~ b)) = F))"}
{"_id": "2357", "title": "ABSOLUTELY_CONTINUOUS_ISOMETRIC", "text": "|- (!f. (!g. (!s. ((!x. (!y. ((distance ((f x) , (f y))) = (distance ((g x) , (g y)))))) ==> (((absolutely_continuous_on f) s) = ((absolutely_continuous_on g) s))))))"}
{"_id": "2358", "title": "CLOSURE_SPHERE", "text": "|- (!a. (!r. ((closure (sphere (a , r))) = (sphere (a , r)))))"}
{"_id": "2359", "title": "COMPLEX_MUL_RID", "text": "|- (!x. (((complex_mul x) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = x))"}
{"_id": "2360", "title": "MATRIX_SUB_RZERO", "text": "|- (!A. (((matrix_sub A) (mat (NUMERAL _0))) = A))"}
{"_id": "2361", "title": "vector_mul", "text": "|- (!c. (!x. (((% c) x) = (lambdai. ((real_mul c) (x $ i))))))"}
{"_id": "2362", "title": "absolutely_integrable_on", "text": "|- (!f. (!s. (((absolutely_integrable_on f) s) = (((integrable_on f) s) /\\ ((integrable_on (\\x. (lift (vector_norm (f x))))) s)))))"}
{"_id": "2363", "title": "REAL_MUL_SYM", "text": "|- (!x. (!y. (((real_mul x) y) = ((real_mul y) x))))"}
{"_id": "2364", "title": "SUMMABLE_SUBSET", "text": "|- (!x. (!s. (!t. (((s SUBSET t) /\\ ((summable t) (\\i. (((COND (i IN s)) (x i)) (vec (NUMERAL _0)))))) ==> ((summable s) x)))))"}
{"_id": "2365", "title": "Ssrnat.maxnC", "text": "|- (!m. (!n. (((maxn m) n) = ((maxn n) m))))"}
{"_id": "2366", "title": "Ldurdpn.VECTOR_SCALE_CHANGE", "text": "|- ((~ (a = (real_of_num (NUMERAL _0)))) ==> ((((% a) x) = y) = (x = ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) a)) y))))"}
{"_id": "2367", "title": "PRODUCT_NEG_NUMSEG", "text": "|- (!f. (!m. (!n. (((product (m .. n)) (\\i. (real_neg (f i)))) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((n + (NUMERAL (BIT1 _0))) - m))) ((product (m .. n)) f))))))"}
{"_id": "2368", "title": "Hypermap_iso.darts_of_fan_disj1", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((DISJOINT (d1_fan ((vec (NUMERAL _0)) , (V , E)))) (d20_fan ((vec (NUMERAL _0)) , (V , E)))))))"}
{"_id": "2369", "title": "REAL_EQ_ADD_RCANCEL_0", "text": "|- (!x. (!y. ((((real_add x) y) = y) = (x = (real_of_num (NUMERAL _0))))))"}
{"_id": "2370", "title": "TRANSP_COMPONENT", "text": "|- (!A. (!i. (!j. ((((transp A) $ i) $ j) = ((A $ j) $ i)))))"}
{"_id": "2371", "title": "Seq.rot_oversize", "text": "|- (!n. (!s. (((sizel s) <= n) ==> (((rot n) s) = s))))"}
{"_id": "2372", "title": "DIV_UNIQ", "text": "|- (!m. (!n. (!q. (!r. (((m = ((q * n) + r)) /\\ (r < n)) ==> ((m DIV n) = q))))))"}
{"_id": "2373", "title": "CONTINUOUS_IMP_MEASURABLE_ON", "text": "|- (!f. (((continuous_on f) UNIV) ==> ((measurable_on f) UNIV)))"}
{"_id": "2374", "title": "HAS_REAL_INTEGRAL_ON_SUPERSET", "text": "|- (!f. (!s. (!t. (((!x. ((~ (x IN s)) ==> ((f x) = (real_of_num (NUMERAL _0))))) /\\ ((s SUBSET t) /\\ (((has_real_integral f) i) s))) ==> (((has_real_integral f) i) t)))))"}
{"_id": "2375", "title": "SERIES_COMPLEX_RMUL", "text": "|- (!f. (!l. (!c. (!s. ((((sums f) l) s) ==> (((sums (\\x. ((complex_mul (f x)) c))) ((complex_mul l) c)) s))))))"}
{"_id": "2376", "title": "LIFT_ADD", "text": "|- (!x. (!y. ((lift ((real_add x) y)) = ((vector_add (lift x)) (lift y)))))"}
{"_id": "2377", "title": "Hypermap.power_inverse_element_lemma", "text": "|- (!s. (!p. (!n. (((FINITE s) /\\ ((permutes p) s)) ==> (?j. (((POWER (inverse p)) n) = ((POWER p) j)))))))"}
{"_id": "2378", "title": "BIT0", "text": "|- (!n. ((BIT0 n) = (n + n)))"}
{"_id": "2379", "title": "RE_II", "text": "|- ((Re ii) = (real_of_num (NUMERAL _0)))"}
{"_id": "2380", "title": "NSUM_EQ", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((nsum s) f) = ((nsum s) g))))))"}
{"_id": "2381", "title": "LINEAR_CMUL", "text": "|- (!f. (!c. (!x. ((linear f) ==> ((f ((% c) x)) = ((% c) (f x)))))))"}
{"_id": "2382", "title": "LINEAR_SCALING", "text": "|- (!c. (linear (\\x. ((% c) x))))"}
{"_id": "2383", "title": "OPEN_INSIDE", "text": "|- (!s. ((closed s) ==> (open (inside s))))"}
{"_id": "2384", "title": "REAL_POW_ZERO", "text": "|- (!n. (((real_pow (real_of_num (NUMERAL _0))) n) = (((COND (n = (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))"}
{"_id": "2385", "title": "CONVEX_HULL_REDUNDANT_SUBSET_GEN", "text": "|- (!s. (!t. (((compact s) /\\ ((t SUBSET s) /\\ ((DISJOINT (s DIFF t)) (GSPEC (\\GEN%PVAR%2254. (?x. (((SETSPEC GEN%PVAR%2254) ((extreme_point_of x) ((hull convex) s))) x))))))) ==> (((hull convex) s) = ((hull convex) t)))))"}
{"_id": "2386", "title": "HAS_SIZE_FINITE_IMAGE", "text": "|- (!s. (UNIV HAS_SIZE (dimindex s)))"}
{"_id": "2387", "title": "MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN_INTERVAL", "text": "|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!a. (!b. (lebesgue_measurable (GSPEC (\\GEN%PVAR%3638. (?x. (((SETSPEC GEN%PVAR%3638) ((x IN s) /\\ ((f x) IN (open_interval (a , b))))) x)))))))))))"}
{"_id": "2388", "title": "EXISTS_IN_GSPEC", "text": "|- ((!P. (!f. ((?z. ((z IN (GSPEC (\\GEN%PVAR%25. (?x. (((SETSPEC GEN%PVAR%25) (P x)) (f x)))))) /\\ (Q z))) = (?x. ((P x) /\\ (Q (f x))))))) /\\ ((!P. (!f. ((?z. ((z IN (GSPEC (\\GEN%PVAR%26. (?x. (?y. (((SETSPEC GEN%PVAR%26) ((P x) y)) ((f x) y))))))) /\\ (Q z))) = (?x. (?y. (((P x) y) /\\ (Q ((f x) y)))))))) /\\ ((!P. (!f. ((?z. ((z IN (GSPEC (\\GEN%PVAR%27. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%27) (((P w) x) y)) (((f w) x) y)))))))) /\\ (Q z))) = (?w. (?x. (?y. ((((P w) x) y) /\\ (Q (((f w) x) y))))))))) /\\ (!P. (!f. ((?z. ((z IN (GSPEC (\\GEN%PVAR%28. (?v. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%28) ((((P v) w) x) y)) ((((f v) w) x) y))))))))) /\\ (Q z))) = (?v. (?w. (?x. (?y. (((((P v) w) x) y) /\\ (Q ((((f v) w) x) y)))))))))))))"}
{"_id": "2389", "title": "CONVEX_CONTAINS_SEGMENT", "text": "|- (!s. ((convex s) = (!a. (!b. (((a IN s) /\\ (b IN s)) ==> ((closed_segment ((CONS (a , b)) NIL)) SUBSET s))))))"}
{"_id": "2390", "title": "IN_INTER", "text": "|- (!s. (!t. (!x. ((x IN (s INTER t)) = ((x IN s) /\\ (x IN t))))))"}
{"_id": "2391", "title": "COMPLEX_MUL_ASSOC", "text": "|- (!x. (!y. (!z. (((complex_mul x) ((complex_mul y) z)) = ((complex_mul ((complex_mul x) y)) z)))))"}
{"_id": "2392", "title": "AFF_DIM_EQ_0", "text": "|- (!s. (((aff_dim s) = (int_of_num (NUMERAL _0))) = (?a. (s = (a INSERT EMPTY)))))"}
{"_id": "2393", "title": "List_hypermap_iso.e_fan_pair_ext_explicit", "text": "|- (!V. (!E. (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> (((e_fan_pair_ext (V , E)) (v , w)) = (w , v)))))))"}
{"_id": "2394", "title": "INTEGRABLE_ON_SUBINTERVAL", "text": "|- (!f. (!s. (!a. (!b. ((((integrable_on f) s) /\\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "2395", "title": "MATRIX_CMUL_LZERO", "text": "|- (!A. (((%% (real_of_num (NUMERAL _0))) A) = (mat (NUMERAL _0))))"}
{"_id": "2396", "title": "PATH_JOIN", "text": "|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((path ((++ g1) g2)) = ((path g1) /\\ (path g2))))))"}
{"_id": "2397", "title": "Lp_ineqs_proofs.dart4_y4'_hi", "text": "|- (!V. (!E. ((lp_fan (V , E)) ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ ((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le ((y4'_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))"}
{"_id": "2398", "title": "Conforming.FAN80_FANADD", "text": "|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (!ds1. (!ds2. (!f10. (!f20. (!f30. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((fan80 (x , (V , E))) /\\ ((ds IN (face_set (hypermap1_of_fanx (x , (V , E))))) /\\ (((CARD ds) > (NUMERAL (BIT1 (BIT1 _0)))) /\\ (((f1 INSERT (f2 INSERT (f3 INSERT EMPTY))) SUBSET ds) /\\ ((((((f1_fan x) V) E) f1) = f2) /\\ ((((((f1_fan x) V) E) f2) = f3) /\\ ((~ (((((f1_fan x) V) E) f3) = f1)) /\\ (((pr2 f1) = v) /\\ (((pr2 f2) = u) /\\ (((pr2 f3) = w) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ (((pr3 f1) = u) /\\ (((pr3 f2) = w) /\\ ((((face (hypermap1_of_fanx (x , (V , E1)))) (x , (v , (w , (((((sigma_fan x) V) E1) v) w))))) = ds1) /\\ ((((face (hypermap1_of_fanx (x , (V , E1)))) (x , (w , (v , (((((sigma_fan x) V) E1) w) v))))) = ds2) /\\ (((x , (w , (v , u))) = f10) /\\ (((x , (v , (u , w))) = f20) /\\ (((x , (u , (w , v))) = f30) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1)))))))))))))))))))))))) ==> (fan80 (x , (V , E1))))))))))))))))))))"}
{"_id": "2399", "title": "LIFT_DROP", "text": "|- ((!x. ((lift (drop x)) = x)) /\\ (!x. ((drop (lift x)) = x)))"}
{"_id": "2400", "title": "Functional_equation.proj_x2", "text": "|- (((((((proj_x2 x1) x2) x3) x4) x5) x6) = x2)"}
{"_id": "2401", "title": "is_metric_space", "text": "|- (!s. (!d. ((is_metric_space (s , d)) = ((!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (real_of_num (NUMERAL _0))) (d (x , y)))))) /\\ ((!x. (!y. (((x IN s) /\\ (y IN s)) ==> (((d (x , y)) = (real_of_num (NUMERAL _0))) = (x = y))))) /\\ ((!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((d (x , y)) = (d (y , x)))))) /\\ (!x. (!y. (!z. (((x IN s) /\\ ((y IN s) /\\ (z IN s))) ==> ((real_le (d (x , z))) ((real_add (d (x , y))) (d (y , z))))))))))))))"}
{"_id": "2402", "title": "Seq.ncons", "text": "|- (!n. (!x. (((ncons n) x) = ((iter n) (CONS x)))))"}
{"_id": "2403", "title": "Localization.h_dart", "text": "|- (!x. ((h_dart x) = ((real_div (vector_norm (FST x))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))"}
{"_id": "2404", "title": "REAL_POW_EQ_1", "text": "|- (!x. (!n. ((((real_pow x) n) = (real_of_num (NUMERAL (BIT1 _0)))) = ((((real_abs x) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((real_lt x) (real_of_num (NUMERAL _0))) ==> (EVEN n))) \\/ (n = (NUMERAL _0))))))"}
{"_id": "2405", "title": "NSUM_OFFSET", "text": "|- (!p. (!f. (!m. (!n. (((nsum ((m + p) .. (n + p))) f) = ((nsum (m .. n)) (\\i. (f (i + p)))))))))"}
{"_id": "2406", "title": "REAL_EXP_LE_X", "text": "|- (!x. ((real_le ((real_add (real_of_num (NUMERAL (BIT1 _0)))) x)) (exp x)))"}
{"_id": "2407", "title": "BETWEEN_LINEAR_IMAGE_EQ", "text": "|- (!f. (!x. (!y. (!z. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((between (f x)) ((f y) , (f z))) = ((between x) (y , z))))))))"}
{"_id": "2408", "title": "REAL_INV_1", "text": "|- ((real_inv (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "2409", "title": "POSITIVE_SEMIDEFINITE_IMP_SYMMETRIC", "text": "|- (!A. ((positive_semidefinite A) ==> ((transp A) = A)))"}
{"_id": "2410", "title": "Seq.filter_uniq", "text": "|- (!s. (!a. ((uniq s) ==> (uniq ((filter a) s)))))"}
{"_id": "2411", "title": "DECREASING_LEFT_LIMIT_1", "text": "|- (!f. (!a. (!b. (!c. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f y))) (drop (f x)))))) /\\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (?l. (((--> f) l) ((within (at c)) (closed_interval ((CONS (a , c)) NIL))))))))))"}
{"_id": "2412", "title": "EXISTS_DOUBLE_ARC_EXPLICIT", "text": "|- (!g. (!a. (!b. (((simple_path g) /\\ (((pathfinish g) = (pathstart g)) /\\ ((a IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ ((b IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ (((real_le (drop a)) (drop b)) /\\ (~ ((g a) = (g b)))))))) ==> (?u. (?d. ((arc u) /\\ ((arc d) /\\ (((pathstart u) = (g a)) /\\ (((pathfinish u) = (g b)) /\\ (((pathstart d) = (g b)) /\\ (((pathfinish d) = (g a)) /\\ (((path_image u) = ((IMAGE g) (closed_interval ((CONS (a , b)) NIL)))) /\\ (((path_image d) = ((IMAGE g) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) DIFF (open_interval (a , b))))) /\\ ((((path_image u) INTER (path_image d)) = ((g a) INSERT ((g b) INSERT EMPTY))) /\\ (((path_image u) UNION (path_image d)) = (path_image g)))))))))))))))))"}
{"_id": "2413", "title": "COS_PI", "text": "|- ((cos pi) = (real_neg (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "2414", "title": "BETWEEN_TRANS_2", "text": "|- (!a. (!b. (!c. (!d. ((((between a) (b , c)) /\\ ((between d) (a , b))) ==> ((between a) (c , d)))))))"}
{"_id": "2415", "title": "LIM_CMUL", "text": "|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\\x. ((% c) (f x)))) ((% c) l)) net)))))"}
{"_id": "2416", "title": "PATHSTART_LINEAR_IMAGE_EQ", "text": "|- (!f. (!g. ((linear f) ==> ((pathstart (f o g)) = (f (pathstart g))))))"}
{"_id": "2417", "title": "Ssrnat.leq_pred", "text": "|- (!n. ((n - (NUMERAL (BIT1 _0))) <= n))"}
{"_id": "2418", "title": "CLOSEST_POINT_LE", "text": "|- (!s. (!a. (!x. (((closed s) /\\ (x IN s)) ==> ((real_le (distance (a , ((closest_point s) a)))) (distance (a , x)))))))"}
{"_id": "2419", "title": "RADON_PARTITION", "text": "|- (!c. (((FINITE c) /\\ (affine_dependent c)) ==> (?m. (?p. (((DISJOINT m) p) /\\ (((m UNION p) = c) /\\ (~ ((DISJOINT ((hull convex) m)) ((hull convex) p)))))))))"}
{"_id": "2420", "title": "Tame_list.indexf", "text": "|- (!y. (!x. (!s. ((((indexf x) y) s) = (((COND (((MEM x) s) /\\ ((MEM y) s))) ((((indexl y) s) + ((sizel s) - ((indexl x) s))) MOD (sizel s))) (sizel s))))))"}
{"_id": "2421", "title": "IMAGE", "text": "|- (!s. (!f. (((IMAGE f) s) = (GSPEC (\\GEN%PVAR%7. (?y. (((SETSPEC GEN%PVAR%7) (?x. ((x IN s) /\\ (y = (f x))))) y)))))))"}
{"_id": "2422", "title": "mdist", "text": "|- (!m. ((mdist m) = (SND (dest_metric m))))"}
{"_id": "2423", "title": "Hypermap.lemma_in_orbit", "text": "|- (!f. (!n. (!x. ((((POWER f) n) x) IN ((orbit_map f) x)))))"}
{"_id": "2424", "title": "CLOSED_IN_REFL", "text": "|- (!s. ((closed_in ((subtopology euclidean) s)) s))"}
{"_id": "2425", "title": "REAL_LE_RDIV_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le x) ((real_div y) z)) = ((real_le ((real_mul x) z)) y))))))"}
{"_id": "2426", "title": "REAL_VECTOR_POLYNOMIAL_FUNCTION_o", "text": "|- (!f. (!g. (((vector_polynomial_function f) /\\ (real_polynomial_function g)) ==> (real_polynomial_function (g o f)))))"}
{"_id": "2427", "title": "Ssrbool.andbT", "text": "|- (!b. ((b /\\ T) = b))"}
{"_id": "2428", "title": "SPAN_OF_SUBSPACE", "text": "|- (!s. ((subspace s) ==> ((span s) = s)))"}
{"_id": "2429", "title": "FINITE_SET_OF_LIST", "text": "|- (!l. (FINITE (set_of_list l)))"}
{"_id": "2430", "title": "COMPACT_EQ_HEINE_BOREL", "text": "|- (!s. ((compact s) = (!f. (((!t. ((t IN f) ==> (open t))) /\\ (s SUBSET (UNIONS f))) ==> (?f'. ((f' SUBSET f) /\\ ((FINITE f') /\\ (s SUBSET (UNIONS f')))))))))"}
{"_id": "2431", "title": "Ssrnat.maxn_mulr", "text": "|- (!m. (!n1. (!n2. ((m * ((maxn n1) n2)) = ((maxn (m * n1)) (m * n2))))))"}
{"_id": "2432", "title": "IN_SEGMENT_CX_GEN", "text": "|- (!a. (!b. (!x. ((x IN (closed_segment ((CONS ((Cx a) , (Cx b))) NIL))) = (((Im x) = (real_of_num (NUMERAL _0))) /\\ ((((real_le a) (Re x)) /\\ ((real_le (Re x)) b)) \\/ (((real_le b) (Re x)) /\\ ((real_le (Re x)) a))))))))"}
{"_id": "2433", "title": "REAL_LT_NEGTOTAL", "text": "|- (!x. ((x = (real_of_num (NUMERAL _0))) \\/ (((real_lt (real_of_num (NUMERAL _0))) x) \\/ ((real_lt (real_of_num (NUMERAL _0))) (real_neg x)))))"}
{"_id": "2434", "title": "MATRIX_MUL_RID", "text": "|- (!A. (((matrix_mul A) (mat (NUMERAL (BIT1 _0)))) = A))"}
{"_id": "2435", "title": "LOCALLY_COMPACT_PCROSS", "text": "|- (!s. (!t. ((((locally compact) s) /\\ ((locally compact) t)) ==> ((locally compact) (s PCROSS t)))))"}
{"_id": "2436", "title": "INVERTIBLE_DET_NZ", "text": "|- (!A. ((invertible A) = (~ ((det A) = (real_of_num (NUMERAL _0))))))"}
{"_id": "2437", "title": "HAS_VECTOR_DERIVATIVE_REAL_COMPLEX", "text": "|- ((((has_complex_derivative f) f') (at (Cx (drop a)))) ==> (((has_vector_derivative (\\x. (f (Cx (drop x))))) f') (at a)))"}
{"_id": "2438", "title": "BOUNDED_INSIDE", "text": "|- (!s. ((bounded s) ==> (bounded (inside s))))"}
{"_id": "2439", "title": "Nkezbfc_local.order", "text": "|- (!f. (!x. (!y. ((((order f) x) y) = (@n. (((((ITER n) f) x) = y) /\\ (!i. ((((NUMERAL _0) < i) /\\ (i < n)) ==> (~ ((((ITER i) f) x) = y))))))))))"}
{"_id": "2440", "title": "VECTOR_MATRIX_MUL_TRANSP", "text": "|- (!A. (!x. (((vector_matrix_mul x) A) = ((matrix_vector_mul (transp A)) x))))"}
{"_id": "2441", "title": "MULT_SUC", "text": "|- (!m. (!n. ((m * (SUC n)) = (m + (m * n)))))"}
{"_id": "2442", "title": "Appendix.scs_J_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_J_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = J))))))))))"}
{"_id": "2443", "title": "Fan.d20_fan", "text": "|- (!x. (!V. (!E. ((d20_fan (x , (V , E))) = (GSPEC (\\GEN%PVAR%5471. (?x'. (?v. (((SETSPEC GEN%PVAR%5471) ((x' = x) /\\ ((V v) /\\ ((((set_of_edge v) V) E) = EMPTY)))) (x' , (v , (v , v))))))))))))"}
{"_id": "2444", "title": "AFFINE_TRANSLATION", "text": "|- (!s. (!a. ((affine s) ==> (affine ((IMAGE (\\x. ((vector_add a) x))) s)))))"}
{"_id": "2445", "title": "CONNECTED_NEST", "text": "|- (!s. (((!n. ((compact (s n)) /\\ (connected (s n)))) /\\ (!m. (!n. ((m <= n) ==> ((s n) SUBSET (s m)))))) ==> (connected (INTERS (GSPEC (\\GEN%PVAR%1154. (?n. (((SETSPEC GEN%PVAR%1154) (n IN UNIV)) (s n)))))))))"}
{"_id": "2446", "title": "NOT_NEGLIGIBLE_UNIV", "text": "|- (~ (negligible UNIV))"}
{"_id": "2447", "title": "Seq.rot0", "text": "|- (!s. (((rot (NUMERAL _0)) s) = s))"}
{"_id": "2448", "title": "CLOG_MUL", "text": "|- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> ((clog ((complex_mul w) z)) = (((COND ((real_le (Im ((vector_add (clog w)) (clog z)))) (real_neg pi))) ((vector_add ((vector_add (clog w)) (clog z))) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))) (((COND ((real_gt (Im ((vector_add (clog w)) (clog z)))) pi)) ((vector_sub ((vector_add (clog w)) (clog z))) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))) ((vector_add (clog w)) (clog z))))))))"}
{"_id": "2449", "title": "Ssrnat.fact0", "text": "|- ((FACT (NUMERAL _0)) = (NUMERAL (BIT1 _0)))"}
{"_id": "2450", "title": "Pack2.NEGLIGIBLE_INTER_VORONOI_CLOSED", "text": "|- (!s. (!u. (!v. (((u IN s) /\\ ((v IN s) /\\ (~ (u = v)))) ==> (negligible (((voronoi_closed s) u) INTER ((voronoi_closed s) v)))))))"}
{"_id": "2451", "title": "Tame_defs.tame_planar_hypermap", "text": "|- (!H. ((tame_planar_hypermap H) = ((tame_1 H) /\\ ((tame_2 H) /\\ ((tame_3 H) /\\ ((tame_4 H) /\\ ((tame_5a H) /\\ ((tame_8 H) /\\ ((tame_9a H) /\\ ((tame_10 H) /\\ ((tame_11a H) /\\ ((tame_11b H) /\\ ((tame_12o H) /\\ (tame_13a H))))))))))))))"}
{"_id": "2452", "title": "BOUNDED_UNIONS", "text": "|- (!f. (((FINITE f) /\\ (!s. ((s IN f) ==> (bounded s)))) ==> (bounded (UNIONS f))))"}
{"_id": "2453", "title": "simple_path", "text": "|- (!g. ((simple_path g) = ((path g) /\\ (!x. (!y. (((x IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ ((y IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ ((g x) = (g y)))) ==> ((x = y) \\/ (((x = (vec (NUMERAL _0))) /\\ (y = (vec (NUMERAL (BIT1 _0))))) \\/ ((x = (vec (NUMERAL (BIT1 _0)))) /\\ (y = (vec (NUMERAL _0))))))))))))"}
{"_id": "2454", "title": "Lp_ineqs_proofs.lp_tau", "text": "|- (!V. (!E. ((lp_tau (V , E)) = (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL _0))) (tauVEF (V , (E , f)))))))))"}
{"_id": "2455", "title": "SEPARATING_HYPERPLANE_SETS", "text": "|- (!s. (!t. (((convex s) /\\ ((convex t) /\\ ((~ (s = EMPTY)) /\\ ((~ (t = EMPTY)) /\\ ((DISJOINT s) t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\\ ((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\\ (!x. ((x IN t) ==> ((real_ge ((dot a) x)) b))))))))))"}
{"_id": "2456", "title": "WF_UREC", "text": "|- ((WF <<) ==> (!H. ((!f. (!g. (!x. ((!z. (((<< z) x) ==> ((f z) = (g z)))) ==> (((H f) x) = ((H g) x)))))) ==> (!f. (!g. (((!x. ((f x) = ((H f) x))) /\\ (!x. ((g x) = ((H g) x)))) ==> (f = g)))))))"}
{"_id": "2457", "title": "Functional_equation.mul6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((mul6 f) g) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "2458", "title": "ORTHOGONAL_TRANSFORMATION_ORTHOGONAL_EIGENVECTORS", "text": "|- (!f. (!v. (!w. (!a. (!b. (((orthogonal_transformation f) /\\ (((f v) = ((% a) v)) /\\ (((f w) = ((% b) w)) /\\ (~ (a = b))))) ==> ((orthogonal v) w)))))))"}
{"_id": "2459", "title": "CONVEX_CONNECTED", "text": "|- (!s. ((convex s) ==> (connected s)))"}
{"_id": "2460", "title": "Lp_ineqs_proofs.face5_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((LET (\\w'. (LET_END ((LET (\\u. (LET_END ((LET (\\z. (LET_END (((v , w) IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((((f_fan_pair_ext (V , E)) (v , w)) = (w , z)) /\\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) (v , w)) = (z , u)) /\\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) (v , w)) = (u , w')) /\\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (v , w)) = (w' , v)) /\\ ((((f_fan_pair_ext (V , E)) (w , z)) = (z , u)) /\\ ((((f_fan_pair_ext (V , E)) (z , u)) = (u , w')) /\\ ((((f_fan_pair_ext (V , E)) (u , w')) = (w' , v)) /\\ ((((f_fan_pair_ext (V , E)) (w' , v)) = (v , w)) /\\ (((inverse (f_fan_pair_ext (V , E))) (v , w)) = (w' , v)))))))))) /\\ ((((v , w) IN (dart1_of_fan (V , E))) /\\ (((w , z) IN (dart1_of_fan (V , E))) /\\ (((z , u) IN (dart1_of_fan (V , E))) /\\ (((u , w') IN (dart1_of_fan (V , E))) /\\ ((w' , v) IN (dart1_of_fan (V , E))))))) /\\ (((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , z) INSERT ((z , u) INSERT ((u , w') INSERT ((w' , v) INSERT EMPTY)))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) u) w'))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))"}
{"_id": "2461", "title": "HOMOTOPIC_PATHS_RCANCEL", "text": "|- (!p. (!q. (!r. (!s. (((((homotopic_paths s) ((++ p) r)) ((++ q) r)) /\\ (((pathfinish p) = (pathstart r)) /\\ ((pathfinish q) = (pathstart r)))) ==> (((homotopic_paths s) p) q))))))"}
{"_id": "2462", "title": "SUBSET_HALFSPACES_IMP_COLLINEAR", "text": "|- ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%388. (?x. (((SETSPEC GEN%PVAR%388) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%389. (?x. (((SETSPEC GEN%PVAR%389) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%390. (?x. (((SETSPEC GEN%PVAR%390) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%391. (?x. (((SETSPEC GEN%PVAR%391) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%392. (?x. (((SETSPEC GEN%PVAR%392) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%393. (?x. (((SETSPEC GEN%PVAR%393) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%394. (?x. (((SETSPEC GEN%PVAR%394) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%395. (?x. (((SETSPEC GEN%PVAR%395) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%396. (?x. (((SETSPEC GEN%PVAR%396) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%397. (?x. (((SETSPEC GEN%PVAR%397) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%398. (?x. (((SETSPEC GEN%PVAR%398) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%399. (?x. (((SETSPEC GEN%PVAR%399) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%400. (?x. (((SETSPEC GEN%PVAR%400) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%401. (?x. (((SETSPEC GEN%PVAR%401) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%402. (?x. (((SETSPEC GEN%PVAR%402) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%403. (?x. (((SETSPEC GEN%PVAR%403) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%404. (?x. (((SETSPEC GEN%PVAR%404) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%405. (?x. (((SETSPEC GEN%PVAR%405) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%406. (?x. (((SETSPEC GEN%PVAR%406) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%407. (?x. (((SETSPEC GEN%PVAR%407) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%408. (?x. (((SETSPEC GEN%PVAR%408) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%409. (?x. (((SETSPEC GEN%PVAR%409) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%410. (?x. (((SETSPEC GEN%PVAR%410) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%411. (?x. (((SETSPEC GEN%PVAR%411) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%412. (?x. (((SETSPEC GEN%PVAR%412) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%413. (?x. (((SETSPEC GEN%PVAR%413) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%414. (?x. (((SETSPEC GEN%PVAR%414) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%415. (?x. (((SETSPEC GEN%PVAR%415) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%416. (?x. (((SETSPEC GEN%PVAR%416) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%417. (?x. (((SETSPEC GEN%PVAR%417) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%418. (?x. (((SETSPEC GEN%PVAR%418) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%419. (?x. (((SETSPEC GEN%PVAR%419) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%420. (?x. (((SETSPEC GEN%PVAR%420) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%421. (?x. (((SETSPEC GEN%PVAR%421) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%422. (?x. (((SETSPEC GEN%PVAR%422) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%423. (?x. (((SETSPEC GEN%PVAR%423) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%424. (?x. (((SETSPEC GEN%PVAR%424) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%425. (?x. (((SETSPEC GEN%PVAR%425) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%426. (?x. (((SETSPEC GEN%PVAR%426) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%427. (?x. (((SETSPEC GEN%PVAR%427) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%428. (?x. (((SETSPEC GEN%PVAR%428) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%429. (?x. (((SETSPEC GEN%PVAR%429) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%430. (?x. (((SETSPEC GEN%PVAR%430) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%431. (?x. (((SETSPEC GEN%PVAR%431) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%432. (?x. (((SETSPEC GEN%PVAR%432) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%433. (?x. (((SETSPEC GEN%PVAR%433) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ ((!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%434. (?x. (((SETSPEC GEN%PVAR%434) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%435. (?x. (((SETSPEC GEN%PVAR%435) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\\ (!a. (!b. (!c. (!d. (((GSPEC (\\GEN%PVAR%436. (?x. (((SETSPEC GEN%PVAR%436) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\\GEN%PVAR%437. (?x. (((SETSPEC GEN%PVAR%437) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY)))))))))))))))))))))))))))))))))"}
{"_id": "2463", "title": "NSUM_SUBSET", "text": "|- (!u. (!v. (!f. (((FINITE u) /\\ ((FINITE v) /\\ (!x. ((x IN (u DIFF v)) ==> ((f x) = (NUMERAL _0)))))) ==> (((nsum u) f) <= ((nsum v) f))))))"}
{"_id": "2464", "title": "HAS_REAL_INTEGRAL_EMPTY", "text": "|- (!f. (((has_real_integral f) (real_of_num (NUMERAL _0))) EMPTY))"}
{"_id": "2465", "title": "LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED", "text": "|- (!s. (((locally path_connected) s) ==> ((locally connected) s)))"}
{"_id": "2466", "title": "DOT_LMUL_MATRIX", "text": "|- (!A. (!x. (!y. (((dot ((vector_matrix_mul x) A)) y) = ((dot x) ((matrix_vector_mul A) y))))))"}
{"_id": "2467", "title": "RESTRICTION_FIXPOINT", "text": "|- (!s. (!f. ((((RESTRICTION s) f) = f) = (f IN (EXTENSIONAL s)))))"}
{"_id": "2468", "title": "COMMUTING_WITH_DIAGONAL_MATRIX", "text": "|- (!A. (!D. ((diagonal_matrix D) ==> ((((matrix_mul A) D) = ((matrix_mul D) A)) = (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ (j <= (dimindex UNIV))))) ==> ((((A $ i) $ j) = (real_of_num (NUMERAL _0))) \\/ (((D $ i) $ i) = ((D $ j) $ j))))))))))"}
{"_id": "2469", "title": "REAL_LE_RNEG", "text": "|- (!x. (!y. (((real_le x) (real_neg y)) = ((real_le ((real_add x) y)) (real_of_num (NUMERAL _0))))))"}
{"_id": "2470", "title": "OPEN_INTERIOR", "text": "|- (!s. (open (interior s)))"}
{"_id": "2471", "title": "INT_POS", "text": "|- (!n. ((int_le (int_of_num (NUMERAL _0))) (int_of_num n)))"}
{"_id": "2472", "title": "Fan.fan2", "text": "|- (!E. (!x. (!V. ((fan2 (x , (V , E))) = (~ (x IN V))))))"}
{"_id": "2473", "title": "homeomorphic", "text": "|- (!s. (!t. (((homeomorphic s) t) = (?f. (?g. ((homeomorphism (s , t)) (f , g)))))))"}
{"_id": "2474", "title": "BALL_MIN_INTER", "text": "|- (!a. (!r. (!s. ((ball (a , ((real_min r) s))) = ((ball (a , r)) INTER (ball (a , s)))))))"}
{"_id": "2475", "title": "LE_CASES", "text": "|- (!m. (!n. ((m <= n) \\/ (n <= m))))"}
{"_id": "2476", "title": "Fan.e_fan", "text": "|- (!x. (!V. (!E. ((((e_fan x) V) E) = (GABS (\\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (w , (v , (((((sigma_fan x) V) E) w) v)))))))))))))))"}
{"_id": "2477", "title": "Hypermap.edge", "text": "|- (!H. (!x. (((edge H) x) = ((orbit_map (edge_map H)) x))))"}
{"_id": "2478", "title": "ARC_JOIN_EQ", "text": "|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((arc ((++ g1) g2)) = ((arc g1) /\\ ((arc g2) /\\ (((path_image g1) INTER (path_image g2)) SUBSET ((pathstart g2) INSERT EMPTY))))))))"}
{"_id": "2479", "title": "HAS_REAL_VECTOR_DERIVATIVE_AT", "text": "|- ((((has_real_derivative f) f') (atreal x)) = (((has_vector_derivative (lift o (f o drop))) (lift f')) (at (lift x))))"}
{"_id": "2480", "title": "IN_IMAGE_LIFT_DROP", "text": "|- ((!x. (!s. ((x IN ((IMAGE lift) s)) = ((drop x) IN s)))) /\\ (!x. (!s. ((x IN ((IMAGE drop) s)) = ((lift x) IN s)))))"}
{"_id": "2481", "title": "CONTINUOUS_ON_MUL", "text": "|- (!s. (!c. (!f. ((((continuous_on (lift o c)) s) /\\ ((continuous_on f) s)) ==> ((continuous_on (\\x. ((% (c x)) (f x)))) s)))))"}
{"_id": "2482", "title": "HAS_BOUNDED_VARIATION_ON_ADD", "text": "|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\\x. ((vector_add (f x)) (g x)))) s)))))"}
{"_id": "2483", "title": "Seq.nth_drop", "text": "|- (!x0. (!n0. (!s. (!i. ((((nth x0) ((dropl n0) s)) i) = (((nth x0) s) (n0 + i)))))))"}
{"_id": "2484", "title": "PASTECART_CMUL", "text": "|- (!x1. (!y1. (!c. (((pastecart ((% c) x1)) ((% c) y1)) = ((% c) ((pastecart x1) y1))))))"}
{"_id": "2485", "title": "TC_INC", "text": "|- (!R. (!x. (!y. (((R x) y) ==> (((TC R) x) y)))))"}
{"_id": "2486", "title": "Packing3.HULL_INTER_SUBSET_INTER_HULL", "text": "|- (!P. (!s. (!t. (((hull P) (s INTER t)) SUBSET (((hull P) s) INTER ((hull P) t))))))"}
{"_id": "2487", "title": "AFFINE_HULL_CONVEX_INTER_OPEN_IN", "text": "|- (!s. (!t. (((convex s) /\\ (((open_in ((subtopology euclidean) ((hull affine) s))) t) /\\ (~ ((s INTER t) = EMPTY)))) ==> (((hull affine) (s INTER t)) = ((hull affine) s)))))"}
{"_id": "2488", "title": "Nonlin_def.ups_126", "text": "|- (ups_126 = (((((((compose6 (promote3_to_6 ups_x)) proj_x1) proj_x2) proj_x6) dummy6) dummy6) dummy6))"}
{"_id": "2489", "title": "Merge_ineq.delta_min_corner_4_784", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1) /\\ (((real_le x1) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x2) /\\ (((real_le x2) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x3) /\\ (((real_le x3) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4) /\\ (((real_le x4) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x5) /\\ (((real_le x5) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6) /\\ ((real_le x6) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) ==> ((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "2490", "title": "iterate", "text": "|- (!f. (!s. (!op. ((((iterate op) s) f) = (((COND (FINITE (((support op) f) s))) (((ITSET (\\x. (\\a. ((op (f x)) a)))) (((support op) f) s)) (neutral op))) (neutral op))))))"}
{"_id": "2491", "title": "OPEN_IN_REFL", "text": "|- (!s. ((open_in ((subtopology euclidean) s)) s))"}
{"_id": "2492", "title": "REAL_LT_ADDL", "text": "|- (!x. (!y. (((real_lt y) ((real_add x) y)) = ((real_lt (real_of_num (NUMERAL _0))) x))))"}
{"_id": "2493", "title": "Nonlinear_lemma.vol3f_x_lfun_alt", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\\ (((real_le (real_of_num (NUMERAL _0))) x2) /\\ ((real_le (real_of_num (NUMERAL _0))) x6))) ==> (((((((vol3f_x_lfun x1) x2) x3) x4) x5) x6) = ((real_sub ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((((((dih3_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_add ((((((dih4_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_sub ((((((dih5_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi))))))))) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi))) ((real_add ((((((ldih_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_add ((((((ldih2_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((((((ldih6_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)))))))))))))"}
{"_id": "2494", "title": "Miqmcsn.scs_4M8_13", "text": "|- (scs_4M8_13 = ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))"}
{"_id": "2495", "title": "NORM_VSUM_PYTHAGOREAN", "text": "|- (!k. (!u. (((FINITE k) /\\ ((pairwise (\\i. (\\j. ((orthogonal (u i)) (u j))))) k)) ==> (((real_pow (vector_norm ((vsum k) u))) (NUMERAL (BIT0 (BIT1 _0)))) = ((sum k) (\\i. ((real_pow (vector_norm (u i))) (NUMERAL (BIT0 (BIT1 _0))))))))))"}
{"_id": "2496", "title": "Merge_ineq.nonfunctional_mu6_x_y", "text": "|- (!x1. (!x2. (!x3. (!a. (!b. (!c. (((((((mu6_x x1) x2) x3) a) b) c) = (((mu_y (sqrt x1)) (sqrt x2)) (sqrt x3)))))))))"}
{"_id": "2497", "title": "VSUM_CLAUSES_NUMSEG", "text": "|- ((!m. (((vsum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (vec (NUMERAL _0))))) /\\ (!m. (!n. (((vsum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((vector_add ((vsum (m .. n)) f)) (f (SUC n)))) ((vsum (m .. n)) f))))))"}
{"_id": "2498", "title": "HOMOTOPIC_WITH_IMP_CONTINUOUS", "text": "|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> (((continuous_on f) X) /\\ ((continuous_on g) X))))))))"}
{"_id": "2499", "title": "SIMPLE_PATH_TRANSLATION_EQ", "text": "|- (!a. (!g. ((simple_path ((\\x. ((vector_add a) x)) o g)) = (simple_path g))))"}
{"_id": "2500", "title": "Constants_approx.acs3_interval", "text": "|- ((interval_arith (acs ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))))))))))))))))))))))"}
{"_id": "2501", "title": "COMPLEX_NORM_MUL", "text": "|- (!w. (!z. ((vector_norm ((complex_mul w) z)) = ((real_mul (vector_norm w)) (vector_norm z)))))"}
{"_id": "2502", "title": "Ssrbool.andbAC", "text": "|- (!b. (!c. (!p. (((b /\\ c) /\\ p) = ((b /\\ p) /\\ c)))))"}
{"_id": "2503", "title": "TAN_PI4", "text": "|- ((tan ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "2504", "title": "COUNTABLE_ANR_PATH_COMPONENTS", "text": "|- (!s. (!t. ((ANR s) ==> (COUNTABLE (GSPEC (\\GEN%PVAR%2677. (?x. (((SETSPEC GEN%PVAR%2677) (x IN t)) ((path_component s) x)))))))))"}
{"_id": "2505", "title": "CROSS_LMUL", "text": "|- (!c. (!x. (!y. (((cross ((% c) x)) y) = ((% c) ((cross x) y))))))"}
{"_id": "2506", "title": "EXP_ONE", "text": "|- (!n. (((NUMERAL (BIT1 _0)) EXP n) = (NUMERAL (BIT1 _0))))"}
{"_id": "2507", "title": "EMPTY_INTERIOR_FINITE", "text": "|- (!s. ((FINITE s) ==> ((interior s) = EMPTY)))"}
{"_id": "2508", "title": "ENDS_IN_INTERVAL", "text": "|- ((!a. (!b. ((a IN (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\\ ((!a. (!b. ((b IN (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\\ ((!a. (!b. (~ (a IN (open_interval (a , b)))))) /\\ (!a. (!b. (~ (b IN (open_interval (a , b)))))))))"}
{"_id": "2509", "title": "Trigonometry.KGLLRQT2", "text": "|- (!x. ((sin (real_neg x)) = (real_neg (sin x))))"}
{"_id": "2510", "title": "sphere", "text": "|- (!x. (!e. ((sphere (x , e)) = (GSPEC (\\GEN%PVAR%555. (?y. (((SETSPEC GEN%PVAR%555) ((distance (x , y)) = e)) y)))))))"}
{"_id": "2511", "title": "CONNECTED_CONNECTED_IMP_CLOSED_POINTIMAGES", "text": "|- (!f. (!s. (((!c. (((c SUBSET s) /\\ (connected c)) ==> (connected ((IMAGE f) c)))) /\\ (!y. (connected (GSPEC (\\GEN%PVAR%1938. (?x. (((SETSPEC GEN%PVAR%1938) ((x IN s) /\\ ((f x) = y))) x))))))) ==> (!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%1939. (?x. (((SETSPEC GEN%PVAR%1939) ((x IN s) /\\ ((f x) = y))) x)))))))))"}
{"_id": "2512", "title": "HAS_BOUNDED_VARIATION_ON_CONST", "text": "|- (!s. (!c. ((has_bounded_variation_on (\\x. c)) s)))"}
{"_id": "2513", "title": "OPEN_UNIV", "text": "|- (open UNIV)"}
{"_id": "2514", "title": "COLLINEAR_TRANSLATION_EQ", "text": "|- (!a. (!s. ((collinear ((IMAGE (\\x. ((vector_add a) x))) s)) = (collinear s))))"}
{"_id": "2515", "title": "ORTHOGONAL_MATRIX_TRANSFORMATION", "text": "|- (!A. ((orthogonal_matrix A) = (orthogonal_transformation (\\x. ((matrix_vector_mul A) x)))))"}
{"_id": "2516", "title": "retract_of", "text": "|- (!s. (!t. (((retract_of t) s) = (?r. ((retraction (s , t)) r)))))"}
{"_id": "2517", "title": "PRODUCT_CLAUSES", "text": "|- ((!f. (((product EMPTY) f) = (real_of_num (NUMERAL (BIT1 _0))))) /\\ (!x. (!f. (!s. ((FINITE s) ==> (((product (x INSERT s)) f) = (((COND (x IN s)) ((product s) f)) ((real_mul (f x)) ((product s) f)))))))))"}
{"_id": "2518", "title": "Nonlin_def.sqrt_x6", "text": "|- (((((((sqrt_x6 x1) x2) x3) x4) x5) x6) = (sqrt x6))"}
{"_id": "2519", "title": "Tame_defs2.mapAt", "text": "|- (((((mapAt NIL) f) xs) = xs) /\\ ((((mapAt ((CONS n) ns)) f) xs) = (((COND (n < (LENGTH xs))) (((mapAt ns) f) (((list_update xs) n) (f ((EL n) xs))))) (((mapAt ns) f) xs))))"}
{"_id": "2520", "title": "Packing3.EL_TABLE", "text": "|- (!f. (!n. (!i. ((i < n) ==> (((EL i) ((TABLE f) n)) = (f i))))))"}
{"_id": "2521", "title": "COMPLEX_DIFFERENTIABLE_AT_CSIN", "text": "|- (!z. ((complex_differentiable csin) (at z)))"}
{"_id": "2522", "title": "CONVEX_IMP_BOUNDED_ON_INTERVAL", "text": "|- (!f. (!a. (!b. (((convex_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_abs (f x))) B)))))))))"}
{"_id": "2523", "title": "SUM_CLAUSES", "text": "|- ((!f. (((sum EMPTY) f) = (real_of_num (NUMERAL _0)))) /\\ (!x. (!f. (!s. ((FINITE s) ==> (((sum (x INSERT s)) f) = (((COND (x IN s)) ((sum s) f)) ((real_add (f x)) ((sum s) f)))))))))"}
{"_id": "2524", "title": "CONVEX_CONVEX_HULL", "text": "|- (!s. (convex ((hull convex) s)))"}
{"_id": "2525", "title": "REAL_INV_NEG", "text": "|- (!x. ((real_inv (real_neg x)) = (real_neg (real_inv x))))"}
{"_id": "2526", "title": "EVEN_MULT", "text": "|- (!m. (!n. ((EVEN (m * n)) = ((EVEN m) \\/ (EVEN n)))))"}
{"_id": "2527", "title": "COMPLEX_POW_2", "text": "|- (!x. (((complex_pow x) (NUMERAL (BIT0 (BIT1 _0)))) = ((complex_mul x) x)))"}
{"_id": "2528", "title": "CLOSURE_IMAGE_CLOSURE", "text": "|- (!f. (!s. (((continuous_on f) (closure s)) ==> ((closure ((IMAGE f) (closure s))) = (closure ((IMAGE f) s))))))"}
{"_id": "2529", "title": "Conforming.SIGMA_FAN_OF_FANADD1", "text": "|- (!x. (!V. (!E. (!E1. (!v. (!w. (((FAN (x , (V , E))) /\\ ((FAN (x , (V , E1))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((~ ((v INSERT (w INSERT EMPTY)) IN E)) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))) ==> (!v1. (!w1. ((((v1 INSERT (w1 INSERT EMPTY)) IN E) /\\ (~ (v1 IN (v INSERT (w INSERT EMPTY))))) ==> ((((((sigma_fan x) V) E1) v1) w1) = (((((sigma_fan x) V) E) v1) w1))))))))))))"}
{"_id": "2530", "title": "Lp_ineqs_def.ln_fan", "text": "|- (!v. ((ln_fan v) = (lmfun ((real_div (yn_fan v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "2531", "title": "Local_lemmas1.HAFL_CIRCLE_FORM_LOCAL_FAN", "text": "|- ((((local_fan (V , (E , FF))) /\\ ((v IN V) /\\ ((w IN V) /\\ ((~ (v = w)) /\\ ((!z. (!t. (((z IN (v INSERT (w INSERT EMPTY))) /\\ (t IN (V DIFF (z INSERT EMPTY)))) ==> (~ (collinear ((vec (NUMERAL _0)) INSERT (z INSERT (t INSERT EMPTY)))))))) /\\ (!x. ((x IN FF) ==> (((aff_gt ((vec (NUMERAL _0)) INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) SUBSET ((wedge_in_fan_gt x) E))))))))) /\\ ((HS = (hypermap (HYP ((vec (NUMERAL _0)) , (V , (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))))))) /\\ (fv = ((face HS) (v , ((rho_node1 FF) v)))))) ==> (local_fan (((v_prime V) fv) , (((e_prime (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))) fv) , fv))))"}
{"_id": "2532", "title": "HAUSDIST_STILL_INSIDE_INTERIOR", "text": "|- (!s. (!t. (!x. (((bounded s) /\\ ((bounded t) /\\ ((convex s) /\\ ((convex t) /\\ ((~ (t = EMPTY)) /\\ ((real_lt (hausdist (s , t))) (setdist ((x INSERT EMPTY) , (UNIV DIFF s))))))))) ==> (x IN (interior t))))))"}
{"_id": "2533", "title": "CONNECTED_LINEAR_IMAGE", "text": "|- (!f. (!s. (((connected s) /\\ (linear f)) ==> (connected ((IMAGE f) s)))))"}
{"_id": "2534", "title": "HAS_REAL_DERIVATIVE_ATREAL_WITHIN", "text": "|- (!f. (!f'. (!x. (!s. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative f) f') ((within (atreal x)) s)))))))"}
{"_id": "2535", "title": "SIN_COS_INJ", "text": "|- (!x. (!y. ((((sin x) = (sin y)) /\\ (((cos x) = (cos y)) /\\ ((real_lt (real_abs ((real_sub x) y))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) ==> (x = y))))"}
{"_id": "2536", "title": "Miqmcsn.SCS_3T7_OPP_IS_SCS", "text": "|- (is_scs_v39 (scs_opp_v39 scs_3T7))"}
{"_id": "2537", "title": "MEASURABLE_ON_MAX", "text": "|- (!f. (!g. (!s. ((((measurable_on f) s) /\\ ((measurable_on g) s)) ==> ((measurable_on (\\x. (lambdai. ((real_max ((f x) $ i)) ((g x) $ i))))) s)))))"}
{"_id": "2538", "title": "REAL_SUB_LNEG", "text": "|- (!x. (!y. (((real_sub (real_neg x)) y) = (real_neg ((real_add x) y)))))"}
{"_id": "2539", "title": "HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_LEFT_EQ", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homotopy_equivalent ((IMAGE f) s)) t) = ((homotopy_equivalent s) t))))))"}
{"_id": "2540", "title": "Hypermap_iso.iso_components", "text": "|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!d. ((d IN (dart H)) ==> ((((node G) (f d)) = ((IMAGE f) ((node H) d))) /\\ ((((face G) (f d)) = ((IMAGE f) ((face H) d))) /\\ (((edge G) (f d)) = ((IMAGE f) ((edge H) d)))))))))))"}
{"_id": "2541", "title": "INT_GE", "text": "|- (!x. (!y. (((int_ge x) y) = ((int_le y) x))))"}
{"_id": "2542", "title": "CONTINUOUS_ON_SUB", "text": "|- (!f. (!g. (!s. ((((continuous_on f) s) /\\ ((continuous_on g) s)) ==> ((continuous_on (\\x. ((vector_sub (f x)) (g x)))) s)))))"}
{"_id": "2543", "title": "LIMPT_OF_OPEN_IN", "text": "|- (!s. (!t. (!x. ((((open_in ((subtopology euclidean) s)) t) /\\ (((limit_point_of x) s) /\\ (x IN t))) ==> ((limit_point_of x) t)))))"}
{"_id": "2544", "title": "BOUNDED_PATH_IMAGE", "text": "|- (!g. ((path g) ==> (bounded (path_image g))))"}
{"_id": "2545", "title": "ABSOLUTELY_INTEGRABLE_SUB", "text": "|- (!f. (!g. (!s. ((((absolutely_integrable_on f) s) /\\ ((absolutely_integrable_on g) s)) ==> ((absolutely_integrable_on (\\x. ((vector_sub (f x)) (g x)))) s)))))"}
{"_id": "2546", "title": "REAL_FLOOR_NEG", "text": "|- (!x. ((floor (real_neg x)) = (((COND (integer x)) (real_neg x)) (real_neg ((real_add (floor x)) (real_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "2547", "title": "CONVEX_ON_DIRECTIONAL_DERIVATIVE_FUNCTIONS", "text": "|- (!f. (!s. ((((convex_on f) s) /\\ ((is_interval s) /\\ (open s))) ==> (?l. (?r. ((!x. ((x IN s) ==> (((has_vector_derivative (lift o f)) (l x)) ((within (at x)) (GSPEC (\\GEN%PVAR%3307. (?t. (((SETSPEC GEN%PVAR%3307) ((real_le (drop t)) (drop x))) t)))))))) /\\ ((!x. ((x IN s) ==> (((has_vector_derivative (lift o f)) (r x)) ((within (at x)) (GSPEC (\\GEN%PVAR%3308. (?t. (((SETSPEC GEN%PVAR%3308) ((real_le (drop x)) (drop t))) t)))))))) /\\ ((!x. ((x IN s) ==> ((real_le (drop (l x))) (drop (r x))))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (drop (r x))) (drop (l y))))))))))))))"}
{"_id": "2548", "title": "VSUM_IMAGE", "text": "|- (!f. (!g. (!s. (((FINITE s) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y))))) ==> (((vsum ((IMAGE f) s)) g) = ((vsum s) (g o f)))))))"}
{"_id": "2549", "title": "SWAP_IDEMPOTENT", "text": "|- (!a. (!b. (((swap (a , b)) o (swap (a , b))) = I)))"}
{"_id": "2550", "title": "convex_on", "text": "|- (!s. (!f. (((convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\\ ((y IN s) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))"}
{"_id": "2551", "title": "CONIC_LINEAR_IMAGE", "text": "|- (!f. (!s. (((conic s) /\\ (linear f)) ==> (conic ((IMAGE f) s)))))"}
{"_id": "2552", "title": "SEPARATING_HYPERPLANE_CLOSED_0", "text": "|- (!s. (((convex s) /\\ ((closed s) /\\ (~ ((vec (NUMERAL _0)) IN s)))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\\ (((real_lt (real_of_num (NUMERAL _0))) b) /\\ (!x. ((x IN s) ==> ((real_gt ((dot a) x)) b)))))))))"}
{"_id": "2553", "title": "Lp_main_estimate.diag_not_in_dart", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!a. (!b. (!f. (((f IN (face_set (hypermap_of_fan (V , E)))) /\\ ((a IN ((v_prime V) f)) /\\ ((b IN ((v_prime V) f)) /\\ ((~ ((a , b) IN f)) /\\ (~ ((b , a) IN f)))))) ==> (~ ((a , b) IN (dart_of_fan (V , E))))))))))))"}
{"_id": "2554", "title": "Tame_defs.tame_2", "text": "|- (!H. ((tame_2 H) = ((connected_hypermap H) /\\ (simple_hypermap H))))"}
{"_id": "2555", "title": "LENGTH", "text": "|- (((LENGTH NIL) = (NUMERAL _0)) /\\ (!h. (!t. ((LENGTH ((CONS h) t)) = (SUC (LENGTH t))))))"}
{"_id": "2556", "title": "VECTOR_4", "text": "|- ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT1 _0))) = w) /\\ ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT0 (BIT1 _0)))) = x) /\\ ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT1 (BIT1 _0)))) = y) /\\ (((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) = z))))"}
{"_id": "2557", "title": "REAL_SERIES_RMUL", "text": "|- (!x. (!x0. (!c. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\\n. ((real_mul (x n)) c))) ((real_mul x0) c)) s))))))"}
{"_id": "2558", "title": "CANTOR_BENDIXSON_GEN", "text": "|- (!s. (?t. (?u. (((closed_in ((subtopology euclidean) s)) t) /\\ ((!x. ((x IN t) ==> ((condensation_point_of x) t))) /\\ ((COUNTABLE u) /\\ (s = (t UNION u))))))))"}
{"_id": "2559", "title": "SIN_NPI", "text": "|- (!n. ((sin ((real_mul (real_of_num n)) pi)) = (real_of_num (NUMERAL _0))))"}
{"_id": "2560", "title": "FLOOR_FRAC", "text": "|- (!x. ((integer (floor x)) /\\ (((real_le (real_of_num (NUMERAL _0))) (frac x)) /\\ (((real_lt (frac x)) (real_of_num (NUMERAL (BIT1 _0)))) /\\ (x = ((real_add (floor x)) (frac x)))))))"}
{"_id": "2561", "title": "covering_space", "text": "|- (!s. (!c. (!p. (((covering_space (c , p)) s) = (((continuous_on p) c) /\\ ((((IMAGE p) c) = s) /\\ (!x. ((x IN s) ==> (?t. ((x IN t) /\\ (((open_in ((subtopology euclidean) s)) t) /\\ (?v. (((UNIONS v) = (GSPEC (\\GEN%PVAR%2050. (?x. (((SETSPEC GEN%PVAR%2050) ((x IN c) /\\ ((p x) IN t))) x))))) /\\ ((!u. ((u IN v) ==> ((open_in ((subtopology euclidean) c)) u))) /\\ (((pairwise DISJOINT) v) /\\ (!u. ((u IN v) ==> (?q. ((homeomorphism (u , t)) (p , q))))))))))))))))))))"}
{"_id": "2562", "title": "TAN_ATN", "text": "|- (!x. ((((real_lt (real_neg ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) x) /\\ ((real_lt x) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((atn (tan x)) = x)))"}
{"_id": "2563", "title": "Hypermap_iso.hyp_iso_dart", "text": "|- (!H. (!f. (!G. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((f x) IN (dart G))))))))"}
{"_id": "2564", "title": "CARD_IMAGE_INJ", "text": "|- (!f. (!s. (((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) /\\ (FINITE s)) ==> ((CARD ((IMAGE f) s)) = (CARD s)))))"}
{"_id": "2565", "title": "integer", "text": "|- (!x. ((integer x) = (?n. ((real_abs x) = (real_of_num n)))))"}
{"_id": "2566", "title": "Miqmcsn.SCS_4M8_13_BASIC", "text": "|- (scs_basic_v39 scs_4M8_13)"}
{"_id": "2567", "title": "SEPARATING_HYPERPLANE_SET_POINT_INAFF", "text": "|- (!s. (!z. (((convex s) /\\ ((~ (s = EMPTY)) /\\ (~ (z IN s)))) ==> (?a. (?b. ((((vector_add z) a) IN ((hull affine) (z INSERT s))) /\\ ((~ (a = (vec (NUMERAL _0)))) /\\ (((real_le ((dot a) z)) b) /\\ (!x. ((x IN s) ==> ((real_ge ((dot a) x)) b)))))))))))"}
{"_id": "2568", "title": "Constants_approx.INTERVAL_SUB", "text": "|- ((((interval_arith x) (a , b)) /\\ ((interval_arith y) (c , d))) ==> ((interval_arith ((real_sub x) y)) (((real_sub a) d) , ((real_sub b) c))))"}
{"_id": "2569", "title": "Nonlin_def.vol3f_sqrt2_lmplus", "text": "|- (!y3. (!y4. (!y5. (!y1. (!y2. (!y6. (((((((vol3f_sqrt2_lmplus y1) y2) y3) y4) y5) y6) = ((real_sub ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((((((dih3_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_add ((((((dih4_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_sub ((((((dih5_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi))))))))) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi))) ((real_add ((real_mul (lfun ((real_div y2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih2_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_mul (lfun ((real_div y6) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih6_y y1) y2) sqrt2) sqrt2) sqrt2) y6))))))))))))"}
{"_id": "2570", "title": "SUM_EQ", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((sum s) f) = ((sum s) g))))))"}
{"_id": "2571", "title": "Hypermap_iso.dart_of_fan_eq", "text": "|- (!V. (!E. ((dart_of_fan (V , E)) = ((dart1_of_fan (V , E)) UNION (GSPEC (\\GEN%PVAR%7628. (?v. (((SETSPEC GEN%PVAR%7628) ((v IN V) /\\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))))))"}
{"_id": "2572", "title": "REAL_ABS_INV", "text": "|- (!x. ((real_abs (real_inv x)) = (real_inv (real_abs x))))"}
{"_id": "2573", "title": "RC_CASES", "text": "|- (!R. (!a0. (!a1. ((((RC R) a0) a1) = (((R a0) a1) \\/ (a1 = a0))))))"}
{"_id": "2574", "title": "AFF_DIM_TRANSLATION_EQ", "text": "|- (!a. (!s. ((aff_dim ((IMAGE (\\x. ((vector_add a) x))) s)) = (aff_dim s))))"}
{"_id": "2575", "title": "SUM_RMUL", "text": "|- (!f. (!c. (!s. (((sum s) (\\x. ((real_mul (f x)) c))) = ((real_mul ((sum s) f)) c)))))"}
{"_id": "2576", "title": "Pack2.MEASURABLE_VORONOI_CLOSED", "text": "|- (!s. (!v. ((saturated s) ==> (measurable ((voronoi_closed s) v)))))"}
{"_id": "2577", "title": "continuous_atreal", "text": "|- (((continuous f) (atreal x)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x'. (((real_lt (real_abs ((real_sub x') x))) d) ==> ((real_lt (distance ((f x') , (f x)))) e))))))))"}
{"_id": "2578", "title": "CLOSEST_POINT_IN_SET", "text": "|- (!s. (!a. (((closed s) /\\ (~ (s = EMPTY))) ==> (((closest_point s) a) IN s))))"}
{"_id": "2579", "title": "FINITE_POWERSET", "text": "|- (!s. ((FINITE s) ==> (FINITE (GSPEC (\\GEN%PVAR%109. (?t. (((SETSPEC GEN%PVAR%109) (t SUBSET s)) t)))))))"}
{"_id": "2580", "title": "Collect_geom2.COLLINEAR3_IMP_COPLANAR", "text": "|- (!v1. (!v2. (!v3. (!v. ((collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY)))) ==> (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v INSERT EMPTY))))))))))"}
{"_id": "2581", "title": "CONTINUOUS_TRANSFORM_WITHIN_SET_IMP", "text": "|- (!f. (!a. (!s. (!t. ((((eventually (\\x. ((x IN t) ==> (x IN s)))) (at a)) /\\ ((continuous f) ((within (at a)) s))) ==> ((continuous f) ((within (at a)) t)))))))"}
{"_id": "2582", "title": "REAL_LE_DIV", "text": "|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_div x) y)))))"}
{"_id": "2583", "title": "IM_CLOG_EQ_PI", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((Im (clog z)) = pi) = (((real_lt (Re z)) (real_of_num (NUMERAL _0))) /\\ ((Im z) = (real_of_num (NUMERAL _0)))))))"}
{"_id": "2584", "title": "Float.POW_MINUS1", "text": "|- (!n. (((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) * n)) = (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "2585", "title": "DIST_POS_LE", "text": "|- (!x. (!y. ((real_le (real_of_num (NUMERAL _0))) (distance (x , y)))))"}
{"_id": "2586", "title": "LEBESGUE_MEASURABLE_UNION", "text": "|- (!s. (!t. (((lebesgue_measurable s) /\\ (lebesgue_measurable t)) ==> (lebesgue_measurable (s UNION t)))))"}
{"_id": "2587", "title": "rpow", "text": "|- (!y. (!x. (((rpow x) y) = (((COND ((real_lt (real_of_num (NUMERAL _0))) x)) (exp ((real_mul y) (log x)))) (((COND (x = (real_of_num (NUMERAL _0)))) (((COND (y = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) (((COND (?m. (?n. ((ODD m) /\\ ((ODD n) /\\ ((real_abs y) = ((real_div (real_of_num m)) (real_of_num n)))))))) (real_neg (exp ((real_mul y) (log (real_neg x)))))) (exp ((real_mul y) (log (real_neg x))))))))))"}
{"_id": "2588", "title": "CONTINUOUS_ON_CNJ", "text": "|- (!s. ((continuous_on cnj) s))"}
{"_id": "2589", "title": "REAL_ADD_LINV", "text": "|- (!x. (((real_add (real_neg x)) x) = (real_of_num (NUMERAL _0))))"}
{"_id": "2590", "title": "int_lt", "text": "|- (!x. (!y. (((int_lt x) y) = ((real_lt (real_of_int x)) (real_of_int y)))))"}
{"_id": "2591", "title": "VSUM_CLAUSES", "text": "|- ((!f. (((vsum EMPTY) f) = (vec (NUMERAL _0)))) /\\ (!x. (!f. (!s. ((FINITE s) ==> (((vsum (x INSERT s)) f) = (((COND (x IN s)) ((vsum s) f)) ((vector_add (f x)) ((vsum s) f)))))))))"}
{"_id": "2592", "title": "Ssrnat.add1n", "text": "|- (!n. (((NUMERAL (BIT1 _0)) + n) = (SUC n)))"}
{"_id": "2593", "title": "Lp_ineqs_proofs.y5_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y5_fan (V , E)) d))))))))))"}
{"_id": "2594", "title": "INVERTIBLE_CMUL", "text": "|- (!A. (!c. ((invertible ((%% c) A)) = ((~ (c = (real_of_num (NUMERAL _0)))) /\\ (invertible A)))))"}
{"_id": "2595", "title": "REAL_SUB_0", "text": "|- (!x. (!y. ((((real_sub x) y) = (real_of_num (NUMERAL _0))) = (x = y))))"}
{"_id": "2596", "title": "REAL_EQ_ADD_LCANCEL", "text": "|- (!x. (!y. (!z. ((((real_add x) y) = ((real_add x) z)) = (y = z)))))"}
{"_id": "2597", "title": "REAL_POW_DIV", "text": "|- (!x. (!y. (!n. (((real_pow ((real_div x) y)) n) = ((real_div ((real_pow x) n)) ((real_pow y) n))))))"}
{"_id": "2598", "title": "INTEGRABLE_RESTRICT_UNIV", "text": "|- (!f. (!s. (((integrable_on (\\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) UNIV) = ((integrable_on f) s))))"}
{"_id": "2599", "title": "CSIN_EQ_1", "text": "|- (!z. (((csin z) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) = (?n. ((integer n) /\\ (z = (Cx ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) pi)))))))"}
{"_id": "2600", "title": "Ssrbool.andbb", "text": "|- (!b. ((b /\\ b) = b))"}
{"_id": "2601", "title": "DET_NEG", "text": "|- (!A. ((det (matrix_neg A)) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (dimindex UNIV))) (det A))))"}
{"_id": "2602", "title": "LINEAR_VSUM", "text": "|- (!f. (!g. (!s. (((linear f) /\\ (FINITE s)) ==> ((f ((vsum s) g)) = ((vsum s) (f o g)))))))"}
{"_id": "2603", "title": "BOUNDED_HAS_SUP", "text": "|- (!s. (((bounded ((IMAGE lift) s)) /\\ (~ (s = EMPTY))) ==> ((!x. ((x IN s) ==> ((real_le x) (sup s)))) /\\ (!b. ((!x. ((x IN s) ==> ((real_le x) b))) ==> ((real_le (sup s)) b))))))"}
{"_id": "2604", "title": "tagged_partial_division_of", "text": "|- (!i. (!s. (((tagged_partial_division_of s) i) = ((FINITE s) /\\ ((!x. (!k. (((x , k) IN s) ==> ((x IN k) /\\ ((k SUBSET i) /\\ (?a. (?b. (k = (closed_interval ((CONS (a , b)) NIL)))))))))) /\\ (!x1. (!k1. (!x2. (!k2. ((((x1 , k1) IN s) /\\ (((x2 , k2) IN s) /\\ (~ ((x1 , k1) = (x2 , k2))))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))))))))))"}
{"_id": "2605", "title": "Functional_equation.dih_all_atn", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((dih_x x1) x2) x3) x4) x5) x6) = ((((((dihatn_x x1) x2) x3) x4) x5) x6)) /\\ ((((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih2atn_x x1) x2) x3) x4) x5) x6)) /\\ ((((((((dih3_x x1) x2) x3) x4) x5) x6) = ((((((dih3atn_x x1) x2) x3) x4) x5) x6)) /\\ ((((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4atn_x x1) x2) x3) x4) x5) x6)) /\\ ((((((((dih5_x x1) x2) x3) x4) x5) x6) = ((((((dih5atn_x x1) x2) x3) x4) x5) x6)) /\\ (((((((dih6_x x1) x2) x3) x4) x5) x6) = ((((((dih6atn_x x1) x2) x3) x4) x5) x6))))))))))))))"}
{"_id": "2606", "title": "CX_INJ", "text": "|- (!x. (!y. (((Cx x) = (Cx y)) = (x = y))))"}
{"_id": "2607", "title": "AFFINE_IMP_CONVEX", "text": "|- (!s. ((affine s) ==> (convex s)))"}
{"_id": "2608", "title": "REAL_LT_RDIV_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_lt x) ((real_div y) z)) = ((real_lt ((real_mul x) z)) y))))))"}
{"_id": "2609", "title": "REAL_ABS_0", "text": "|- ((real_abs (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))"}
{"_id": "2610", "title": "Ssrnat.leqNgt", "text": "|- (!m. (!n. ((m <= n) = (~ (n < m)))))"}
{"_id": "2611", "title": "CLOSED_IN_CLOSED_INTER", "text": "|- (!u. (!s. ((closed s) ==> ((closed_in ((subtopology euclidean) u)) (u INTER s)))))"}
{"_id": "2612", "title": "INVERTIBLE_I", "text": "|- (invertible (mat (NUMERAL (BIT1 _0))))"}
{"_id": "2613", "title": "Constants_approx.pi_interval", "text": "|- ((interval_arith pi) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))))))))))))))))))))))"}
{"_id": "2614", "title": "real_polynomial_function_INDUCT", "text": "|- (!real_polynomial_function'. (((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (real_polynomial_function' (\\x. (x $ i))))) /\\ ((!c. (real_polynomial_function' (\\x. c))) /\\ ((!f. (!g. (((real_polynomial_function' f) /\\ (real_polynomial_function' g)) ==> (real_polynomial_function' (\\x. ((real_add (f x)) (g x))))))) /\\ (!f. (!g. (((real_polynomial_function' f) /\\ (real_polynomial_function' g)) ==> (real_polynomial_function' (\\x. ((real_mul (f x)) (g x)))))))))) ==> (!a. ((real_polynomial_function a) ==> (real_polynomial_function' a)))))"}
{"_id": "2615", "title": "has_complex_derivative", "text": "|- (!f. (!f'. (!net. ((((has_complex_derivative f) f') net) = (((has_derivative f) (\\x. ((complex_mul f') x))) net)))))"}
{"_id": "2616", "title": "REAL_MEASURE_UNIONS_LE_IMAGE", "text": "|- (!f. (!s. (((FINITE f) /\\ (!a. ((a IN f) ==> (real_measurable (s a))))) ==> ((real_le (real_measure (UNIONS ((IMAGE s) f)))) ((sum f) (\\a. (real_measure (s a))))))))"}
{"_id": "2617", "title": "NSUM_UNION_RZERO", "text": "|- (!f. (!u. (!v. (((FINITE u) /\\ (!x. (((x IN v) /\\ (~ (x IN u))) ==> ((f x) = (NUMERAL _0))))) ==> (((nsum (u UNION v)) f) = ((nsum u) f))))))"}
{"_id": "2618", "title": "HAS_REAL_INTEGRAL_NEG", "text": "|- (!f. (!k. (!s. ((((has_real_integral f) k) s) ==> (((has_real_integral (\\x. (real_neg (f x)))) (real_neg k)) s)))))"}
{"_id": "2619", "title": "SQRT_POS_LT", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) (sqrt x))))"}
{"_id": "2620", "title": "COUNTABLE_SING", "text": "|- (!x. (COUNTABLE (x INSERT EMPTY)))"}
{"_id": "2621", "title": "Fan_defs.n_fan_pair", "text": "|- (!V. (!E. (!v. (!w. (((n_fan_pair (V , E)) (v , w)) = (v , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))"}
{"_id": "2622", "title": "ROTATION_LOWDIM_HORIZONTAL", "text": "|- (!s. (((dim s) < (dimindex UNIV)) ==> (?f. ((orthogonal_transformation f) /\\ (((det (matrix f)) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((IMAGE f) s) SUBSET (GSPEC (\\GEN%PVAR%456. (?z. (((SETSPEC GEN%PVAR%456) ((z $ (dimindex UNIV)) = (real_of_num (NUMERAL _0)))) z))))))))))"}
{"_id": "2623", "title": "matrix_sub", "text": "|- (!A. (!B. (((matrix_sub A) B) = (lambdai. (lambdaj. ((real_sub ((A $ i) $ j)) ((B $ i) $ j)))))))"}
{"_id": "2624", "title": "ITERATE_PAIR", "text": "|- (!op. ((monoidal op) ==> (!f. (!m. (!n. ((((iterate op) (((NUMERAL (BIT0 (BIT1 _0))) * m) .. (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))) f) = (((iterate op) (m .. n)) (\\i. ((op (f ((NUMERAL (BIT0 (BIT1 _0))) * i))) (f (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))))))"}
{"_id": "2625", "title": "DISJOINT_EMPTY", "text": "|- (!s. (((DISJOINT EMPTY) s) /\\ ((DISJOINT s) EMPTY)))"}
{"_id": "2626", "title": "NEUTRAL_MUL", "text": "|- ((neutral *) = (NUMERAL (BIT1 _0)))"}
{"_id": "2627", "title": "CONTINUOUS_LE_ON_CLOSURE", "text": "|- (!f. (!s. (!a. ((((continuous_on (lift o f)) (closure s)) /\\ (!x. ((x IN s) ==> ((real_le (f x)) a)))) ==> (!x. ((x IN (closure s)) ==> ((real_le (f x)) a)))))))"}
{"_id": "2628", "title": "HAS_COMPLEX_DERIVATIVE_CONST", "text": "|- (!c. (!net. (((has_complex_derivative (\\x. c)) (Cx (real_of_num (NUMERAL _0)))) net)))"}
{"_id": "2629", "title": "Conforming.NEGLIGIBLE_AFF_3", "text": "|- (!x. (!v. (!u. (negligible (aff (x INSERT (v INSERT (u INSERT EMPTY))))))))"}
{"_id": "2630", "title": "le_c", "text": "|- (!t. (!s. ((s <=_c t) = (?f. ((!x. ((x IN s) ==> ((f x) IN t))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))))))))"}
{"_id": "2631", "title": "Seq.has_predC", "text": "|- (!a. (!s. (((has (predC a)) s) = (~ ((all a) s)))))"}
{"_id": "2632", "title": "COVERING_SPACE_FINITE_SHEETS", "text": "|- (!p. (!c. (!s. (!b. ((((covering_space (c , p)) s) /\\ (compact c)) ==> (FINITE (GSPEC (\\GEN%PVAR%2084. (?x. (((SETSPEC GEN%PVAR%2084) ((x IN c) /\\ ((p x) = b))) x))))))))))"}
{"_id": "2633", "title": "SHIFTPATH_LINEAR_IMAGE", "text": "|- (!f. (!t. (!g. ((linear f) ==> (((shiftpath t) (f o g)) = (f o ((shiftpath t) g)))))))"}
{"_id": "2634", "title": "FRONTIER_UNION_SUBSET", "text": "|- (!s. (!t. ((frontier (s UNION t)) SUBSET ((frontier s) UNION (frontier t)))))"}
{"_id": "2635", "title": "real_INFINITE", "text": "|- (INFINITE UNIV)"}
{"_id": "2636", "title": "BOUNDED_SUMS_IMAGE", "text": "|- (!f. (!g. (!t. (((bounded (GSPEC (\\GEN%PVAR%588. (?x. (((SETSPEC GEN%PVAR%588) (x IN t)) (f x)))))) /\\ (bounded (GSPEC (\\GEN%PVAR%589. (?x. (((SETSPEC GEN%PVAR%589) (x IN t)) (g x))))))) ==> (bounded (GSPEC (\\GEN%PVAR%590. (?x. (((SETSPEC GEN%PVAR%590) (x IN t)) ((vector_add (f x)) (g x)))))))))))"}
{"_id": "2637", "title": "CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP", "text": "|- (!f. (!g. (!s. (!t. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((continuous_on g) t) /\\ ((((IMAGE g) t) SUBSET s) /\\ (!y. ((y IN t) ==> ((f (g y)) = y))))))) ==> (!u. ((u SUBSET t) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%651. (?x. (((SETSPEC GEN%PVAR%651) ((x IN s) /\\ ((f x) IN u))) x))))) = ((open_in ((subtopology euclidean) t)) u)))))))))"}
{"_id": "2638", "title": "Dih2k_hypermap.EXISTS_POINT_DART_OF_HYP", "text": "|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\\ (((NUMERAL (BIT0 (BIT1 _0))) < (dimindex UNIV)) /\\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ ((j <= (dimindex UNIV)) /\\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (?x. (x IN ((darts_of_hyp (E_SY (vecmats l))) (V_SY (vecmats l))))))"}
{"_id": "2639", "title": "FSIGMA_TRANSLATION", "text": "|- (!a. (!s. ((fsigma ((IMAGE (\\x. ((vector_add a) x))) s)) = (fsigma s))))"}
{"_id": "2640", "title": "COBOUNDED_UNIQUE_UNBOUNDED_COMPONENTS", "text": "|- (!s. (!c. (!c'. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ ((bounded (UNIV DIFF s)) /\\ ((c IN (components s)) /\\ ((~ (bounded c)) /\\ ((c' IN (components s)) /\\ (~ (bounded c'))))))) ==> (c' = c)))))"}
{"_id": "2641", "title": "Ssrbool.orbF", "text": "|- (!b. ((b \\/ F) = b))"}
{"_id": "2642", "title": "real_variation", "text": "|- (!s. (!f. (((real_variation s) f) = ((vector_variation ((IMAGE lift) s)) (lift o (f o drop))))))"}
{"_id": "2643", "title": "Fan.d1_fan", "text": "|- (!x. (!V. (!E. ((d1_fan (x , (V , E))) = (GSPEC (\\GEN%PVAR%5465. (?x'. (?v. (?w. (?w1. (((SETSPEC GEN%PVAR%5465) ((x' = x) /\\ (((v INSERT (w INSERT EMPTY)) IN E) /\\ (w1 = (((((sigma_fan x) V) E) v) w))))) (x' , (v , (w , w1))))))))))))))"}
{"_id": "2644", "title": "REAL_ABS_NZ", "text": "|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) = ((real_lt (real_of_num (NUMERAL _0))) (real_abs x))))"}
{"_id": "2645", "title": "UNIONS_GSPEC", "text": "|- ((!P. (!f. ((UNIONS (GSPEC (\\GEN%PVAR%31. (?x. (((SETSPEC GEN%PVAR%31) (P x)) (f x)))))) = (GSPEC (\\GEN%PVAR%32. (?a. (((SETSPEC GEN%PVAR%32) (?x. ((P x) /\\ (a IN (f x))))) a))))))) /\\ ((!P. (!f. ((UNIONS (GSPEC (\\GEN%PVAR%33. (?x. (?y. (((SETSPEC GEN%PVAR%33) ((P x) y)) ((f x) y))))))) = (GSPEC (\\GEN%PVAR%34. (?a. (((SETSPEC GEN%PVAR%34) (?x. (?y. (((P x) y) /\\ (a IN ((f x) y)))))) a))))))) /\\ (!P. (!f. ((UNIONS (GSPEC (\\GEN%PVAR%35. (?x. (?y. (?z. (((SETSPEC GEN%PVAR%35) (((P x) y) z)) (((f x) y) z)))))))) = (GSPEC (\\GEN%PVAR%36. (?a. (((SETSPEC GEN%PVAR%36) (?x. (?y. (?z. ((((P x) y) z) /\\ (a IN (((f x) y) z))))))) a)))))))))"}
{"_id": "2646", "title": "COMPLEX_ADD_ASSOC", "text": "|- (!x. (!y. (!z. (((vector_add x) ((vector_add y) z)) = ((vector_add ((vector_add x) y)) z)))))"}
{"_id": "2647", "title": "SNDCART_PASTECART", "text": "|- (!x. (!y. ((sndcart ((pastecart x) y)) = y)))"}
{"_id": "2648", "title": "INTEGRABLE_INTEGRAL", "text": "|- (!f. (!i. (((integrable_on f) i) ==> (((has_integral f) ((integral i) f)) i))))"}
{"_id": "2649", "title": "RETRACTION", "text": "|- (!s. (!t. (!r. (((retraction (s , t)) r) = ((t SUBSET s) /\\ (((continuous_on r) s) /\\ ((((IMAGE r) s) = t) /\\ (!x. ((x IN t) ==> ((r x) = x))))))))))"}
{"_id": "2650", "title": "HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT", "text": "|- (!p. (!f. (!g. (!h. (!X. (!Y. (!Z. ((((((homotopic_with (\\f. (p (h o f)))) (X , Y)) f) g) /\\ (((continuous_on h) Y) /\\ (((IMAGE h) Y) SUBSET Z))) ==> ((((homotopic_with p) (X , Z)) (h o f)) (h o g))))))))))"}
{"_id": "2651", "title": "REAL_OF_NUM_SUB", "text": "|- (!m. (!n. ((m <= n) ==> (((real_sub (real_of_num n)) (real_of_num m)) = (real_of_num (n - m))))))"}
{"_id": "2652", "title": "UNION_SUBSET", "text": "|- (!s. (!t. (!u. (((s UNION t) SUBSET u) = ((s SUBSET u) /\\ (t SUBSET u))))))"}
{"_id": "2653", "title": "TRANSP_MATRIX_INV", "text": "|- (!A. ((transp (matrix_inv A)) = (matrix_inv (transp A))))"}
{"_id": "2654", "title": "REAL_EQ_SUB_RADD", "text": "|- (!x. (!y. (!z. ((((real_sub x) y) = z) = (x = ((real_add z) y))))))"}
{"_id": "2655", "title": "BOUNDED_INTERVAL", "text": "|- ((!a. (!b. (bounded (closed_interval ((CONS (a , b)) NIL))))) /\\ (!a. (!b. (bounded (open_interval (a , b))))))"}
{"_id": "2656", "title": "COVERING_SPACE_LIFT_STRONG", "text": "|- (!p. (!c. (!s. (!f. (!u. (!a. (!z. ((((covering_space (c , p)) s) /\\ ((a IN c) /\\ ((z IN u) /\\ ((simply_connected u) /\\ (((locally path_connected) u) /\\ (((continuous_on f) u) /\\ ((((IMAGE f) u) SUBSET s) /\\ ((f z) = (p a))))))))) ==> (?g. (((continuous_on g) u) /\\ ((((IMAGE g) u) SUBSET c) /\\ (((g z) = a) /\\ (!y. ((y IN u) ==> ((p (g y)) = (f y))))))))))))))))"}
{"_id": "2657", "title": "MEASURABLE_MEASURE_POS_LT", "text": "|- (!s. ((measurable s) ==> (((real_lt (real_of_num (NUMERAL _0))) (measure s)) = (~ (negligible s)))))"}
{"_id": "2658", "title": "FSIGMA_CONTINUOUS_IMAGE", "text": "|- (!f. (!s. ((((continuous_on f) s) /\\ (fsigma s)) ==> (fsigma ((IMAGE f) s)))))"}
{"_id": "2659", "title": "MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE", "text": "|- (!s. (!t. (((measurable s) /\\ (lebesgue_measurable t)) ==> (measurable (s INTER t)))))"}
{"_id": "2660", "title": "Nonlin_def.div6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((div6 f) g) x1) x2) x3) x4) x5) x6) = ((real_div ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "2661", "title": "STARLIKE_CONVEX_TWEAK_BOUNDARY_POINTS", "text": "|- (!s. (!t. (((convex s) /\\ ((~ (s = EMPTY)) /\\ (((relative_interior s) SUBSET t) /\\ (t SUBSET (closure s))))) ==> (starlike t))))"}
{"_id": "2662", "title": "Hypermap.lemma_face_subset", "text": "|- (!H. (!x. ((x IN (dart H)) ==> (((face H) x) SUBSET (dart H)))))"}
{"_id": "2663", "title": "SURJECTIVE_IMAGE_EQ", "text": "|- (!s. (!t. (((!y. ((y IN t) ==> (?x. ((f x) = y)))) /\\ (!x. (((f x) IN t) = (x IN s)))) ==> (((IMAGE f) s) = t))))"}
{"_id": "2664", "title": "ARG_ROTATE2D_UNIQUE", "text": "|- (!t. (!a. (!z. (((~ (z = (Cx (real_of_num (NUMERAL _0))))) /\\ ((Arg ((rotate2d t) z)) = a)) ==> (?n. ((integer n) /\\ (t = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul n) pi))) ((real_sub a) (Arg z))))))))))"}
{"_id": "2665", "title": "HYPERPLANE_EQ_UNIV", "text": "|- (!a. (!b. (((GSPEC (\\GEN%PVAR%172. (?x. (((SETSPEC GEN%PVAR%172) (((dot a) x) = b)) x)))) = UNIV) = ((a = (vec (NUMERAL _0))) /\\ (b = (real_of_num (NUMERAL _0)))))))"}
{"_id": "2666", "title": "BOUNDED_HAS_INF", "text": "|- (!s. (((bounded ((IMAGE lift) s)) /\\ (~ (s = EMPTY))) ==> ((!x. ((x IN s) ==> ((real_le (inf s)) x))) /\\ (!b. ((!x. ((x IN s) ==> ((real_le b) x))) ==> ((real_le b) (inf s)))))))"}
{"_id": "2667", "title": "Nonlinear_lemma.lmfun0", "text": "|- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) ==> ((lmfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0)))))"}
{"_id": "2668", "title": "SUBSET_REFL", "text": "|- (!s. (s SUBSET s))"}
{"_id": "2669", "title": "Collect_geom.pow_g", "text": "|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))))"}
{"_id": "2670", "title": "COMPLEX_NORM_DIV", "text": "|- (!w. (!z. ((vector_norm ((complex_div w) z)) = ((real_div (vector_norm w)) (vector_norm z)))))"}
{"_id": "2671", "title": "SUM_DIFFS", "text": "|- (!m. (!n. (((sum (m .. n)) (\\k. ((real_sub (f k)) (f (k + (NUMERAL (BIT1 _0))))))) = (((COND (m <= n)) ((real_sub (f m)) (f (n + (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL _0))))))"}
{"_id": "2672", "title": "CARD_EQ_SIMPLE_PATH_IMAGE", "text": "|- (!g. ((simple_path g) ==> ((path_image g) =_c UNIV)))"}
{"_id": "2673", "title": "INTEGRABLE_SUBINTERVALS_IMP_MEASURABLE", "text": "|- (!f. ((!a. (!b. ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> ((measurable_on f) UNIV)))"}
{"_id": "2674", "title": "CLOSED_IN_SUBTOPOLOGY_EMPTY", "text": "|- (!top. (!s. (((closed_in ((subtopology top) EMPTY)) s) = (s = EMPTY))))"}
{"_id": "2675", "title": "ITERATE_INJECTION", "text": "|- (!op. ((monoidal op) ==> (!f. (!p. (!s. (((FINITE s) /\\ ((!x. ((x IN s) ==> ((p x) IN s))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((p x) = (p y)))) ==> (x = y)))))) ==> ((((iterate op) s) (f o p)) = (((iterate op) s) f))))))))"}
{"_id": "2676", "title": "PERMUTES_INSERT_LEMMA", "text": "|- (!p. (!a. (!s. (((permutes p) (a INSERT s)) ==> ((permutes ((swap (a , (p a))) o p)) s)))))"}
{"_id": "2677", "title": "REAL_LTE_TOTAL", "text": "|- (!x. (!y. (((real_lt x) y) \\/ ((real_le y) x))))"}
{"_id": "2678", "title": "ORTHOGONAL_MATRIX_IMP_INVERTIBLE", "text": "|- (!A. ((orthogonal_matrix A) ==> (invertible A)))"}
{"_id": "2679", "title": "Seq.all_filterP", "text": "|- (!a. (!s. ((((filter a) s) = s) = ((all a) s))))"}
{"_id": "2680", "title": "INTEGRAL_0", "text": "|- (!s. (((integral s) (\\x. (vec (NUMERAL _0)))) = (vec (NUMERAL _0))))"}
{"_id": "2681", "title": "CONDENSATION_POINT_IMP_LIMPT", "text": "|- (!x. (!s. (((condensation_point_of x) s) ==> ((limit_point_of x) s))))"}
{"_id": "2682", "title": "Real_ext.REAL_PROP_POS_ADD2", "text": "|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))"}
{"_id": "2683", "title": "VECTOR_VARIATION_AFFINITY2", "text": "|- (!m. (!c. (!f. (!s. (((vector_variation ((IMAGE (\\x. ((vector_add ((% (real_inv m)) x)) (vector_neg ((% (real_inv m)) c))))) s)) (\\x. (f ((vector_add ((% m) x)) c)))) = (((COND (m = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) ((vector_variation s) f)))))))"}
{"_id": "2684", "title": "Rogers.KHEJKCI_GEN", "text": "|- (!V. (!k. (!r. (!ul. (!vl. (((saturated V) /\\ ((packing V) /\\ ((((barV V) k) ul) /\\ ((((barV V) r) vl) /\\ ((initial_sublist ul) vl))))) ==> ((face_of ((voronoi_list V) vl)) ((voronoi_list V) ul))))))))"}
{"_id": "2685", "title": "REAL_LT_ADDNEG", "text": "|- (!x. (!y. (!z. (((real_lt y) ((real_add x) (real_neg z))) = ((real_lt ((real_add y) z)) x)))))"}
{"_id": "2686", "title": "LIM_ATREAL", "text": "|- (!f. (!l. (!a. ((((--> f) l) (atreal a)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\\ ((real_lt (real_abs ((real_sub x) a))) d)) ==> ((real_lt (distance ((f x) , l))) e)))))))))))"}
{"_id": "2687", "title": "FINITE_CARD_LT", "text": "|- (!s. ((FINITE s) = (s <_c UNIV)))"}
{"_id": "2688", "title": "Ssrbool.negb_and", "text": "|- (!a. (!b. ((~ (a /\\ b)) = ((~ a) \\/ (~ b)))))"}
{"_id": "2689", "title": "Fan_defs.e_fan_pair_ext", "text": "|- (!V. (!E. (!x. (((e_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((e_fan_pair (V , E)) x)) x)))))"}
{"_id": "2690", "title": "FUN_IN_IMAGE", "text": "|- (!f. (!s. (!x. ((x IN s) ==> ((f x) IN ((IMAGE f) s))))))"}
{"_id": "2691", "title": "Trigonometry2.ARG_DIFF_SUCCESSIBLE_IN_FIRST_PERIOD", "text": "|- (!W. (!xicm. (((FINITE W) /\\ (((CARD W) = n) /\\ ((!x. ((W x) ==> (~ (x = (vec (NUMERAL _0)))))) /\\ ((polar_cycle_on xicm) W)))) ==> (!p. (!i. (!j. (((W p) /\\ (((NUMERAL _0) <= i) /\\ ((i <= j) /\\ (j < n)))) ==> (((real_add ((arg_diff p) (((ITER i) xicm) p))) ((arg_diff (((ITER i) xicm) p)) (((ITER j) xicm) p))) = ((arg_diff p) (((ITER j) xicm) p))))))))))"}
{"_id": "2692", "title": "Lp_ineqs_proofs.y3_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y3_fan (V , E)) d))))))))))"}
{"_id": "2693", "title": "REAL_LNEG_UNIQ", "text": "|- (!x. (!y. ((((real_add x) y) = (real_of_num (NUMERAL _0))) = (x = (real_neg y)))))"}
{"_id": "2694", "title": "LIM_CONG_AT", "text": "|- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\\x. (f x))) l) (at a)) = (((--> g) l) (at a))))"}
{"_id": "2695", "title": "POWER_SERIES_CONV_IMP_ABSCONV", "text": "|- (!a. (!k. (!w. (!z. ((((summable k) (\\n. ((complex_mul (a n)) ((complex_pow z) n)))) /\\ ((real_lt (vector_norm w)) (vector_norm z))) ==> ((summable k) (\\n. (Cx (vector_norm ((complex_mul (a n)) ((complex_pow w) n)))))))))))"}
{"_id": "2696", "title": "INVERTIBLE_TRANSP", "text": "|- (!A. ((invertible (transp A)) = (invertible A)))"}
{"_id": "2697", "title": "Lp_ineqs_proofs.f_fan_pair_ext_in_dart", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((f_fan_pair_ext (V , E)) d) IN (dart_of_fan (V , E))))))))"}
{"_id": "2698", "title": "Ssrnat.ltn0", "text": "|- (!n. ((n < (NUMERAL _0)) = F))"}
{"_id": "2699", "title": "CLOSED_IN_CLOSED_EQ", "text": "|- (!s. (!t. ((closed s) ==> (((closed_in ((subtopology euclidean) s)) t) = ((closed t) /\\ (t SUBSET s))))))"}
{"_id": "2700", "title": "AFF_DIM_EQ_AFFINE_HULL", "text": "|- (!s. (!t. (((s SUBSET t) /\\ ((int_le (aff_dim t)) (aff_dim s))) ==> (((hull affine) s) = ((hull affine) t)))))"}
{"_id": "2701", "title": "IMAGE_SUBSET", "text": "|- (!f. (!s. (!t. ((s SUBSET t) ==> (((IMAGE f) s) SUBSET ((IMAGE f) t))))))"}
{"_id": "2702", "title": "AFFSIGN_EQ_AFFINE_HULL", "text": "|- (!sgn. (!s. (!t. ((((affsign sgn) s) EMPTY) = ((hull affine) s)))))"}
{"_id": "2703", "title": "PATH_IMAGE_TRANSLATION", "text": "|- (!a. (!g. ((path_image ((\\x. ((vector_add a) x)) o g)) = ((IMAGE (\\x. ((vector_add a) x))) (path_image g)))))"}
{"_id": "2704", "title": "REAL_LE_RSQRT", "text": "|- (!x. (!y. (((real_le ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) y) ==> ((real_le x) (sqrt y)))))"}
{"_id": "2705", "title": "REAL_ABS_MUL", "text": "|- (!x. (!y. ((real_abs ((real_mul x) y)) = ((real_mul (real_abs x)) (real_abs y)))))"}
{"_id": "2706", "title": "SEGMENT_SYM", "text": "|- ((!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = (closed_segment ((CONS (b , a)) NIL))))) /\\ (!a. (!b. ((open_segment (a , b)) = (open_segment (b , a))))))"}
{"_id": "2707", "title": "REAL_EUCLIDEAN_METRIC", "text": "|- (((mspace real_euclidean_metric) = UNIV) /\\ (!x. (!y. (((mdist real_euclidean_metric) (x , y)) = (real_abs ((real_sub y) x))))))"}
{"_id": "2708", "title": "Appendix.BBprime_v39", "text": "|- (!s. (!vv. (((BBprime_v39 s) vv) = (((BBs_v39 s) vv) /\\ ((!ww. (((BBs_v39 s) ww) ==> ((real_le ((taustar_v39 s) vv)) ((taustar_v39 s) ww)))) /\\ ((real_lt ((taustar_v39 s) vv)) (real_of_num (NUMERAL _0))))))))"}
{"_id": "2709", "title": "MATRIX_VECTOR_MUL_RMUL", "text": "|- (!A. (!x. (!c. (((matrix_vector_mul A) ((% c) x)) = ((% c) ((matrix_vector_mul A) x))))))"}
{"_id": "2710", "title": "Nonlinear_lemma.gcy_high", "text": "|- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) ==> ((gcy y) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_div mm1) pi)))))"}
{"_id": "2711", "title": "REAL_SUB_SUB2", "text": "|- (!x. (!y. (((real_sub x) ((real_sub x) y)) = y)))"}
{"_id": "2712", "title": "COMPACT_IMP_TOTALLY_BOUNDED", "text": "|- (!s. ((compact s) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?k. ((FINITE k) /\\ ((k SUBSET s) /\\ (s SUBSET (UNIONS ((IMAGE (\\x. (ball (x , e)))) k))))))))))"}
{"_id": "2713", "title": "Constants_approx.interval_arith", "text": "|- (!lo. (!x. (!hi. (((interval_arith x) (lo , hi)) = (((real_le lo) x) /\\ ((real_le x) hi))))))"}
{"_id": "2714", "title": "SETDIST_CLOSURE", "text": "|- ((!s. (!t. ((setdist ((closure s) , t)) = (setdist (s , t))))) /\\ (!s. (!t. ((setdist (s , (closure t))) = (setdist (s , t))))))"}
{"_id": "2715", "title": "CONVEX_HULL_SING", "text": "|- (!a. (((hull convex) (a INSERT EMPTY)) = (a INSERT EMPTY)))"}
{"_id": "2716", "title": "Hypermap_iso.hyp_iso_inv", "text": "|- (!f. (!G. (!H. (((hyp_iso f) (H , G)) ==> ((hyp_iso ((res_inv f) (dart H))) (G , H))))))"}
{"_id": "2717", "title": "OPEN_CONTAINS_CBALL", "text": "|- (!s. ((open s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ ((cball (x , e)) SUBSET s)))))))"}
{"_id": "2718", "title": "COMPLEX_MUL_LNEG", "text": "|- (!x. (!y. (((complex_mul (vector_neg x)) y) = (vector_neg ((complex_mul x) y)))))"}
{"_id": "2719", "title": "EVENTUALLY_AT_POSINFINITY", "text": "|- (!p. (((eventually p) at_posinfinity) = (?b. (!x. (((real_ge x) b) ==> (p x))))))"}
{"_id": "2720", "title": "SERIES_NEG", "text": "|- (!x. (!x0. (!s. ((((sums x) x0) s) ==> (((sums (\\n. (vector_neg (x n)))) (vector_neg x0)) s)))))"}
{"_id": "2721", "title": "DECIMAL", "text": "|- (!x. (!y. (((DECIMAL x) y) = ((real_div (real_of_num x)) (real_of_num y)))))"}
{"_id": "2722", "title": "DIAGONAL_MATRIX_MUL_COMPONENT", "text": "|- (!A. (!B. (!i. (!j. (((diagonal_matrix A) /\\ ((diagonal_matrix B) /\\ (((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((NUMERAL (BIT1 _0)) <= j) /\\ (j <= (dimindex UNIV))))))) ==> (((((matrix_mul A) B) $ i) $ j) = ((real_mul ((A $ i) $ j)) ((B $ i) $ j))))))))"}
{"_id": "2723", "title": "LIM_ATREAL_AT", "text": "|- ((((--> f) a) (atreal x)) = (((--> (f o drop)) a) (at (lift x))))"}
{"_id": "2724", "title": "ABSOLUTELY_REAL_INTEGRABLE_ON", "text": "|- (((absolutely_real_integrable_on f) s) = ((absolutely_integrable_on (lift o (f o drop))) ((IMAGE lift) s)))"}
{"_id": "2725", "title": "HAS_INTEGRAL_SPIKE_FINITE", "text": "|- (!f. (!g. (!s. (!t. (!y. (((FINITE s) /\\ ((!x. ((x IN (t DIFF s)) ==> ((g x) = (f x)))) /\\ (((has_integral f) y) t))) ==> (((has_integral g) y) t)))))))"}
{"_id": "2726", "title": "COLLINEAR_AFFINE_HULL_COLLINEAR", "text": "|- (!s. ((collinear ((hull affine) s)) = (collinear s)))"}
{"_id": "2727", "title": "CBALL_SCALING", "text": "|- (!c. (((real_lt (real_of_num (NUMERAL _0))) c) ==> (!x. (!r. ((cball (((% c) x) , ((real_mul c) r))) = ((IMAGE (\\x. ((% c) x))) (cball (x , r))))))))"}
{"_id": "2728", "title": "EXTREME_POINT_OF_LINEAR_IMAGE", "text": "|- (!f. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((extreme_point_of (f x)) ((IMAGE f) s)) = ((extreme_point_of x) s))))"}
{"_id": "2729", "title": "OPEN_IN_TOPSPACE", "text": "|- (!top. ((open_in top) (topspace top)))"}
{"_id": "2730", "title": "SUBSET_RESTRICT", "text": "|- (!s. (!P. ((GSPEC (\\GEN%PVAR%8. (?x. (((SETSPEC GEN%PVAR%8) ((x IN s) /\\ (P x))) x)))) SUBSET s)))"}
{"_id": "2731", "title": "TAYLOR_CSIN", "text": "|- (!n. (!z. ((real_le (vector_norm ((vector_sub (csin z)) ((vsum ((NUMERAL _0) .. n)) (\\k. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) k)) ((complex_div ((complex_pow z) (((NUMERAL (BIT0 (BIT1 _0))) * k) + (NUMERAL (BIT1 _0))))) (Cx (real_of_num (FACT (((NUMERAL (BIT0 (BIT1 _0))) * k) + (NUMERAL (BIT1 _0))))))))))))) ((real_mul (exp (real_abs (Im z)))) ((real_div ((real_pow (vector_norm z)) (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 (BIT1 _0)))))) (real_of_num (FACT (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT0 (BIT1 _0)))))))))))"}
{"_id": "2732", "title": "VECTOR_ANGLE_SYM", "text": "|- (!x. (!y. (((vector_angle x) y) = ((vector_angle y) x))))"}
{"_id": "2733", "title": "CX_LOG", "text": "|- (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> ((Cx (log z)) = (clog (Cx z)))))"}
{"_id": "2734", "title": "Fan_defs.f_fan_pair", "text": "|- (!V. (!E. (!w. (!v. (((f_fan_pair (V , E)) (v , w)) = (w , (((((inverse_sigma_fan (vec (NUMERAL _0))) V) E) w) v)))))))"}
{"_id": "2735", "title": "EXISTS_IN_IMAGE", "text": "|- (!f. (!s. ((?y. ((y IN ((IMAGE f) s)) /\\ (P y))) = (?x. ((x IN s) /\\ (P (f x)))))))"}
{"_id": "2736", "title": "REAL_DIV_EQ_0", "text": "|- (!x. (!y. ((((real_div x) y) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) \\/ (y = (real_of_num (NUMERAL _0)))))))"}
{"_id": "2737", "title": "Fnjlbxs.vector_of_list_comp0", "text": "|- (!d. (!list. (((dimindex UNIV) = d) ==> (((d * (LENGTH list)) <= (dimindex UNIV)) ==> (!i. (((((d * (LENGTH list)) + (NUMERAL (BIT1 _0))) <= i) /\\ (i <= (dimindex UNIV))) ==> ((((vector_of_list d) list) $ i) = (real_of_num (NUMERAL _0)))))))))"}
{"_id": "2738", "title": "Trigonometry.OWECYNV1", "text": "|- (!s. ((convex s) = (!x. (!y. (!u. (!v. (((x IN s) /\\ ((y IN s) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> (((vector_add ((% u) x)) ((% v) y)) IN s))))))))"}
{"_id": "2739", "title": "LOCALLY_CONNECTED_UNIV", "text": "|- ((locally connected) UNIV)"}
{"_id": "2740", "title": "ROTATE2D_ADD", "text": "|- (!s. (!t. (!x. (((rotate2d ((real_add s) t)) x) = ((rotate2d s) ((rotate2d t) x))))))"}
{"_id": "2741", "title": "pair_INDUCT", "text": "|- (!P. ((!x. (!y. (P (x , y)))) ==> (!p. (P p))))"}
{"_id": "2742", "title": "Tame_list.rev1", "text": "|- (!h. ((rev ((CONS h) NIL)) = ((CONS h) NIL)))"}
{"_id": "2743", "title": "IM_CCOS", "text": "|- (!z. ((Im (ccos z)) = ((real_mul ((real_div ((real_sub (exp (real_neg (Im z)))) (exp (Im z)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (sin (Re z)))))"}
{"_id": "2744", "title": "NEGLIGIBLE_IFF_LEBESGUE_MEASURABLE_SUBSETS", "text": "|- (!s. ((negligible s) = (!t. ((t SUBSET s) ==> (lebesgue_measurable t)))))"}
{"_id": "2745", "title": "CNJ_CLOG", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((cnj (clog z)) = (clog (cnj z)))))"}
{"_id": "2746", "title": "CONTINUOUS_WITHIN_CX_NORM", "text": "|- (!z. (!s. ((continuous (\\z. (Cx (vector_norm z)))) ((within (at z)) s))))"}
{"_id": "2747", "title": "BALL_INTERVAL", "text": "|- (!x. (!e. ((ball (x , e)) = (open_interval (((vector_sub x) (lift e)) , ((vector_add x) (lift e)))))))"}
{"_id": "2748", "title": "REAL_LE_INF", "text": "|- (!b. (((~ (s = EMPTY)) /\\ (!x. ((x IN s) ==> ((real_le b) x)))) ==> ((real_le b) (inf s))))"}
{"_id": "2749", "title": "Hdplygy.J1_SY", "text": "|- (!s. ((J1_SY s) = (GSPEC (\\GEN%PVAR%7598. (?x. (((SETSPEC GEN%PVAR%7598) (?i. ((((i MOD (k_sy s)) INSERT (((f_sy s) (i MOD (k_sy s))) INSERT EMPTY)) IN (J_SY s)) /\\ ((i IN ((NUMERAL (BIT1 _0)) .. (k_sy s))) /\\ (x = (i , (SUC (i MOD (k_sy s))))))))) x))))))"}
{"_id": "2750", "title": "ABSOLUTE_RETRACT_IMP_AR", "text": "|- (!s. (!s'. ((((retract_of s) UNIV) /\\ (((homeomorphic s) s') /\\ (closed s'))) ==> ((retract_of s') UNIV))))"}
{"_id": "2751", "title": "SUM_2", "text": "|- (!t. (((sum ((NUMERAL (BIT1 _0)) .. (NUMERAL (BIT0 (BIT1 _0))))) t) = ((real_add (t (NUMERAL (BIT1 _0)))) (t (NUMERAL (BIT0 (BIT1 _0)))))))"}
{"_id": "2752", "title": "REAL_MAX_LT", "text": "|- (!x. (!y. (!z. (((real_lt ((real_max x) y)) z) = (((real_lt x) z) /\\ ((real_lt y) z))))))"}
{"_id": "2753", "title": "Wrgcvdr_cizmrrh.ee_of_hyp2", "text": "|- (((ee_of_hyp (x , (V , E))) u) = (((COND (u IN ((darts_of_hyp E) V))) ((SND u) , (FST u))) u))"}
{"_id": "2754", "title": "MULT_2", "text": "|- (!n. (((NUMERAL (BIT0 (BIT1 _0))) * n) = (n + n)))"}
{"_id": "2755", "title": "Ssrnat.expn0", "text": "|- (!m. ((m EXP (NUMERAL _0)) = (NUMERAL (BIT1 _0))))"}
{"_id": "2756", "title": "IM_DIV_CX", "text": "|- (!z. (!x. ((Im ((complex_div z) (Cx x))) = ((real_div (Im z)) x))))"}
{"_id": "2757", "title": "REAL_MAX_LE", "text": "|- (!x. (!y. (!z. (((real_le ((real_max x) y)) z) = (((real_le x) z) /\\ ((real_le y) z))))))"}
{"_id": "2758", "title": "MATRIX_OF_MATRIX_VECTOR_MUL", "text": "|- (!A. ((matrix (\\x. ((matrix_vector_mul A) x))) = A))"}
{"_id": "2759", "title": "BOUNDED_CLOSURE", "text": "|- (!s. ((bounded s) ==> (bounded (closure s))))"}
{"_id": "2760", "title": "AFFINE_HULL_TRANSLATION", "text": "|- (!a. (!s. (((hull affine) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) ((hull affine) s)))))"}
{"_id": "2761", "title": "INJECTIVE_ON_ALT", "text": "|- (!P. (!f. ((!x. (!y. (((P x) /\\ ((P y) /\\ ((f x) = (f y)))) ==> (x = y)))) = (!x. (!y. (((P x) /\\ (P y)) ==> (((f x) = (f y)) = (x = y))))))))"}
{"_id": "2762", "title": "INTERIOR_COMPLEMENT", "text": "|- (!s. ((interior (UNIV DIFF s)) = (UNIV DIFF (closure s))))"}
{"_id": "2763", "title": "Add_triangle.BIJ_IMAGE", "text": "|- (!f. (!s. (!t. ((((BIJ f) s) t) ==> (t = ((IMAGE f) s))))))"}
{"_id": "2764", "title": "AFF_DIM_CONVEX_HULL", "text": "|- (!s. ((aff_dim ((hull convex) s)) = (aff_dim s)))"}
{"_id": "2765", "title": "HAS_REAL_DERIVATIVE_POW_ATREAL", "text": "|- (!f. (!f'. (!x. (!n. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative (\\x. ((real_pow (f x)) n))) ((real_mul (real_of_num n)) ((real_mul ((real_pow (f x)) (n - (NUMERAL (BIT1 _0))))) f'))) (atreal x)))))))"}
{"_id": "2766", "title": "POSITIVE_DEFINITE_TRANSP", "text": "|- (!A. ((positive_definite (transp A)) = (positive_definite A)))"}
{"_id": "2767", "title": "Add_triangle.sigma_fan_not_fixed", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\\ (fully_surrounded (V , E))) ==> (~ ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) = w))))))))"}
{"_id": "2768", "title": "real_integral", "text": "|- (!f. (!i. (((real_integral i) f) = (@y. (((has_real_integral f) y) i)))))"}
{"_id": "2769", "title": "Hypermap.face", "text": "|- (!H. (!x. (((face H) x) = ((orbit_map (face_map H)) x))))"}
{"_id": "2770", "title": "CARD_EQ_SYM", "text": "|- (!s. (!t. ((s =_c t) = (t =_c s))))"}
{"_id": "2771", "title": "Nonlinear_lemma.lmdih5_0", "text": "|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y5) ==> ((((((((y_of_x lmdih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6) = (real_of_num (NUMERAL _0))))))))))"}
{"_id": "2772", "title": "INTERIOR_BIJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> ((interior ((IMAGE f) s)) = ((IMAGE f) (interior s))))))"}
{"_id": "2773", "title": "Trigonometry.SCEZKRH1", "text": "|- (!x. ((sin ((real_sub ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x)) = (cos x)))"}
{"_id": "2774", "title": "NORM_BOUND_COMPONENT_LT", "text": "|- (!x. (!e. (((real_lt (vector_norm x)) e) ==> (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_lt (real_abs (x $ i))) e))))))"}
{"_id": "2775", "title": "IVT_INCREASING_IM", "text": "|- (!f. (!a. (!b. (!y. ((((real_le a) b) /\\ ((!x. ((((real_le a) x) /\\ ((real_le x) b)) ==> ((continuous f) (at (Cx x))))) /\\ (((real_le (Im (f (Cx a)))) y) /\\ ((real_le y) (Im (f (Cx b))))))) ==> (?x. (((real_le a) x) /\\ (((real_le x) b) /\\ ((Im (f (Cx x))) = y)))))))))"}
{"_id": "2776", "title": "Seq2.take", "text": "|- (!x. (!s. (!n. ((((take (SUC n)) ((CONS x) s)) = ((CONS x) ((take n) s))) /\\ ((((take (NUMERAL _0)) s) = NIL) /\\ (((take n) NIL) = NIL))))))"}
{"_id": "2777", "title": "Sphere.azim_cycle", "text": "|- (!W. (!p. (!w. (!v. (((((azim_cycle W) v) w) p) = (((COND (W SUBSET (p INSERT EMPTY))) p) (@u. ((~ (u = p)) /\\ ((W u) /\\ (!q. (((~ (q = p)) /\\ (W q)) ==> (((real_lt ((((azim v) w) p) u)) ((((azim v) w) p) q)) \\/ ((((((azim v) w) p) u) = ((((azim v) w) p) q)) /\\ ((real_le (vector_norm ((projection ((vector_sub w) v)) ((vector_sub u) v)))) (vector_norm ((projection ((vector_sub w) v)) ((vector_sub q) v)))))))))))))))))"}
{"_id": "2778", "title": "INT_LE_LNEG", "text": "|- (!x. (!y. (((int_le (int_neg x)) y) = ((int_le (int_of_num (NUMERAL _0))) ((int_add x) y)))))"}
{"_id": "2779", "title": "Lp_ineqs_proofs.const1_ge0", "text": "|- ((real_le (real_of_num (NUMERAL _0))) const1)"}
{"_id": "2780", "title": "HAS_REAL_INTEGRAL_SUBSET_LE", "text": "|- (!f. (!s. (!t. (!i. (!j. (((s SUBSET t) /\\ ((((has_real_integral f) i) s) /\\ ((((has_real_integral f) j) t) /\\ (!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (f x))))))) ==> ((real_le i) j)))))))"}
{"_id": "2781", "title": "LIM_CONST_EQ", "text": "|- (!net. (!c. (!d. ((((--> (\\x. c)) d) net) = ((trivial_limit net) \\/ (c = d))))))"}
{"_id": "2782", "title": "SIN_VECTOR_ANGLE_LMUL", "text": "|- (!a. (!x. (!y. ((sin ((vector_angle ((% a) x)) y)) = (((COND (a = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (sin ((vector_angle x) y)))))))"}
{"_id": "2783", "title": "REAL_ABS_NORM", "text": "|- (!x. ((real_abs (vector_norm x)) = (vector_norm x)))"}
{"_id": "2784", "title": "CONVEX_ON_COUNTABLE_NONDIFFERENTIABLE", "text": "|- (!f. (!s. ((((convex_on f) s) /\\ (COUNTABLE (components s))) ==> (COUNTABLE (GSPEC (\\GEN%PVAR%3315. (?x. (((SETSPEC GEN%PVAR%3315) ((x IN s) /\\ (~ ((differentiable (lift o f)) (at x))))) x))))))))"}
{"_id": "2785", "title": "SPAN_CLAUSES", "text": "|- ((!a. (!s. ((a IN s) ==> (a IN (span s))))) /\\ (((vec (NUMERAL _0)) IN (span s)) /\\ ((!x. (!y. (!s. (((x IN (span s)) /\\ (y IN (span s))) ==> (((vector_add x) y) IN (span s)))))) /\\ (!x. (!c. (!s. ((x IN (span s)) ==> (((% c) x) IN (span s)))))))))"}
{"_id": "2786", "title": "HAS_MEASURE_TRANSLATION", "text": "|- (!s. (!m. (!a. (((has_measure s) m) ==> ((has_measure ((IMAGE (\\x. ((vector_add a) x))) s)) m)))))"}
{"_id": "2787", "title": "SUBPATH_LINEAR_IMAGE", "text": "|- (!f. (!g. (!u. (!v. ((linear f) ==> ((((subpath u) v) (f o g)) = (f o (((subpath u) v) g))))))))"}
{"_id": "2788", "title": "REAL_SGN_INEQS", "text": "|- ((!x. (((real_le (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_le (real_of_num (NUMERAL _0))) x))) /\\ ((!x. (((real_lt (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_lt (real_of_num (NUMERAL _0))) x))) /\\ ((!x. (((real_ge (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_ge (real_of_num (NUMERAL _0))) x))) /\\ ((!x. (((real_gt (real_of_num (NUMERAL _0))) (real_sgn x)) = ((real_gt (real_of_num (NUMERAL _0))) x))) /\\ ((!x. (((real_of_num (NUMERAL _0)) = (real_sgn x)) = ((real_of_num (NUMERAL _0)) = x))) /\\ ((!x. (((real_le (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_le x) (real_of_num (NUMERAL _0))))) /\\ ((!x. (((real_lt (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_lt x) (real_of_num (NUMERAL _0))))) /\\ ((!x. (((real_ge (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_ge x) (real_of_num (NUMERAL _0))))) /\\ ((!x. (((real_gt (real_sgn x)) (real_of_num (NUMERAL _0))) = ((real_gt x) (real_of_num (NUMERAL _0))))) /\\ (!x. (((real_sgn x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))))))))))))"}
{"_id": "2789", "title": "COMPACT_FRONTIER", "text": "|- (!s. ((compact s) ==> (compact (frontier s))))"}
{"_id": "2790", "title": "Vol1.radial_norm_deprecated", "text": "|- (radial_norm = radial)"}
{"_id": "2791", "title": "Fan_misc.FAN_IN_SET_OF_EDGE", "text": "|- (!x. (!V. (!E. (!v. (!w. (((FAN (x , (V , E))) /\\ ((v INSERT (w INSERT EMPTY)) IN E)) ==> ((v IN V) /\\ ((w IN V) /\\ ((w IN (((set_of_edge v) V) E)) /\\ (v IN (((set_of_edge w) V) E)))))))))))"}
{"_id": "2792", "title": "SETDIST_CLOSED_COMPACT", "text": "|- (!s. (!t. (((closed s) /\\ ((compact t) /\\ ((~ (s = EMPTY)) /\\ (~ (t = EMPTY))))) ==> (?x. (?y. ((x IN s) /\\ ((y IN t) /\\ ((distance (x , y)) = (setdist (s , t))))))))))"}
{"_id": "2793", "title": "Appendix.scs_hi_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_hi_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = hi))))))))))"}
{"_id": "2794", "title": "ANALYTIC_ON_NEG", "text": "|- (!f. (!s. (((analytic_on f) s) ==> ((analytic_on (\\z. (vector_neg (f z)))) s))))"}
{"_id": "2795", "title": "FORALL_PASTECART", "text": "|- ((!p. (P p)) = (!x. (!y. (P ((pastecart x) y)))))"}
{"_id": "2796", "title": "REAL_ABS_BETWEEN2", "text": "|- (!x0. (!x. (!y0. (!y. ((((real_lt x0) y0) /\\ (((real_lt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_abs ((real_sub x) x0)))) ((real_sub y0) x0)) /\\ ((real_lt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_abs ((real_sub y) y0)))) ((real_sub y0) x0)))) ==> ((real_lt x) y))))))"}
{"_id": "2797", "title": "REAL_DIFFERENTIABLE_IMP_CONTINUOUS_ATREAL", "text": "|- (!f. (!x. (((real_differentiable f) (atreal x)) ==> ((real_continuous f) (atreal x)))))"}
{"_id": "2798", "title": "REAL_LE_NEGL", "text": "|- (!x. (((real_le (real_neg x)) x) = ((real_le (real_of_num (NUMERAL _0))) x)))"}
{"_id": "2799", "title": "VECTOR_MUL_RNEG", "text": "|- (((% c) (vector_neg x)) = (vector_neg ((% c) x)))"}
{"_id": "2800", "title": "BAIRE", "text": "|- (!g. (!s. ((((locally compact) s) /\\ ((COUNTABLE g) /\\ (!t. ((t IN g) ==> (((open_in ((subtopology euclidean) s)) t) /\\ (s SUBSET (closure t))))))) ==> (s SUBSET (closure (INTERS g))))))"}
{"_id": "2801", "title": "LIM_AT", "text": "|- (!f. (!l. (!a. ((((--> f) l) (at a)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\\ ((real_lt (distance (x , a))) d)) ==> ((real_lt (distance ((f x) , l))) e)))))))))))"}
{"_id": "2802", "title": "MATRIX_TRANSP_MUL", "text": "|- (!A. (!B. ((transp ((matrix_mul A) B)) = ((matrix_mul (transp B)) (transp A)))))"}
{"_id": "2803", "title": "SUM_INTEGRAL_UBOUND_INCREASING", "text": "|- (!f. (!g. (!m. (!n. (((m <= n) /\\ ((!x. ((x IN (closed_segment ((CONS ((Cx (real_of_num m)) , (Cx ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) NIL))) ==> (((has_complex_derivative g) (f x)) (at x)))) /\\ (!x. (!y. ((((real_le (real_of_num m)) x) /\\ (((real_le x) y) /\\ ((real_le y) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((real_le (Re (f (Cx x)))) (Re (f (Cx y))))))))) ==> ((real_le ((sum (m .. n)) (\\k. (Re (f (Cx (real_of_num k))))))) (Re ((vector_sub (g (Cx ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) (g (Cx (real_of_num m)))))))))))"}
{"_id": "2804", "title": "HAS_REAL_DERIVATIVE_SEQUENCE", "text": "|- (!s. (!f. (!f'. (!g'. (((is_realinterval s) /\\ ((!n. (!x. ((x IN s) ==> (((has_real_derivative (f n)) ((f' n) x)) ((within (atreal x)) s))))) /\\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\\ (x IN s)) ==> ((real_le (real_abs ((real_sub ((f' n) x)) (g' x)))) e))))))) /\\ (?x. (?l. ((x IN s) /\\ (((---> (\\n. ((f n) x))) l) sequentially))))))) ==> (?g. (!x. ((x IN s) ==> ((((---> (\\n. ((f n) x))) (g x)) sequentially) /\\ (((has_real_derivative g) (g' x)) ((within (atreal x)) s)))))))))))"}
{"_id": "2805", "title": "ANR_CLOSED_UNION", "text": "|- (!s. (!t. (((closed s) /\\ ((closed t) /\\ ((ANR s) /\\ ((ANR t) /\\ (ANR (s INTER t)))))) ==> (ANR (s UNION t)))))"}
{"_id": "2806", "title": "Ssrnat.expnS", "text": "|- (!m. (!n. ((m EXP (SUC n)) = (m * (m EXP n)))))"}
{"_id": "2807", "title": "gt_c", "text": "|- (!t. (!s. ((s >_c t) = (t <_c s))))"}
{"_id": "2808", "title": "WITHIN_UNIV", "text": "|- (!x. (((within (at x)) UNIV) = (at x)))"}
{"_id": "2809", "title": "CONTINUOUS_CLOSED_GRAPH_EQ", "text": "|- (!f. (!s. (!t. (((compact t) /\\ (((IMAGE f) s) SUBSET t)) ==> (((continuous_on f) s) = ((closed_in ((subtopology euclidean) (s PCROSS t))) (GSPEC (\\GEN%PVAR%965. (?x. (((SETSPEC GEN%PVAR%965) (x IN s)) ((pastecart x) (f x))))))))))))"}
{"_id": "2810", "title": "AFFINE_HULL_2_ALT", "text": "|- (!a. (!b. (((hull affine) (a INSERT (b INSERT EMPTY))) = (GSPEC (\\GEN%PVAR%1416. (?u. (((SETSPEC GEN%PVAR%1416) (u IN UNIV)) ((vector_add a) ((% u) ((vector_sub b) a))))))))))"}
{"_id": "2811", "title": "IN_UNIV", "text": "|- (!x. (x IN UNIV))"}
{"_id": "2812", "title": "path_connected", "text": "|- (!s. ((path_connected s) = (!x. (!y. (((x IN s) /\\ (y IN s)) ==> (?g. ((path g) /\\ (((path_image g) SUBSET s) /\\ (((pathstart g) = x) /\\ ((pathfinish g) = y))))))))))"}
{"_id": "2813", "title": "CART_EQ", "text": "|- (!x. (!y. ((x = y) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((x $ i) = (y $ i)))))))"}
{"_id": "2814", "title": "DIM_SUBSET", "text": "|- (!s. (!t. ((s SUBSET t) ==> ((dim s) <= (dim t)))))"}
{"_id": "2815", "title": "REAL_MEASURABLE_ON_ADD", "text": "|- (!f. (!g. (!s. ((((real_measurable_on f) s) /\\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\\x. ((real_add (f x)) (g x)))) s)))))"}
{"_id": "2816", "title": "MEASURABLE_ON_COMPOSE_CONTINUOUS", "text": "|- (!f. (!g. ((((measurable_on f) UNIV) /\\ ((continuous_on g) UNIV)) ==> ((measurable_on (g o f)) UNIV))))"}
{"_id": "2817", "title": "PASTECART_VEC", "text": "|- (!n. (((pastecart (vec n)) (vec n)) = (vec n)))"}
{"_id": "2818", "title": "PATH_CONNECTED_IMP_CONNECTED", "text": "|- (!s. ((path_connected s) ==> (connected s)))"}
{"_id": "2819", "title": "Real_ext.REAL_PROP_ZERO_ABS", "text": "|- (!x. (((real_abs x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0)))))"}
{"_id": "2820", "title": "Trigonometry2.POW2_COND", "text": "|- (!a. (!b. ((((real_le (real_of_num (NUMERAL _0))) a) /\\ ((real_le (real_of_num (NUMERAL _0))) b)) ==> (((real_le a) b) = ((real_le ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow b) (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "2821", "title": "ABSOLUTELY_CONTINUOUS_ON_VMUL_EQ", "text": "|- (!f. (!s. (!v. (((absolutely_continuous_on (\\x. ((% (f x)) v))) s) = ((v = (vec (NUMERAL _0))) \\/ ((absolutely_continuous_on (\\x. (lift (f x)))) s))))))"}
{"_id": "2822", "title": "LET_TRANS", "text": "|- (!m. (!n. (!p. (((m <= n) /\\ (n < p)) ==> (m < p)))))"}
{"_id": "2823", "title": "IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT", "text": "|- (!s. (!x. (!a. (((convex s) /\\ ((x IN (relative_interior s)) /\\ ((a IN ((hull affine) s)) /\\ (~ (x = a))))) ==> (?b. ((b IN s) /\\ (x IN (open_segment (a , b)))))))))"}
{"_id": "2824", "title": "LOG_RPOW", "text": "|- (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log ((rpow x) y)) = ((real_mul y) (log x))))))"}
{"_id": "2825", "title": "LINEAR_SUB", "text": "|- (!f. (!x. (!y. ((linear f) ==> ((f ((vector_sub x) y)) = ((vector_sub (f x)) (f y)))))))"}
{"_id": "2826", "title": "Sphere.rotate6", "text": "|- (!f. (!x6. (!x1. (!x5. (!x3. (!x4. (!x2. ((((((((rotate6 f) x1) x2) x3) x4) x5) x6) = ((((((f x6) x1) x5) x3) x4) x2)))))))))"}
{"_id": "2827", "title": "Fnjlbxs.length_list_of_vector", "text": "|- (!d. (!n. (!v. ((LENGTH (((list_of_vector d) n) v)) = n))))"}
{"_id": "2828", "title": "Lp_main_estimate.ineq_tau4_tauVEF", "text": "|- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\\ ((packing V) /\\ ((V SUBSET ball_annulus) /\\ (((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) <= (CARD V)) /\\ (((CARD V) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) /\\ ((!v. (!w. (((~ (v = w)) /\\ ((v IN V) /\\ ((w IN V) /\\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (distance (v , w)))))) /\\ (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v , w))) /\\ ((real_le (distance (v , w))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))))))))) ==> ((real_le (d_tame (CARD V))) (((tau_fun V) E) FF)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ ((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) d))))))))))))"}
{"_id": "2829", "title": "CONNECTED_CONVEX_1_GEN", "text": "|- (!s. (((dimindex UNIV) = (NUMERAL (BIT1 _0))) ==> ((convex s) = (connected s))))"}
{"_id": "2830", "title": "INT_OF_NUM_GT", "text": "|- (!m. (!n. (((int_gt (int_of_num m)) (int_of_num n)) = (m > n))))"}
{"_id": "2831", "title": "REAL_EXP_MONO_LT", "text": "|- (!x. (!y. (((real_lt (exp x)) (exp y)) = ((real_lt x) y))))"}
{"_id": "2832", "title": "REAL_SUB_LE", "text": "|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) ((real_sub x) y)) = ((real_le y) x))))"}
{"_id": "2833", "title": "COND_RATOR", "text": "|- (!b. (!f. (!g. (!x. (((((COND b) f) g) x) = (((COND b) (f x)) (g x)))))))"}
{"_id": "2834", "title": "Functional_equation.add6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((add6 f) g) x1) x2) x3) x4) x5) x6) = ((real_add ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "2835", "title": "BASIS_NONZERO", "text": "|- (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV))) ==> (~ ((basis k) = (vec (NUMERAL _0))))))"}
{"_id": "2836", "title": "Hdplygy.sigma_sy", "text": "|- (!s. ((sigma_sy s) = (((COND (ear_sy s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_neg (real_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "2837", "title": "MEASURABLE_ON_MIN", "text": "|- (!f. (!g. (!s. ((((measurable_on f) s) /\\ ((measurable_on g) s)) ==> ((measurable_on (\\x. (lambdai. ((real_min ((f x) $ i)) ((g x) $ i))))) s)))))"}
{"_id": "2838", "title": "columns", "text": "|- (!A. ((columns A) = (GSPEC (\\GEN%PVAR%126. (?i. (((SETSPEC GEN%PVAR%126) (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) ((column i) A)))))))"}
{"_id": "2839", "title": "COMPLEX_ADD_CCOS", "text": "|- (!w. (!z. (((vector_add (ccos w)) (ccos z)) = ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_mul (ccos ((complex_div ((vector_add w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) (ccos ((complex_div ((vector_sub w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))"}
{"_id": "2840", "title": "Lp_ineqs_proofs.y6_def", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((y6_fan d) = (ye_fan d))))))))))"}
{"_id": "2841", "title": "CONVEX_CLOSURE", "text": "|- (!s. ((convex s) ==> (convex (closure s))))"}
{"_id": "2842", "title": "DIFFERENTIABLE_ADD", "text": "|- (!f. (!g. (!net. ((((differentiable f) net) /\\ ((differentiable g) net)) ==> ((differentiable (\\z. ((vector_add (f z)) (g z)))) net)))))"}
{"_id": "2843", "title": "CARD_LE_FINITE_INFINITE", "text": "|- (!s. (!t. (((FINITE s) /\\ (INFINITE t)) ==> (s <=_c t))))"}
{"_id": "2844", "title": "OPEN_IN_CLOSED_IN_EQ", "text": "|- (!top. (!s. (((open_in top) s) = ((s SUBSET (topspace top)) /\\ ((closed_in top) ((topspace top) DIFF s))))))"}
{"_id": "2845", "title": "CARD_EQ_REFL", "text": "|- (!s. (s =_c s))"}
{"_id": "2846", "title": "REAL_EQ_MUL_RCANCEL", "text": "|- (!x. (!y. (!z. ((((real_mul x) z) = ((real_mul y) z)) = ((x = y) \\/ (z = (real_of_num (NUMERAL _0))))))))"}
{"_id": "2847", "title": "LOCALLY_COMPACT_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((locally compact) ((IMAGE f) s)) = ((locally compact) s)))))"}
{"_id": "2848", "title": "COMPLEX_SQNORM", "text": "|- (((real_pow (vector_norm z)) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_add ((real_pow (Re z)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (Im z)) (NUMERAL (BIT0 (BIT1 _0))))))"}
{"_id": "2849", "title": "Hypermap_iso.dart1_of_fan_eq_image", "text": "|- (!V. (!E. ((dart1_of_fan (V , E)) = ((IMAGE contracted_dart) (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))"}
{"_id": "2850", "title": "CONVEX_ON_CONTINUOUS", "text": "|- (!f. (!s. (((open s) /\\ ((convex_on f) s)) ==> ((continuous_on (lift o f)) s))))"}
{"_id": "2851", "title": "LINEAR_VMUL_DROP", "text": "|- (!f. (!v. ((linear f) ==> (linear (\\x. ((% (drop (f x))) v))))))"}
{"_id": "2852", "title": "Lp_ineqs_proofs.dart4_y4'_lo", "text": "|- (!V. (!E. ((lp_fan (V , E)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) ((y4'_fan (V , E)) d)))))))"}
{"_id": "2853", "title": "ANR_EMPTY", "text": "|- (ANR EMPTY)"}
{"_id": "2854", "title": "Rogers.UNIQUE_SOLUTION_AFFINE_INDEPENDENT", "text": "|- (!S. (!b. (((~ (S = EMPTY)) /\\ (~ (affine_dependent S))) ==> (?!p. ((p IN ((hull affine) S)) /\\ (!x. (!y. (((x IN S) /\\ (y IN S)) ==> (((dot p) ((vector_sub x) y)) = ((real_sub (b x)) (b y)))))))))))"}
{"_id": "2855", "title": "ORTHONORMAL_BASIS_SUBSPACE", "text": "|- (!s. ((subspace s) ==> (?b. ((b SUBSET s) /\\ (((pairwise orthogonal) b) /\\ ((!x. ((x IN b) ==> ((vector_norm x) = (real_of_num (NUMERAL (BIT1 _0)))))) /\\ ((independent b) /\\ ((b HAS_SIZE (dim s)) /\\ ((span b) = s)))))))))"}
{"_id": "2856", "title": "Seq2.prev_el", "text": "|- (!x. (!s. (((prev_el s) x) = (((COND (~ ((MEM x) s))) x) (((COND (((indexl x) s) = (NUMERAL _0))) ((last x) s)) (((nth x) s) (((indexl x) s) - (NUMERAL (BIT1 _0)))))))))"}
{"_id": "2857", "title": "homotopy_equivalent", "text": "|- (!s. (!t. (((homotopy_equivalent s) t) = (?f. (?g. (((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((continuous_on g) t) /\\ ((((IMAGE g) t) SUBSET s) /\\ (((((homotopic_with (\\x. T)) (s , s)) (g o f)) I) /\\ ((((homotopic_with (\\x. T)) (t , t)) (f o g)) I)))))))))))"}
{"_id": "2858", "title": "list_cases", "text": "|- (!P. (((P NIL) /\\ (!h. (!t. (P ((CONS h) t))))) ==> (!l. (P l))))"}
{"_id": "2859", "title": "SIMPLE_PATH_IMP_PATH", "text": "|- (!g. ((simple_path g) ==> (path g)))"}
{"_id": "2860", "title": "Ssrnat.maxnK", "text": "|- (!m. (!n. (((minn ((maxn m) n)) m) = m)))"}
{"_id": "2861", "title": "COMPLEX_INV_MUL", "text": "|- (!w. (!z. ((complex_inv ((complex_mul w) z)) = ((complex_mul (complex_inv w)) (complex_inv z)))))"}
{"_id": "2862", "title": "NORM_POW_2", "text": "|- (!x. (((real_pow (vector_norm x)) (NUMERAL (BIT0 (BIT1 _0)))) = ((dot x) x)))"}
{"_id": "2863", "title": "VECTOR_ADD_LDISTRIB", "text": "|- (((% c) ((vector_add x) y)) = ((vector_add ((% c) x)) ((% c) y)))"}
{"_id": "2864", "title": "LIM_TRANSFORM_EVENTUALLY", "text": "|- (!net. (!f. (!g. (!l. ((((eventually (\\x. ((f x) = (g x)))) net) /\\ (((--> f) l) net)) ==> (((--> g) l) net))))))"}
{"_id": "2865", "title": "HAS_INTEGRAL_COMPONENT_LE_AE", "text": "|- (!f. (!g. (!s. (!i. (!j. (!k. (!t. ((((NUMERAL (BIT1 _0)) <= k) /\\ ((k <= (dimindex UNIV)) /\\ ((negligible t) /\\ ((((has_integral f) i) s) /\\ ((((has_integral g) j) s) /\\ (!x. ((x IN (s DIFF t)) ==> ((real_le ((f x) $ k)) ((g x) $ k))))))))) ==> ((real_le (i $ k)) (j $ k))))))))))"}
{"_id": "2866", "title": "Vol1.sol_spec", "text": "|- (!x. (!C. (!r. ((((real_gt r) (real_of_num (NUMERAL _0))) /\\ ((measurable (C INTER (ball (x , r)))) /\\ (((radial r) x) (C INTER (ball (x , r)))))) ==> (((sol x) C) = ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_div (measure (C INTER (ball (x , r))))) ((real_pow r) (NUMERAL (BIT1 (BIT1 _0)))))))))))"}
{"_id": "2867", "title": "AFFINE_HULL_CONVEX_HULL", "text": "|- (!s. (((hull affine) ((hull convex) s)) = ((hull affine) s)))"}
{"_id": "2868", "title": "EXISTS_PAIRED_THM", "text": "|- (!P. ((? (GABS (\\f. (!x. (!y. ((GEQ (f (x , y))) ((P x) y))))))) = (?x. (?y. ((P x) y)))))"}
{"_id": "2869", "title": "GDELTA_COMPLEMENT", "text": "|- (!s. ((gdelta (UNIV DIFF s)) = (fsigma s)))"}
{"_id": "2870", "title": "COND_CLAUSES", "text": "|- (!t1. (!t2. (((((COND T) t1) t2) = t1) /\\ ((((COND F) t1) t2) = t2))))"}
{"_id": "2871", "title": "HAS_BOUNDED_SETVARIATION_ON_NEG", "text": "|- (!f. (!s. (((has_bounded_setvariation_on (\\x. (vector_neg (f x)))) s) = ((has_bounded_setvariation_on f) s))))"}
{"_id": "2872", "title": "IN_RELATIVE_INTERIOR_CLOSURE_CONVEX_SEGMENT", "text": "|- (!s. (!a. (!b. (((convex s) /\\ ((a IN (relative_interior s)) /\\ (b IN (closure s)))) ==> ((open_segment (a , b)) SUBSET (relative_interior s))))))"}
{"_id": "2873", "title": "CONTINUOUS_ON_CMUL", "text": "|- (!f. (!c. (!s. (((continuous_on f) s) ==> ((continuous_on (\\x. ((% c) (f x)))) s)))))"}
{"_id": "2874", "title": "FSIGMA_EMPTY", "text": "|- (fsigma EMPTY)"}
{"_id": "2875", "title": "LIM_COMPONENT_UBOUND", "text": "|- (!net. (!f. (!l. (!b. (!k. (((~ (trivial_limit net)) /\\ ((((--> f) l) net) /\\ (((eventually (\\x. ((real_le ((f x) $ k)) b))) net) /\\ (((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV)))))) ==> ((real_le (l $ k)) b)))))))"}
{"_id": "2876", "title": "ITER", "text": "|- ((!f. ((((ITER (NUMERAL _0)) f) x) = x)) /\\ (!f. (!n. ((((ITER (SUC n)) f) x) = (f (((ITER n) f) x))))))"}
{"_id": "2877", "title": "DIM_SPAN", "text": "|- (!s. ((dim (span s)) = (dim s)))"}
{"_id": "2878", "title": "Tame_lemmas.fully_surrounded_azim_eq_dih_y", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\\w'. (LET_END ((LET (\\y1. (LET_END ((LET (\\y2. (LET_END ((LET (\\y3. (LET_END ((LET (\\y4. (LET_END ((LET (\\y5. (LET_END ((LET (\\y6. (LET_END (((azim_dart (V , E)) (v , w)) = ((((((dih_y y1) y2) y3) y4) y5) y6))))) (distance (v , w)))))) (distance (v , w')))))) (distance (w , w')))))) (vector_norm w'))))) (vector_norm w))))) (vector_norm v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))"}
{"_id": "2879", "title": "CONVEX_POSITIVE_ORTHANT", "text": "|- (convex (GSPEC (\\GEN%PVAR%1454. (?x. (((SETSPEC GEN%PVAR%1454) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))"}
{"_id": "2880", "title": "mspace", "text": "|- (!m. ((mspace m) = (FST (dest_metric m))))"}
{"_id": "2881", "title": "REAL_SERIES_RESTRICT", "text": "|- (!f. (!k. (!l. ((((real_sums (\\n. (((COND (n IN k)) (f n)) (real_of_num (NUMERAL _0))))) l) UNIV) = (((real_sums f) l) k)))))"}
{"_id": "2882", "title": "POLYTOPE_IMP_COMPACT", "text": "|- (!s. ((polytope s) ==> (compact s)))"}
{"_id": "2883", "title": "CONVEX_CONE_HULL_SEPARATE", "text": "|- (!s. (((hull convex_cone) s) = ((vec (NUMERAL _0)) INSERT ((hull conic) ((hull convex) s)))))"}
{"_id": "2884", "title": "SND", "text": "|- (!x. (!y. ((SND (x , y)) = y)))"}
{"_id": "2885", "title": "CONTRAPOS_THM", "text": "|- (!t1. (!t2. (((~ t1) ==> (~ t2)) = (t2 ==> t1))))"}
{"_id": "2886", "title": "RE_MUL_II", "text": "|- (!z. (((Re ((complex_mul z) ii)) = (real_neg (Im z))) /\\ ((Re ((complex_mul ii) z)) = (real_neg (Im z)))))"}
{"_id": "2887", "title": "Hypermap.atom", "text": "|- (!H. (!L. (!x. ((((atom H) L) x) = (GSPEC (\\GEN%PVAR%5421. (?y. (((SETSPEC GEN%PVAR%5421) (((((is_node_going H) L) x) y) \\/ ((((is_node_going H) L) y) x))) y))))))))"}
{"_id": "2888", "title": "Nonlinear_lemma.halfbump_x1", "text": "|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((halfbump_x1 x1) x2) x3) x4) x5) x6) = (halfbump_x x1))))))))"}
{"_id": "2889", "title": "WF_REC", "text": "|- ((WF <<) ==> (!H. ((!f. (!g. (!x. ((!z. (((<< z) x) ==> ((f z) = (g z)))) ==> (((H f) x) = ((H g) x)))))) ==> (?f. (!x. ((f x) = ((H f) x)))))))"}
{"_id": "2890", "title": "TL", "text": "|- ((TL ((CONS h) t)) = t)"}
{"_id": "2891", "title": "STARLIKE_TRANSLATION_EQ", "text": "|- (!a. (!s. ((starlike ((IMAGE (\\x. ((vector_add a) x))) s)) = (starlike s))))"}
{"_id": "2892", "title": "PATH_COMPONENT_EQ_CONNECTED_COMPONENT", "text": "|- (!s. (!x. (((locally path_connected) s) ==> (((path_component s) x) = ((connected_component s) x)))))"}
{"_id": "2893", "title": "Pqcsxwg.PQCSXWG2_ATREAL", "text": "|- (!v0. (!v1. (!v2. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (((~ (collinear ((v0 a) INSERT ((v1 a) INSERT ((v2 a) INSERT EMPTY))))) /\\ (((x1 a) = ((real_pow (distance ((v1 a) , (v0 a)))) (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((x2 a) = ((real_pow (distance ((v2 a) , (v0 a)))) (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((x6 a) = ((real_pow (distance ((v1 a) , (v2 a)))) (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x (x1 a)) (x2 a)) (x3 a)) (x4 a)) (x5 a)) (x6 a))) /\\ (((continuous v0) (atreal a)) /\\ (((continuous v1) (atreal a)) /\\ (((continuous v2) (atreal a)) /\\ (((real_continuous x1) (atreal a)) /\\ (((real_continuous x2) (atreal a)) /\\ (((real_continuous x3) (atreal a)) /\\ (((real_continuous x4) (atreal a)) /\\ (((real_continuous x5) (atreal a)) /\\ ((real_continuous x6) (atreal a))))))))))))))) ==> ((continuous (\\t. (((((((((mk_simplex1 (v0 t)) (v1 t)) (v2 t)) (x1 t)) (x2 t)) (x3 t)) (x4 t)) (x5 t)) (x6 t)))) (atreal a)))))))))))))"}
{"_id": "2894", "title": "Add_triangle.fully_surrounded_imp_conforming", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (conforming_fan ((vec (NUMERAL _0)) , (V , E)))))))"}
{"_id": "2895", "title": "CARD_EQ_CLOSED", "text": "|- (!s. ((closed s) ==> ((s <=_c UNIV) \\/ (s =_c UNIV))))"}
{"_id": "2896", "title": "CONTRACTIBLE_IMP_PATH_CONNECTED", "text": "|- (!s. ((contractible s) ==> (path_connected s)))"}
{"_id": "2897", "title": "INTERIOR_CLOSURE", "text": "|- (!s. ((interior s) = (UNIV DIFF (closure (UNIV DIFF s)))))"}
{"_id": "2898", "title": "Lp_ineqs_def.AZIM_DART_POS", "text": "|- ((real_le (real_of_num (NUMERAL _0))) ((azim_dart (V , E)) x))"}
{"_id": "2899", "title": "Local_lemmas.AFF_GE_MONO_TRANS", "text": "|- ((S SUBSET X) ==> (((aff_ge (X DIFF S)) (Y UNION S)) SUBSET ((aff_ge X) Y)))"}
{"_id": "2900", "title": "MEASURE_CLOSURE", "text": "|- (!s. (((bounded s) /\\ (negligible (frontier s))) ==> ((measure (closure s)) = (measure s))))"}
{"_id": "2901", "title": "Hypermap.lemma_two_series_eq", "text": "|- (!p. (!q. (!n. ((!i. ((i <= n) ==> ((p i) = (q i)))) ==> ((GSPEC (\\GEN%PVAR%5391. (?i. (((SETSPEC GEN%PVAR%5391) (i <= n)) (p i))))) = (GSPEC (\\GEN%PVAR%5392. (?i. (((SETSPEC GEN%PVAR%5392) (i <= n)) (q i))))))))))"}
{"_id": "2902", "title": "EMPTY_NOT_UNIV", "text": "|- (~ (EMPTY = UNIV))"}
{"_id": "2903", "title": "RE_COMPLEX_DIV_GT_0", "text": "|- (!a. (!b. (((real_lt (real_of_num (NUMERAL _0))) (Re ((complex_div a) b))) = ((real_lt (real_of_num (NUMERAL _0))) (Re ((complex_mul a) (cnj b)))))))"}
{"_id": "2904", "title": "real_sums", "text": "|- (!s. (!f. (!l. ((((real_sums f) l) s) = (((---> (\\n. ((sum (s INTER ((NUMERAL _0) .. n))) f))) l) sequentially)))))"}
{"_id": "2905", "title": "DOT_RSUB", "text": "|- (!x. (!y. (!z. (((dot x) ((vector_sub y) z)) = ((real_sub ((dot x) y)) ((dot x) z))))))"}
{"_id": "2906", "title": "UNIONS_COMPONENTS", "text": "|- (!u. (u = (UNIONS (components u))))"}
{"_id": "2907", "title": "OPEN_IN_CONTAINS_BALL", "text": "|- (!s. (!t. (((open_in ((subtopology euclidean) t)) s) = ((s SUBSET t) /\\ (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (((ball (x , e)) INTER t) SUBSET s)))))))))"}
{"_id": "2908", "title": "REAL_LOG_CONVEX_MUL", "text": "|- (!f. (!g. (!s. ((((real_log_convex_on f) s) /\\ ((real_log_convex_on g) s)) ==> ((real_log_convex_on (\\x. ((real_mul (f x)) (g x)))) s)))))"}
{"_id": "2909", "title": "Nonlin_def.ups234_x", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ups234_x x1) x2) x3) x4) x5) x6) = (((ups_x x2) x3) x4))))))))"}
{"_id": "2910", "title": "RSTC_REFL", "text": "|- (!R. (!x. (((RSTC R) x) x)))"}
{"_id": "2911", "title": "ANGLE_EQ_0_DIST", "text": "|- (!A. (!B. (!C. (((angle (A , (B , C))) = (real_of_num (NUMERAL _0))) = ((~ (A = B)) /\\ ((~ (C = B)) /\\ (((distance (A , B)) = ((real_add (distance (A , C))) (distance (C , B)))) \\/ ((distance (B , C)) = ((real_add (distance (A , C))) (distance (A , B)))))))))))"}
{"_id": "2912", "title": "LAMBDA_BETA", "text": "|- (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (((lambda g) $ i) = (g i))))"}
{"_id": "2913", "title": "CONTINUOUS_AT", "text": "|- (!f. (!x. (((continuous f) (at x)) = (((--> f) (f x)) (at x)))))"}
{"_id": "2914", "title": "CBALL_TRANSLATION", "text": "|- (!a. (!x. (!r. ((cball (((vector_add a) x) , r)) = ((IMAGE (\\y. ((vector_add a) y))) (cball (x , r)))))))"}
{"_id": "2915", "title": "CONNECTED_EMPTY", "text": "|- (connected EMPTY)"}
{"_id": "2916", "title": "ARITH_ADD", "text": "|- ((!m. (!n. (((NUMERAL m) + (NUMERAL n)) = (NUMERAL (m + n))))) /\\ (((_0 + _0) = _0) /\\ ((!n. ((_0 + (BIT0 n)) = (BIT0 n))) /\\ ((!n. ((_0 + (BIT1 n)) = (BIT1 n))) /\\ ((!n. (((BIT0 n) + _0) = (BIT0 n))) /\\ ((!n. (((BIT1 n) + _0) = (BIT1 n))) /\\ ((!m. (!n. (((BIT0 m) + (BIT0 n)) = (BIT0 (m + n))))) /\\ ((!m. (!n. (((BIT0 m) + (BIT1 n)) = (BIT1 (m + n))))) /\\ ((!m. (!n. (((BIT1 m) + (BIT0 n)) = (BIT1 (m + n))))) /\\ (!m. (!n. (((BIT1 m) + (BIT1 n)) = (BIT0 (SUC (m + n)))))))))))))))"}
{"_id": "2917", "title": "Tame_defs.heights", "text": "|- (!g. ((heights g) = (SND (SND (SND g)))))"}
{"_id": "2918", "title": "ANR_TRIANGULATION", "text": "|- (!tr. ((triangulation tr) ==> (ANR (UNIONS tr))))"}
{"_id": "2919", "title": "COMPACT_IMP_BOUNDED", "text": "|- (!s. ((compact s) ==> (bounded s)))"}
{"_id": "2920", "title": "CONIC_PCROSS", "text": "|- (!s. (!t. (((conic s) /\\ (conic t)) ==> (conic (s PCROSS t)))))"}
{"_id": "2921", "title": "Ssrnat.muln_subl", "text": "|- (!m. (!n. (!p. (((m - n) * p) = ((m * p) - (n * p))))))"}
{"_id": "2922", "title": "SUMS_EQ", "text": "|- (!f. (!g. (!k. (((!x. ((x IN k) ==> ((f x) = (g x)))) /\\ (((sums f) l) k)) ==> (((sums g) l) k)))))"}
{"_id": "2923", "title": "LIM_AT_INFINITY", "text": "|- (!f. (!l. ((((--> f) l) at_infinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge (vector_norm x)) b) ==> ((real_lt (distance ((f x) , l))) e)))))))))"}
{"_id": "2924", "title": "Fan_defs.dart1_of_fan", "text": "|- (!V. (!E. ((dart1_of_fan (V , E)) = (GSPEC (\\GEN%PVAR%5440. (?v. (?w. (((SETSPEC GEN%PVAR%5440) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w)))))))))"}
{"_id": "2925", "title": "CONVEX_EXPLICIT", "text": "|- (!s. ((convex s) = (!t. (!u. (((FINITE t) /\\ ((t SUBSET s) /\\ ((!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\\ (((sum t) u) = (real_of_num (NUMERAL (BIT1 _0))))))) ==> (((vsum t) (\\x. ((% (u x)) x))) IN s))))))"}
{"_id": "2926", "title": "real_lt", "text": "|- (!y. (!x. (((real_lt x) y) = (~ ((real_le y) x)))))"}
{"_id": "2927", "title": "SIMPLY_CONNECTED_IMP_PATH_CONNECTED", "text": "|- (!s. ((simply_connected s) ==> (path_connected s)))"}
{"_id": "2928", "title": "Add_triangle.fully_surrounded_card_set_of_edge1", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (((fully_surrounded (V , E)) /\\ (v IN V)) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))))))"}
{"_id": "2929", "title": "Appendix.BBs_v39", "text": "|- (!s. (!vv. (((BBs_v39 s) vv) = ((LET (\\V. (LET_END ((LET (\\E. (LET_END ((LET (\\FF. (LET_END ((V SUBSET ball_annulus) /\\ (((periodic vv) (scs_k_v39 s)) /\\ ((!i. (!j. (((real_le (((scs_a_v39 s) i) j)) (distance ((vv i) , (vv j)))) /\\ ((real_le (distance ((vv i) , (vv j)))) (((scs_b_v39 s) i) j))))) /\\ (((scs_k_v39 s) <= (NUMERAL (BIT1 (BIT1 _0)))) \\/ (convex_local_fan (V , (E , FF)))))))))) ((IMAGE (\\i. ((vv i) , (vv (SUC i))))) UNIV))))) ((IMAGE (\\i. ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY)))) UNIV))))) ((IMAGE vv) UNIV)))))"}
{"_id": "2930", "title": "TWO", "text": "|- ((NUMERAL (BIT0 (BIT1 _0))) = (SUC (NUMERAL (BIT1 _0))))"}
{"_id": "2931", "title": "INT_NOT_EQ", "text": "|- (!x. (!y. ((~ (x = y)) = (((int_lt x) y) \\/ ((int_lt y) x)))))"}
{"_id": "2932", "title": "Nonlin_def.compose6", "text": "|- (!f. (!p1. (!p2. (!p3. (!p4. (!p5. (!p6. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((((((compose6 f) p1) p2) p3) p4) p5) p6) x1) x2) x3) x4) x5) x6) = ((((((f ((((((p1 x1) x2) x3) x4) x5) x6)) ((((((p2 x1) x2) x3) x4) x5) x6)) ((((((p3 x1) x2) x3) x4) x5) x6)) ((((((p4 x1) x2) x3) x4) x5) x6)) ((((((p5 x1) x2) x3) x4) x5) x6)) ((((((p6 x1) x2) x3) x4) x5) x6))))))))))))))))"}
{"_id": "2933", "title": "LE", "text": "|- ((!m. ((m <= (NUMERAL _0)) = (m = (NUMERAL _0)))) /\\ (!m. (!n. ((m <= (SUC n)) = ((m = (SUC n)) \\/ (m <= n))))))"}
{"_id": "2934", "title": "LOWDIM_SUBSET_HYPERPLANE", "text": "|- (!s. (((dim s) < (dimindex UNIV)) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\\ ((span s) SUBSET (GSPEC (\\GEN%PVAR%183. (?x. (((SETSPEC GEN%PVAR%183) (((dot a) x) = (real_of_num (NUMERAL _0)))) x)))))))))"}
{"_id": "2935", "title": "Tame_lemmas.fully_surrounded_tau_eq_tauq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> ((LET (\\w'. (LET_END ((LET (\\u. (LET_END ((LET (\\y1. (LET_END ((LET (\\y2. (LET_END ((LET (\\y3. (LET_END ((LET (\\y4. (LET_END ((LET (\\y5. (LET_END ((LET (\\y6. (LET_END ((LET (\\y7. (LET_END ((LET (\\y8. (LET_END ((LET (\\y9. (LET_END ((tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) (v , w))))) = (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9))))) (distance (w , u)))))) (distance (w' , u)))))) (vector_norm u))))) (distance (v , w)))))) (distance (v , w')))))) (distance (w , w')))))) (vector_norm w'))))) (vector_norm w))))) (vector_norm v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))))))))"}
{"_id": "2936", "title": "Seq.perm_eq_size", "text": "|- (!s1. (!s2. (((perm_eq s1) s2) ==> ((sizel s1) = (sizel s2)))))"}
{"_id": "2937", "title": "NOT_INSERT_EMPTY", "text": "|- (!x. (!s. (~ ((x INSERT s) = EMPTY))))"}
{"_id": "2938", "title": "REAL_NEGLIGIBLE_SUBSET", "text": "|- (!s. (!t. (((real_negligible s) /\\ (t SUBSET s)) ==> (real_negligible t))))"}
{"_id": "2939", "title": "REALLIM_NULL_POW_EQ", "text": "|- (!net. (!f. (!n. ((~ (n = (NUMERAL _0))) ==> ((((---> (\\x. ((real_pow (f x)) n))) (real_of_num (NUMERAL _0))) net) = (((---> f) (real_of_num (NUMERAL _0))) net))))))"}
{"_id": "2940", "title": "ABSOLUTELY_INTEGRABLE_ADD", "text": "|- (!f. (!g. (!s. ((((absolutely_integrable_on f) s) /\\ ((absolutely_integrable_on g) s)) ==> ((absolutely_integrable_on (\\x. ((vector_add (f x)) (g x)))) s)))))"}
{"_id": "2941", "title": "CARD_EQ_COUNTABLE_SUBSETS_SUBREAL", "text": "|- (!s. (((INFINITE s) /\\ (s <=_c UNIV)) ==> ((GSPEC (\\GEN%PVAR%72. (?t. (((SETSPEC GEN%PVAR%72) ((t SUBSET s) /\\ (COUNTABLE t))) t)))) =_c UNIV)))"}
{"_id": "2942", "title": "EDELSTEIN_FIX", "text": "|- (!f. (!s. (((compact s) /\\ ((~ (s = EMPTY)) /\\ ((((IMAGE f) s) SUBSET s) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ (~ (x = y)))) ==> ((real_lt (distance ((f x) , (f y)))) (distance (x , y))))))))) ==> (?!x. ((x IN s) /\\ ((f x) = x))))))"}
{"_id": "2943", "title": "IN_BALL_0", "text": "|- (!x. (!e. ((x IN (ball ((vec (NUMERAL _0)) , e))) = ((real_lt (vector_norm x)) e))))"}
{"_id": "2944", "title": "uniformly_continuous_on", "text": "|- (!s. (!f. (((uniformly_continuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (!x'. (((x IN s) /\\ ((x' IN s) /\\ ((real_lt (distance (x' , x))) d))) ==> ((real_lt (distance ((f x') , (f x)))) e)))))))))))"}
{"_id": "2945", "title": "Tame_lemmas.dart_leads_into_eq1", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w) = ((((dartset_leads_into_fan (vec (NUMERAL _0))) V) E) ((IMAGE (ext_dart (V , E))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))))"}
{"_id": "2946", "title": "Lp_details.quad_3862621143_revised", "text": "|- (quad_nonlinear_v10 ==> (!y0. (!y1. (!y2. (!y3. (!y5. (!y6. (!y7. (!y8. (!y9. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y0 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y7 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y8 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y9 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL)))))))))) (((real_lt ((real_add ((((((taum y1) y2) y7) y9) y0) y6)) ((((((taum y1) y3) y7) y8) y0) y5))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) \\/ (((real_gt ((real_add ((((((taum y1) y2) y7) y9) y0) y6)) ((real_add ((real_sub ((((((taum y1) y3) y7) y8) y0) y5)) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_add ((((((dih_y y1) y2) y7) y9) y0) y6)) ((((((dih_y y1) y3) y7) y8) y0) y5))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) \\/ (((real_lt ((((((delta_y y1) y2) y7) y9) y0) y6)) (real_of_num (NUMERAL _0))) \\/ ((real_lt ((((((delta_y y1) y3) y7) y8) y0) y5)) (real_of_num (NUMERAL _0)))))))))))))))))"}
{"_id": "2947", "title": "REAL_CONVERGENT_IMP_BOUNDED", "text": "|- (!s. (!l. ((((---> s) l) sequentially) ==> (real_bounded ((IMAGE s) UNIV)))))"}
{"_id": "2948", "title": "INSIDE_SUBSET_CONVEX", "text": "|- (!s. (!c. (((convex c) /\\ (s SUBSET c)) ==> ((inside s) SUBSET c))))"}
{"_id": "2949", "title": "CONTINUOUS_SUB", "text": "|- (!f. (!g. (!net. ((((continuous f) net) /\\ ((continuous g) net)) ==> ((continuous (\\x. ((vector_sub (f x)) (g x)))) net)))))"}
{"_id": "2950", "title": "JOINPATHS_TRANSLATION", "text": "|- (!a. (!g1. (!g2. (((++ ((\\x. ((vector_add a) x)) o g1)) ((\\x. ((vector_add a) x)) o g2)) = ((\\x. ((vector_add a) x)) o ((++ g1) g2))))))"}
{"_id": "2951", "title": "subpath", "text": "|- (!g. (!v. (!u. ((((subpath u) v) g) = (\\x. (g ((vector_add u) ((% (drop ((vector_sub v) u))) x))))))))"}
{"_id": "2952", "title": "FORALL_2", "text": "|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (NUMERAL (BIT0 (BIT1 _0))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\\ (P (NUMERAL (BIT0 (BIT1 _0)))))))"}
{"_id": "2953", "title": "Hexagons.PROP_EQU_IS_SCS", "text": "|- ((is_scs_v39 s) ==> (is_scs_v39 ((scs_prop_equ_v39 s) i)))"}
{"_id": "2954", "title": "REAL_DIFFERENTIABLE_AT_LOG", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable log) (atreal x))))"}
{"_id": "2955", "title": "VECTOR_SUB_EQ", "text": "|- (!x. (!y. ((((vector_sub x) y) = (vec (NUMERAL _0))) = (x = y))))"}
{"_id": "2956", "title": "Ssrnat.ltn_exp2l", "text": "|- (!m. (!n1. (!n2. (((NUMERAL (BIT1 _0)) < m) ==> (((m EXP n1) < (m EXP n2)) = (n1 < n2))))))"}
{"_id": "2957", "title": "REAL_POW_LT", "text": "|- (!x. (!n. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_pow x) n)))))"}
{"_id": "2958", "title": "Seq.size_cons", "text": "|- (!h. (!t. ((sizel ((CONS h) t)) = (SUC (sizel t)))))"}
{"_id": "2959", "title": "Lp_ineqs_def.rhazim3_fan", "text": "|- (!V. (!E. (!x. (((rhazim3_fan (V , E)) x) = ((rhazim_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))"}
{"_id": "2960", "title": "VECTOR_SUB_RZERO", "text": "|- (((vector_sub x) (vec (NUMERAL _0))) = x)"}
{"_id": "2961", "title": "Tame_defs.GRAPH", "text": "|- (!fs. (!n. (!f. (!h. (((((Graph fs) n) f) h) = (fs , (n , (f , h))))))))"}
{"_id": "2962", "title": "DROP_NEG", "text": "|- (!x. ((drop (vector_neg x)) = (real_neg (drop x))))"}
{"_id": "2963", "title": "COMPLEX_INV_INV", "text": "|- (!x. ((complex_inv (complex_inv x)) = x))"}
{"_id": "2964", "title": "CONTINUOUS_ON_EQ", "text": "|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((continuous_on f) s)) ==> ((continuous_on g) s)))))"}
{"_id": "2965", "title": "RIGHT_OR_FORALL_THM", "text": "|- (!P. (!Q. ((P \\/ (!x. (Q x))) = (!x. (P \\/ (Q x))))))"}
{"_id": "2966", "title": "int_gcd", "text": "|- (!a. (!b. (((int_le (int_of_num (NUMERAL _0))) (int_gcd (a , b))) /\\ (((int_divides (int_gcd (a , b))) a) /\\ (((int_divides (int_gcd (a , b))) b) /\\ (?x. (?y. ((int_gcd (a , b)) = ((int_add ((int_mul a) x)) ((int_mul b) y))))))))))"}
{"_id": "2967", "title": "PCROSS_EMPTY", "text": "|- ((!s. ((s PCROSS EMPTY) = EMPTY)) /\\ (!t. ((EMPTY PCROSS t) = EMPTY)))"}
{"_id": "2968", "title": "Add_triangle.dart1_switch", "text": "|- (!V. (!E. (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) = ((w , v) IN (dart1_of_fan (V , E))))))))"}
{"_id": "2969", "title": "REAL_LE_SQUARE", "text": "|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((real_mul x) x)))"}
{"_id": "2970", "title": "MULT_0", "text": "|- (!m. ((m * (NUMERAL _0)) = (NUMERAL _0)))"}
{"_id": "2971", "title": "FRONTIER_UNIONS_SUBSET_CLOSURE", "text": "|- (!f. ((frontier (UNIONS f)) SUBSET (closure (UNIONS (GSPEC (\\GEN%PVAR%1825. (?t. (((SETSPEC GEN%PVAR%1825) (t IN f)) (frontier t)))))))))"}
{"_id": "2972", "title": "SETDIST_TRANSLATION", "text": "|- (!a. (!s. (!t. ((setdist (((IMAGE (\\x. ((vector_add a) x))) s) , ((IMAGE (\\x. ((vector_add a) x))) t))) = (setdist (s , t))))))"}
{"_id": "2973", "title": "SUM_OVER_PERMUTATIONS_INSERT", "text": "|- (!f. (!a. (!s. (((FINITE s) /\\ (~ (a IN s))) ==> (((sum (GSPEC (\\GEN%PVAR%295. (?p. (((SETSPEC GEN%PVAR%295) ((permutes p) (a INSERT s))) p))))) f) = ((sum (a INSERT s)) (\\b. ((sum (GSPEC (\\GEN%PVAR%296. (?p. (((SETSPEC GEN%PVAR%296) ((permutes p) s)) p))))) (\\q. (f ((swap (a , b)) o q)))))))))))"}
{"_id": "2974", "title": "CONIC_INTERS", "text": "|- ((!s. ((s IN f) ==> (conic s))) ==> (conic (INTERS f)))"}
{"_id": "2975", "title": "COMPLEX_VEC_0", "text": "|- ((vec (NUMERAL _0)) = (Cx (real_of_num (NUMERAL _0))))"}
{"_id": "2976", "title": "Seq.flatten0", "text": "|- ((flatten NIL) = NIL)"}
{"_id": "2977", "title": "REAL_MEASURABLE_ON_UNIV", "text": "|- (((real_measurable_on (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) UNIV) = ((real_measurable_on f) s))"}
{"_id": "2978", "title": "OPEN_SET_COCOUNTABLE_COORDINATES", "text": "|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (COUNTABLE (UNIV DIFF (GSPEC (\\GEN%PVAR%1866. (?x. (((SETSPEC GEN%PVAR%1866) ((P i) x)) x)))))))) ==> (!s. (((open s) /\\ (~ (s = EMPTY))) ==> (?x. ((x IN s) /\\ (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((P i) (x $ i))))))))))"}
{"_id": "2979", "title": "Appendix.periodic2", "text": "|- (!k. (!f. (((periodic2 f) k) = (!i. (!j. ((((f (i + k)) j) = ((f i) j)) /\\ (((f i) (j + k)) = ((f i) j))))))))"}
{"_id": "2980", "title": "NSUM_SUPPORT", "text": "|- (!f. (!s. (((nsum (((support +) f) s)) f) = ((nsum s) f))))"}
{"_id": "2981", "title": "CONVEX_CONE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((convex_cone ((IMAGE f) s)) = (convex_cone s)))))"}
{"_id": "2982", "title": "IS_REALINTERVAL_CONNECTED", "text": "|- (!s. ((is_realinterval s) = (connected ((IMAGE lift) s))))"}
{"_id": "2983", "title": "Functional_equation.scalar6", "text": "|- (!f. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!r. (((((((((scalar6 f) r) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) r))))))))))"}
{"_id": "2984", "title": "LOCALLY_COMPACT_OPEN_INTER_CLOSURE", "text": "|- (!s. (((locally compact) s) ==> (?t. ((open t) /\\ (s = (t INTER (closure s)))))))"}
{"_id": "2985", "title": "RE_ADD", "text": "|- (!x. (!y. ((Re ((vector_add x) y)) = ((real_add (Re x)) (Re y)))))"}
{"_id": "2986", "title": "Conforming.pr23", "text": "|- (pr23 = (GABS (\\f. (!x. (!y. (!z. (!t. ((GEQ (f (x , (y , (z , t))))) (y , z)))))))))"}
{"_id": "2987", "title": "locally", "text": "|- (!s. (!P. (((locally P) s) = (!w. (!x. ((((open_in ((subtopology euclidean) s)) w) /\\ (x IN w)) ==> (?u. (?v. (((open_in ((subtopology euclidean) s)) u) /\\ ((P v) /\\ ((x IN u) /\\ ((u SUBSET v) /\\ (v SUBSET w)))))))))))))"}
{"_id": "2988", "title": "CLOSURE_SUBSET_EQ", "text": "|- (!s. (((closure s) SUBSET s) = (closed s)))"}
{"_id": "2989", "title": "List_hypermap.components_hypermap_of_list", "text": "|- (!L. ((good_list L) ==> (((dart (hypermap_of_list L)) = (darts_of_list L)) /\\ (((edge_map (hypermap_of_list L)) = (e_list_ext L)) /\\ (((node_map (hypermap_of_list L)) = (n_list_ext L)) /\\ ((face_map (hypermap_of_list L)) = (f_list_ext L)))))))"}
{"_id": "2990", "title": "Sphere.x1_delta_x", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((x1_delta_x x1) x2) x3) x4) x5) x6) = ((real_mul x1) ((((((delta_x x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "2991", "title": "COVERING_SPACE_FIBRE_NO_LIMPT", "text": "|- (!p. (!c. (!s. (!a. (!b. ((((covering_space (c , p)) s) /\\ (a IN c)) ==> (~ ((limit_point_of a) (GSPEC (\\GEN%PVAR%2072. (?x. (((SETSPEC GEN%PVAR%2072) ((x IN c) /\\ ((p x) = b))) x))))))))))))"}
{"_id": "2992", "title": "fl", "text": "|- (!l. (!x. (((fl l) x) = (?y. ((l (x , y)) \\/ (l (y , x)))))))"}
{"_id": "2993", "title": "SUMMABLE_EQ", "text": "|- (!f. (!g. (!k. (((!x. ((x IN k) ==> ((f x) = (g x)))) /\\ ((summable k) f)) ==> ((summable k) g)))))"}
{"_id": "2994", "title": "MATRIX_INV_COVARIANCE_LMUL", "text": "|- (!A. (((matrix_mul (transp A)) (matrix_inv ((matrix_mul A) (transp A)))) = (matrix_inv A)))"}
{"_id": "2995", "title": "Hypermap_iso.bij_alt", "text": "|- (!f. (!s. (!t. ((((BIJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\\ ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) /\\ (!y. ((y IN t) ==> (?x. ((x IN s) /\\ ((f x) = y)))))))))))"}
{"_id": "2996", "title": "IS_INTERVAL_CONNECTED_1", "text": "|- (!s. ((is_interval s) = (connected s)))"}
{"_id": "2997", "title": "REAL_SERIES_ADD", "text": "|- (!x. (!x0. (!y. (!y0. (!s. (((((real_sums x) x0) s) /\\ (((real_sums y) y0) s)) ==> (((real_sums (\\n. ((real_add (x n)) (y n)))) ((real_add x0) y0)) s)))))))"}
{"_id": "2998", "title": "Hypermap_iso.darts_of_fan_disj", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((DISJOINT (dart1_of_fan (V , E))) (GSPEC (\\GEN%PVAR%7629. (?v. (((SETSPEC GEN%PVAR%7629) ((v IN V) /\\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))))))"}
{"_id": "2999", "title": "FINITE_DELETE", "text": "|- (!s. (!x. ((FINITE (s DELETE x)) = (FINITE s))))"}
{"_id": "3000", "title": "HOMEOMORPHISM_INTO_1D", "text": "|- (!f. (!s. (!t. (((path_connected s) /\\ (((continuous_on f) s) /\\ ((((IMAGE f) s) = t) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y))))))) ==> (?g. ((homeomorphism (s , t)) (f , g)))))))"}
{"_id": "3001", "title": "BOUNDED_CONVEX_HULL", "text": "|- (!s. ((bounded s) ==> (bounded ((hull convex) s))))"}
{"_id": "3002", "title": "ABSOLUTELY_REAL_INTEGRABLE_CONST", "text": "|- (!a. (!b. (!c. ((absolutely_real_integrable_on (\\x. c)) (closed_real_interval ((CONS (a , b)) NIL))))))"}
{"_id": "3003", "title": "Tame_general.IN_ESTD", "text": "|- (!V. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN (ESTD V)) = ((v IN V) /\\ ((w IN V) /\\ ((~ (v = w)) /\\ ((real_le (distance (v , w))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))"}
{"_id": "3004", "title": "AFF_DIM_SEGMENT", "text": "|- ((!a. (!b. ((aff_dim (closed_segment ((CONS (a , b)) NIL))) = (((COND (a = b)) (int_of_num (NUMERAL _0))) (int_of_num (NUMERAL (BIT1 _0))))))) /\\ (!a. (!b. ((aff_dim (open_segment (a , b))) = (((COND (a = b)) (int_neg (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "3005", "title": "Seq2.list_sum_nil", "text": "|- (!f. (((list_sum NIL) f) = (real_of_num (NUMERAL _0))))"}
{"_id": "3006", "title": "Conforming.SIGMA_FAN_OF_FANADD_AT_POINT4", "text": "|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\\ ((FAN (x , (V , E1))) /\\ ((fan80 (x , (V , E))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ ((~ (u = w1)) /\\ (((v INSERT (w1 INSERT EMPTY)) IN E) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))) ==> ((((((sigma_fan x) V) E1) v) w1) = (((((sigma_fan x) V) E) v) w1))))))))))"}
{"_id": "3007", "title": "COUNTABLE_INTEGER", "text": "|- (COUNTABLE integer)"}
{"_id": "3008", "title": "ANR_PCROSS", "text": "|- (!s. (!t. (((ANR s) /\\ (ANR t)) ==> (ANR (s PCROSS t)))))"}
{"_id": "3009", "title": "Sphere.y_of_x", "text": "|- (!fx. (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((((((y_of_x fx) y1) y2) y3) y4) y5) y6) = ((((((fx ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6))))))))))"}
{"_id": "3010", "title": "Tame_general.sol0_over_pi_EQ_const1", "text": "|- (((real_div sol0) pi) = const1)"}
{"_id": "3011", "title": "LIM_AT_ZERO", "text": "|- (!f. (!l. (!a. ((((--> f) l) (at a)) = (((--> (\\x. (f ((vector_add a) x)))) l) (at (vec (NUMERAL _0))))))))"}
{"_id": "3012", "title": "NOT_ON_PATH_BALL", "text": "|- (!g. (!z. (((path g) /\\ (~ (z IN (path_image g)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (((ball (z , e)) INTER (path_image g)) = EMPTY))))))"}
{"_id": "3013", "title": "Miqmcsn.K_SCS_3M1_prime", "text": "|- ((scs_k_v39 scs_3M1_prime) = (NUMERAL (BIT1 (BIT1 _0))))"}
{"_id": "3014", "title": "Real_ext.REAL_MK_POS_EXP", "text": "|- (!x. ((real_lt (real_of_num (NUMERAL _0))) (exp x)))"}
{"_id": "3015", "title": "REAL_INTERVAL_INTERVAL", "text": "|- (((closed_real_interval ((CONS (a , b)) NIL)) = ((IMAGE drop) (closed_interval ((CONS ((lift a) , (lift b))) NIL)))) /\\ ((open_real_interval (a , b)) = ((IMAGE drop) (open_interval ((lift a) , (lift b))))))"}
{"_id": "3016", "title": "REAL_EQ_IMP_LE", "text": "|- (!x. (!y. ((x = y) ==> ((real_le x) y))))"}
{"_id": "3017", "title": "DIFF_EMPTY", "text": "|- (!s. ((s DIFF EMPTY) = s))"}
{"_id": "3018", "title": "SUBSET_TRANS", "text": "|- (!s. (!t. (!u. (((s SUBSET t) /\\ (t SUBSET u)) ==> (s SUBSET u)))))"}
{"_id": "3019", "title": "Seq.mem_subseq", "text": "|- (!s1. (!s2. (((subseq s1) s2) ==> (!x. (((MEM x) s1) ==> ((MEM x) s2))))))"}
{"_id": "3020", "title": "Functional_equation.ldih_all_atn", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((ldih_x x1) x2) x3) x4) x5) x6) = ((((((ldihatn_x x1) x2) x3) x4) x5) x6)) /\\ ((((((((ldih2_x x1) x2) x3) x4) x5) x6) = ((((((ldih2atn_x x1) x2) x3) x4) x5) x6)) /\\ ((((((((ldih3_x x1) x2) x3) x4) x5) x6) = ((((((ldih3atn_x x1) x2) x3) x4) x5) x6)) /\\ (((((((ldih6_x x1) x2) x3) x4) x5) x6) = ((((((ldih6atn_x x1) x2) x3) x4) x5) x6))))))))))))"}
{"_id": "3021", "title": "SERIES_COMPARISON", "text": "|- (!f. (!g. (!s. (((?l. (((sums (lift o g)) l) s)) /\\ (?N. (!n. (((n >= N) /\\ (n IN s)) ==> ((real_le (vector_norm (f n))) (g n)))))) ==> (?l. (((sums f) l) s))))))"}
{"_id": "3022", "title": "REAL_MUL_LNEG", "text": "|- (!x. (!y. (((real_mul (real_neg x)) y) = (real_neg ((real_mul x) y)))))"}
{"_id": "3023", "title": "BOUNDED_PARTIAL_SUMS", "text": "|- (!f. (!k. ((bounded (GSPEC (\\GEN%PVAR%1123. (?n. (((SETSPEC GEN%PVAR%1123) (n IN UNIV)) ((vsum (k .. n)) f)))))) ==> (bounded (GSPEC (\\GEN%PVAR%1124. (?m. (?n. (((SETSPEC GEN%PVAR%1124) ((m IN UNIV) /\\ (n IN UNIV))) ((vsum (m .. n)) f))))))))))"}
{"_id": "3024", "title": "HOMOTOPIC_PATHS_REFL", "text": "|- (!s. (!p. ((((homotopic_paths s) p) p) = ((path p) /\\ ((path_image p) SUBSET s)))))"}
{"_id": "3025", "title": "TC_TRANS", "text": "|- (!R. (!x. (!y. (!z. (((((TC R) x) y) /\\ (((TC R) y) z)) ==> (((TC R) x) z))))))"}
{"_id": "3026", "title": "SUBSET_HULL", "text": "|- (!P. (!s. (!t. ((P t) ==> ((((hull P) s) SUBSET t) = (s SUBSET t))))))"}
{"_id": "3027", "title": "REAL_POLYNOMIAL_FUNCTION_EXPLICIT_UNIV", "text": "|- (!f. ((real_polynomial_function f) = (?a. ((FINITE (GSPEC (\\GEN%PVAR%5014. (?k. (((SETSPEC GEN%PVAR%5014) (~ ((a k) = (real_of_num (NUMERAL _0))))) k))))) /\\ (f = (\\x. ((sum UNIV) (\\k. ((real_mul (a k)) ((product ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. ((real_pow (x $ i)) (k $ i)))))))))))))"}
{"_id": "3028", "title": "POLYTOPE_IMP_CLOSED", "text": "|- (!s. ((polytope s) ==> (closed s)))"}
{"_id": "3029", "title": "LEBESGUE_MEASURABLE_UNIV", "text": "|- (lebesgue_measurable UNIV)"}
{"_id": "3030", "title": "has_vector_derivative", "text": "|- (!f. (!f'. (!net. ((((has_vector_derivative f) f') net) = (((has_derivative f) (\\x. ((% (drop x)) f'))) net)))))"}
{"_id": "3031", "title": "AFF_DIM_EMPTY", "text": "|- ((aff_dim EMPTY) = (int_neg (int_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "3032", "title": "DET_2", "text": "|- (!A. ((det A) = ((real_sub ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))))))"}
{"_id": "3033", "title": "CONTINUOUS_GE_ON_CLOSURE", "text": "|- (!f. (!s. (!a. ((((continuous_on (lift o f)) (closure s)) /\\ (!x. ((x IN s) ==> ((real_le a) (f x))))) ==> (!x. ((x IN (closure s)) ==> ((real_le a) (f x))))))))"}
{"_id": "3034", "title": "VITALI_COVERING_LEMMA_CBALLS_BALLS", "text": "|- (!a. (!r. (!k. (!B. ((!i. ((i IN k) ==> (((real_lt (real_of_num (NUMERAL _0))) (r i)) /\\ ((real_le (r i)) B)))) ==> (?c. ((COUNTABLE c) /\\ ((c SUBSET k) /\\ (((pairwise (\\i. (\\j. ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))))) c) /\\ (!i. ((i IN k) ==> (?j. ((j IN c) /\\ ((~ ((DISJOINT (cball ((a i) , (r i)))) (cball ((a j) , (r j))))) /\\ ((cball ((a i) , (r i))) SUBSET (ball ((a j) , ((real_mul (real_of_num (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (r j)))))))))))))))))))"}
{"_id": "3035", "title": "RSC", "text": "|- (!R. ((RSC R) = (RC (SC R))))"}
{"_id": "3036", "title": "ARG_EQ_0", "text": "|- (!z. (((Arg z) = (real_of_num (NUMERAL _0))) = ((real z) /\\ ((real_le (real_of_num (NUMERAL _0))) (Re z)))))"}
{"_id": "3037", "title": "Fan_misc.dart1_of_fan", "text": "|- (!V. (!E. ((dart1_of_fan (V , E)) = (GSPEC (\\GEN%PVAR%5545. (?v. (?w. (((SETSPEC GEN%PVAR%5545) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w)))))))))"}
{"_id": "3038", "title": "Vol1.VOLUME_PROPS_MEASURABLE", "text": "|- (!C. ((measurable C) ==> ((real_ge (measure C)) (real_of_num (NUMERAL _0)))))"}
{"_id": "3039", "title": "OPEN_IN_RELATIVE_INTERIOR", "text": "|- (!s. ((open_in ((subtopology euclidean) ((hull affine) s))) (relative_interior s)))"}
{"_id": "3040", "title": "REAL_MUL_RNEG", "text": "|- (!x. (!y. (((real_mul x) (real_neg y)) = (real_neg ((real_mul x) y)))))"}
{"_id": "3041", "title": "Ssrnat.minnA", "text": "|- (!m1. (!m2. (!m3. (((minn m1) ((minn m2) m3)) = ((minn ((minn m1) m2)) m3)))))"}
{"_id": "3042", "title": "MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GE", "text": "|- (!f. (((measurable_on f) UNIV) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV))) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3514. (?x. (((SETSPEC GEN%PVAR%3514) ((real_ge ((f x) $ k)) a)) x))))))))))"}
{"_id": "3043", "title": "IM_II", "text": "|- ((Im ii) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "3044", "title": "PATHSTART_SUBPATH", "text": "|- (!u. (!v. (!g. ((pathstart (((subpath u) v) g)) = (g u)))))"}
{"_id": "3045", "title": "FINITE_INTER_NUMSEG", "text": "|- (!s. (!m. (!n. (FINITE (s INTER (m .. n))))))"}
{"_id": "3046", "title": "Appendix.scs_4M5", "text": "|- (scs_4M5' = ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) NIL))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))"}
{"_id": "3047", "title": "Real_ext.REAL_SV_LMUL_0", "text": "|- (!x. (((real_mul (real_of_num (NUMERAL _0))) x) = (real_of_num (NUMERAL _0))))"}
{"_id": "3048", "title": "Real_ext.REAL_LE", "text": "|- (!m. (!n. (((real_le (real_of_num m)) (real_of_num n)) = (m <= n))))"}
{"_id": "3049", "title": "HAS_BOUNDED_VARIATION_ON_SUB", "text": "|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\\x. ((vector_sub (f x)) (g x)))) s)))))"}
{"_id": "3050", "title": "Nonlinear_lemma.halfbump_x", "text": "|- (!x. ((halfbump_x x) = (bump ((real_div (sqrt x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "3051", "title": "AR_DEFORMATION_RETRACT_OF_CONTRACTIBLE", "text": "|- (!s. (!t. (((contractible s) /\\ ((AR t) /\\ ((closed_in ((subtopology euclidean) s)) t))) ==> (?r. (((((homotopic_with (\\x. T)) (s , s)) (\\x. x)) r) /\\ ((retraction (s , t)) r))))))"}
{"_id": "3052", "title": "LIM_NULL", "text": "|- (!net. (!f. (!l. ((((--> f) l) net) = (((--> (\\x. ((vector_sub (f x)) l))) (vec (NUMERAL _0))) net)))))"}
{"_id": "3053", "title": "REAL_ABS_NEG", "text": "|- (!x. ((real_abs (real_neg x)) = (real_abs x)))"}
{"_id": "3054", "title": "Functional_equation.two6", "text": "|- (two6 = (constant6 (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))"}
{"_id": "3055", "title": "FRONTIER_FRONTIER", "text": "|- (!s. (((open s) \\/ (closed s)) ==> ((frontier (frontier s)) = (frontier s))))"}
{"_id": "3056", "title": "OPEN_BALL", "text": "|- (!x. (!e. (open (ball (x , e)))))"}
{"_id": "3057", "title": "Fan_misc.inverse_sigma_fan", "text": "|- (!x. (!V. (!E. (!v. (((((inverse_sigma_fan x) V) E) v) = (inverse ((((extension_sigma_fan x) V) E) v)))))))"}
{"_id": "3058", "title": "STC_INC", "text": "|- (!R. (!x. (!y. (((R x) y) ==> (((STC R) x) y)))))"}
{"_id": "3059", "title": "CARD_EXP_FINITE", "text": "|- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> (FINITE ((^_c s) t)))))"}
{"_id": "3060", "title": "RE_CLOG", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((Re (clog z)) = (log (vector_norm z)))))"}
{"_id": "3061", "title": "Hypermap_iso.hyp_iso_SURJ", "text": "|- (!f. (!H. (!G. (((hyp_iso f) (H , G)) ==> (((SURJ f) (dart H)) (dart G))))))"}
{"_id": "3062", "title": "FINITE_RULES", "text": "|- ((FINITE EMPTY) /\\ (!x. (!s. ((FINITE s) ==> (FINITE (x INSERT s))))))"}
{"_id": "3063", "title": "AR_IMP_CONTRACTIBLE", "text": "|- (!s. ((AR s) ==> (contractible s)))"}
{"_id": "3064", "title": "REAL_ENTIRE", "text": "|- (!x. (!y. ((((real_mul x) y) = (real_of_num (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) \\/ (y = (real_of_num (NUMERAL _0)))))))"}
{"_id": "3065", "title": "vector_neg", "text": "|- (!x. ((vector_neg x) = (lambdai. (real_neg (x $ i)))))"}
{"_id": "3066", "title": "Sphere.bis_le", "text": "|- (!u. (!v. (((bis_le u) v) = (GSPEC (\\GEN%PVAR%5246. (?x. (((SETSPEC GEN%PVAR%5246) ((real_le (distance (x , u))) (distance (x , v)))) x)))))))"}
{"_id": "3067", "title": "Lp_ineqs_proofs.y78_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((LET (\\w'. (LET_END ((LET (\\u. (LET_END ((((y8_fan (V , E)) (v , w)) = (distance (w' , u))) /\\ (((y7_fan (V , E)) (v , w)) = (vector_norm u)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))))"}
{"_id": "3068", "title": "SUBSET_LIFT_IMAGE", "text": "|- (!s. (!t. ((((IMAGE lift) s) SUBSET ((IMAGE lift) t)) = (s SUBSET t))))"}
{"_id": "3069", "title": "MAP2", "text": "|- (((((MAP2 f) NIL) NIL) = NIL) /\\ ((((MAP2 f) ((CONS h1) t1)) ((CONS h2) t2)) = ((CONS ((f h1) h2)) (((MAP2 f) t1) t2))))"}
{"_id": "3070", "title": "DIM_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((dim ((IMAGE f) s)) = (dim s)))))"}
{"_id": "3071", "title": "closed_in", "text": "|- (!top. (!s. (((closed_in top) s) = ((s SUBSET (topspace top)) /\\ ((open_in top) ((topspace top) DIFF s))))))"}
{"_id": "3072", "title": "Appendix.scs_3T3", "text": "|- (scs_3T3 = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 NIL) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 NIL) cstab) (NUMERAL (BIT1 (BIT1 _0))))))"}
{"_id": "3073", "title": "REAL_LT_MAX", "text": "|- (!x. (!y. (!z. (((real_lt z) ((real_max x) y)) = (((real_lt z) x) \\/ ((real_lt z) y))))))"}
{"_id": "3074", "title": "UNCOUNTABLE_CONTAINS_LIMIT_POINT", "text": "|- (!s. ((~ (COUNTABLE s)) ==> (?x. ((x IN s) /\\ ((limit_point_of x) s)))))"}
{"_id": "3075", "title": "IM_CX", "text": "|- (!x. ((Im (Cx x)) = (real_of_num (NUMERAL _0))))"}
{"_id": "3076", "title": "HAS_SIZE_DIFF", "text": "|- (!s. (!t. (!m. (!n. (((s HAS_SIZE m) /\\ ((t HAS_SIZE n) /\\ (t SUBSET s))) ==> ((s DIFF t) HAS_SIZE (m - n)))))))"}
{"_id": "3077", "title": "REAL_OPEN", "text": "|- (!s. ((real_open s) = (open ((IMAGE lift) s))))"}
{"_id": "3078", "title": "real_derivative", "text": "|- (!f. (!x. (((real_derivative f) x) = (@f'. (((has_real_derivative f) f') (atreal x))))))"}
{"_id": "3079", "title": "HAUSDIST_TRANSLATION", "text": "|- (!a. (!s. (!t. ((hausdist (((IMAGE (\\x. ((vector_add a) x))) s) , ((IMAGE (\\x. ((vector_add a) x))) t))) = (hausdist (s , t))))))"}
{"_id": "3080", "title": "FRONTIER_INTERIOR_SUBSET", "text": "|- (!s. ((frontier (interior s)) SUBSET (frontier s)))"}
{"_id": "3081", "title": "ABSOLUTELY_INTEGRABLE_BOUNDED_SETVARIATION", "text": "|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((has_bounded_setvariation_on (\\k. ((integral k) f))) s))))"}
{"_id": "3082", "title": "REAL_MIN_MAX", "text": "|- (!x. (!y. (((real_min x) y) = (real_neg ((real_max (real_neg x)) (real_neg y))))))"}
{"_id": "3083", "title": "Nonlin_def.flat_term2_234_x", "text": "|- (!d. (!x5. (!x6. ((((flat_term2_234_x d) x5) x6) = (uni (flat_term_x , (((edge2_234_x d) x5) x6)))))))"}
{"_id": "3084", "title": "RSTC_INC", "text": "|- (!R. (!x. (!y. (((R x) y) ==> (((RSTC R) x) y)))))"}
{"_id": "3085", "title": "AFF_DIM_NONEMPTY_INTERIOR", "text": "|- (!s. ((~ ((interior s) = EMPTY)) ==> ((aff_dim s) = (int_of_num (dimindex UNIV)))))"}
{"_id": "3086", "title": "REAL_MEASURABLE_MEASURABLE", "text": "|- (!s. ((real_measurable s) = (measurable ((IMAGE lift) s))))"}
{"_id": "3087", "title": "Seq2.perm_eq_cons_delete1", "text": "|- (!x. (!s. (((MEM x) s) ==> ((perm_eq ((CONS x) ((delete1 x) s))) s))))"}
{"_id": "3088", "title": "complex_mul", "text": "|- (!w. (!z. (((complex_mul w) z) = (complex (((real_sub ((real_mul (Re w)) (Re z))) ((real_mul (Im w)) (Im z))) , ((real_add ((real_mul (Re w)) (Im z))) ((real_mul (Im w)) (Re z))))))))"}
{"_id": "3089", "title": "COPLANAR_TRANSLATION_EQ", "text": "|- (!a. (!s. ((coplanar ((IMAGE (\\x. ((vector_add a) x))) s)) = (coplanar s))))"}
{"_id": "3090", "title": "SUBSPACE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((subspace ((IMAGE f) s)) = (subspace s)))))"}
{"_id": "3091", "title": "Constants_approx.INTERVAL_MUL", "text": "|- ((((interval_arith x) (a , b)) /\\ ((interval_arith y) (c , d))) ==> ((interval_arith ((real_mul x) y)) (((real_min ((real_min ((real_mul a) c)) ((real_mul a) d))) ((real_min ((real_mul b) c)) ((real_mul b) d))) , ((real_max ((real_max ((real_mul a) c)) ((real_mul a) d))) ((real_max ((real_mul b) c)) ((real_mul b) d))))))"}
{"_id": "3092", "title": "WF_FINITE", "text": "|- (!<<. (((!x. (~ ((<< x) x))) /\\ ((!x. (!y. (!z. ((((<< x) y) /\\ ((<< y) z)) ==> ((<< x) z))))) /\\ (!x. (FINITE (GSPEC (\\GEN%PVAR%149. (?y. (((SETSPEC GEN%PVAR%149) ((<< y) x)) y)))))))) ==> (WF <<)))"}
{"_id": "3093", "title": "REAL_LT_ADD1", "text": "|- (!x. (!y. (((real_le x) y) ==> ((real_lt x) ((real_add y) (real_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "3094", "title": "BOUNDED_CLOSURE_EQ", "text": "|- (!s. ((bounded (closure s)) = (bounded s)))"}
{"_id": "3095", "title": "DENSE_IMP_PERFECT", "text": "|- (!s. (((closure s) = UNIV) ==> (!x. ((x IN s) ==> ((limit_point_of x) s)))))"}
{"_id": "3096", "title": "OPEN_IN_SUBSET_TRANS", "text": "|- (!s. (!t. (!u. ((((open_in ((subtopology euclidean) u)) s) /\\ ((s SUBSET t) /\\ (t SUBSET u))) ==> ((open_in ((subtopology euclidean) t)) s)))))"}
{"_id": "3097", "title": "Lp_ineqs_proofs.delta_y_not_neg", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (~ ((real_lt ((((((delta_y (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d))) (real_of_num (NUMERAL _0))))))))))))"}
{"_id": "3098", "title": "CONTINUOUS_ON_COMPONENTS_OPEN", "text": "|- (!f. (!s. (((open s) /\\ (!c. ((c IN (components s)) ==> ((continuous_on f) c)))) ==> ((continuous_on f) s))))"}
{"_id": "3099", "title": "int_gt", "text": "|- (!x. (!y. (((int_gt x) y) = ((real_gt (real_of_int x)) (real_of_int y)))))"}
{"_id": "3100", "title": "REAL_POW_MUL", "text": "|- (!x. (!y. (!n. (((real_pow ((real_mul x) y)) n) = ((real_mul ((real_pow x) n)) ((real_pow y) n))))))"}
{"_id": "3101", "title": "REAL_INF_UNIQUE", "text": "|- (!s. (!b. (((!x. ((x IN s) ==> ((real_le b) x))) /\\ (!b'. (((real_lt b) b') ==> (?x. ((x IN s) /\\ ((real_lt x) b')))))) ==> ((inf s) = b))))"}
{"_id": "3102", "title": "HOMOTOPY_EQUIVALENT_CONTRACTIBLE_SETS", "text": "|- (!s. (!t. (((contractible s) /\\ ((contractible t) /\\ ((s = EMPTY) = (t = EMPTY)))) ==> ((homotopy_equivalent s) t))))"}
{"_id": "3103", "title": "Lp_ineqs_def.y8_fan", "text": "|- (!V. (!E. (!x. (((y8_fan (V , E)) x) = ((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) x))))))"}
{"_id": "3104", "title": "PAIR_SURJECTIVE", "text": "|- (!p. (?x. (?y. (p = (x , y)))))"}
{"_id": "3105", "title": "SEPARATE_COMPACT_CLOSED", "text": "|- (!s. (!t. (((compact s) /\\ ((closed t) /\\ ((s INTER t) = EMPTY))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (!y. (((x IN s) /\\ (y IN t)) ==> ((real_le d) (distance (x , y)))))))))))"}
{"_id": "3106", "title": "Nonlinear_lemma.h0_lt_hplus", "text": "|- ((real_lt h0) hplus)"}
{"_id": "3107", "title": "Tame_defs2.hideDups", "text": "|- (((hideDups NIL) = NIL) /\\ ((hideDups ((CONS b) bs)) = ((CONS (SOME b)) ((hideDupsRec b) bs))))"}
{"_id": "3108", "title": "SUBSET_INTERS", "text": "|- (!s. (!f. ((s SUBSET (INTERS f)) = (!t. ((t IN f) ==> (s SUBSET t))))))"}
{"_id": "3109", "title": "DROP_IN_REAL_INTERVAL", "text": "|- ((!a. (!b. (!x. (((drop x) IN (closed_real_interval ((CONS (a , b)) NIL))) = (x IN (closed_interval ((CONS ((lift a) , (lift b))) NIL))))))) /\\ (!a. (!b. (!x. (((drop x) IN (open_real_interval (a , b))) = (x IN (open_interval ((lift a) , (lift b)))))))))"}
{"_id": "3110", "title": "Lp_ineqs_def.y1_fan", "text": "|- (!x. ((y1_fan x) = (vector_norm (FST x))))"}
{"_id": "3111", "title": "PATHFINISH_SUBPATH", "text": "|- (!u. (!v. (!g. ((pathfinish (((subpath u) v) g)) = (g v)))))"}
{"_id": "3112", "title": "REAL_LE_RMUL_EQ", "text": "|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le ((real_mul x) z)) ((real_mul y) z)) = ((real_le x) y))))))"}
{"_id": "3113", "title": "WOSET_WF", "text": "|- (!l. ((woset l) = ((WF (\\x. (\\y. ((l (x , y)) /\\ (~ (x = y)))))) /\\ (!x. (!y. ((((fl l) x) /\\ ((fl l) y)) ==> ((l (x , y)) \\/ (l (y , x)))))))))"}
{"_id": "3114", "title": "option_RECURSION", "text": "|- (!NONE'. (!SOME'. (?fn. (((fn NONE) = NONE') /\\ (!a. ((fn (SOME a)) = (SOME' a)))))))"}
{"_id": "3115", "title": "REAL_INTEGRAL_UNIQUE", "text": "|- (!f. (!y. (!k. ((((has_real_integral f) y) k) ==> (((real_integral k) f) = y)))))"}
{"_id": "3116", "title": "BILINEAR_DIFFERENTIABLE_ON_COMPOSE", "text": "|- (!f. (!g. (!h. (!s. ((((differentiable_on f) s) /\\ (((differentiable_on g) s) /\\ (bilinear h))) ==> ((differentiable_on (\\x. ((h (f x)) (g x)))) s))))))"}
{"_id": "3117", "title": "AR_ANR", "text": "|- (!s. ((AR s) = ((ANR s) /\\ ((contractible s) /\\ (~ (s = EMPTY))))))"}
{"_id": "3118", "title": "Hypermap_iso.bij_contracted_dart", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (((BIJ contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E)))) (dart_of_fan (V , E))))))"}
{"_id": "3119", "title": "HULL_EQ", "text": "|- (!P. (!s. ((!f. ((!s. ((s IN f) ==> (P s))) ==> (P (INTERS f)))) ==> ((((hull P) s) = s) = (P s)))))"}
{"_id": "3120", "title": "REAL_NEG_LT0", "text": "|- (!x. (((real_lt (real_neg x)) (real_of_num (NUMERAL _0))) = ((real_lt (real_of_num (NUMERAL _0))) x)))"}
{"_id": "3121", "title": "Seq.cons_uniq", "text": "|- (!x. (!s. ((uniq ((CONS x) s)) = ((~ ((MEM x) s)) /\\ (uniq s)))))"}
{"_id": "3122", "title": "HAS_BOUNDED_VARIATION_ON_NEG", "text": "|- (!f. (!s. (((has_bounded_variation_on f) s) ==> ((has_bounded_variation_on (\\x. (vector_neg (f x)))) s))))"}
{"_id": "3123", "title": "Sphere.node3_y", "text": "|- (!f. (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. ((((((((node3_y f) y1) y2) y3) y4) y5) y6) = ((((((f y3) y1) y2) y6) y4) y5)))))))))"}
{"_id": "3124", "title": "Hypermap.hypermap_lemma", "text": "|- (!H. ((FINITE (dart H)) /\\ (((permutes (edge_map H)) (dart H)) /\\ (((permutes (node_map H)) (dart H)) /\\ (((permutes (face_map H)) (dart H)) /\\ (((edge_map H) o ((node_map H) o (face_map H))) = I))))))"}
{"_id": "3125", "title": "COVERING_SPACE_IMP_SURJECTIVE", "text": "|- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> (((IMAGE p) c) = s)))))"}
{"_id": "3126", "title": "INJECTIVE_IMAGE", "text": "|- (!f. ((!s. (!t. ((((IMAGE f) s) = ((IMAGE f) t)) ==> (s = t)))) = (!x. (!y. (((f x) = (f y)) ==> (x = y))))))"}
{"_id": "3127", "title": "ROOT_MONO_LT", "text": "|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\\ ((real_lt x) y)) ==> ((real_lt ((root n) x)) ((root n) y))))))"}
{"_id": "3128", "title": "INSIDE_CONVEX", "text": "|- (!s. ((convex s) ==> ((inside s) = EMPTY)))"}
{"_id": "3129", "title": "Seq.size_eq0", "text": "|- (!s. (((sizel s) = (NUMERAL _0)) = (s = NIL)))"}
{"_id": "3130", "title": "Real_ext.REAL_MK_POS_PI", "text": "|- ((real_lt (real_of_num (NUMERAL _0))) pi)"}
{"_id": "3131", "title": "Seq.index_mem", "text": "|- (!x. (!s. ((((indexl x) s) < (sizel s)) = ((MEM x) s))))"}
{"_id": "3132", "title": "COMPACT_SING", "text": "|- (!a. (compact (a INSERT EMPTY)))"}
{"_id": "3133", "title": "DROP_SUB", "text": "|- (!x. (!y. ((drop ((vector_sub x) y)) = ((real_sub (drop x)) (drop y)))))"}
{"_id": "3134", "title": "CARD_UNION_OVERLAP_EQ", "text": "|- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> (((CARD (s UNION t)) = ((CARD s) + (CARD t))) = ((s INTER t) = EMPTY)))))"}
{"_id": "3135", "title": "Conforming.SIGMA_FAN_OF_FANADD_AT_POINT1", "text": "|- (!x. (!V. (!E. (!E1. (!v. (!u. (!w. (((FAN (x , (V , E))) /\\ ((FAN (x , (V , E1))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((fan80 (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))) ==> ((((((sigma_fan x) V) E1) v) w) = (((((sigma_fan x) V) E) v) u))))))))))"}
{"_id": "3136", "title": "CLOSURE_CLOSED", "text": "|- (!s. ((closed s) ==> ((closure s) = s)))"}
{"_id": "3137", "title": "HULL_MONO", "text": "|- (!P. (!s. (!t. ((s SUBSET t) ==> (((hull P) s) SUBSET ((hull P) t))))))"}
{"_id": "3138", "title": "IS_INTERVAL_TRANSLATION_EQ", "text": "|- (!a. (!s. ((is_interval ((IMAGE (\\x. ((vector_add a) x))) s)) = (is_interval s))))"}
{"_id": "3139", "title": "sums", "text": "|- (!s. (!f. (!l. ((((sums f) l) s) = (((--> (\\n. ((vsum (s INTER ((NUMERAL _0) .. n))) f))) l) sequentially)))))"}
{"_id": "3140", "title": "REAL_LE_NEG", "text": "|- (!x. (!y. (((real_le (real_neg x)) (real_neg y)) = ((real_le y) x))))"}
{"_id": "3141", "title": "PATH_COMPONENT_SYM", "text": "|- (!s. (!x. (!y. ((((path_component s) x) y) ==> (((path_component s) y) x)))))"}
{"_id": "3142", "title": "BOREL_TRANSLATION", "text": "|- (!a. (!s. ((borel ((IMAGE (\\x. ((vector_add a) x))) s)) = (borel s))))"}
{"_id": "3143", "title": "Tame_list.size_list_update", "text": "|- (!xs. (!i. (!v. ((sizel (((list_update xs) i) v)) = (sizel xs)))))"}
{"_id": "3144", "title": "CONVEX_RELATIVE_INTERIOR", "text": "|- (!s. ((convex s) ==> (convex (relative_interior s))))"}
{"_id": "3145", "title": "Optimize.ineq", "text": "|- ((!c. (((ineq NIL) c) = c)) /\\ (!a. (!x. (!b. (!xs. (!c. (((ineq ((CONS (a , (x , b))) xs)) c) = ((((real_le a) x) /\\ ((real_le x) b)) ==> ((ineq xs) c)))))))))"}
{"_id": "3146", "title": "CONTINUOUS_CONST", "text": "|- (!net. (!c. ((continuous (\\x. c)) net)))"}
{"_id": "3147", "title": "LIM_SUB", "text": "|- (!net. (!f. (!g. (!l. (!m. (((((--> f) l) net) /\\ (((--> g) m) net)) ==> (((--> (\\x. ((vector_sub (f x)) (g x)))) ((vector_sub l) m)) net)))))))"}
{"_id": "3148", "title": "INTERVAL_EQ_EMPTY_1", "text": "|- (!a. (!b. ((((closed_interval ((CONS (a , b)) NIL)) = EMPTY) = ((real_lt (drop b)) (drop a))) /\\ (((open_interval (a , b)) = EMPTY) = ((real_le (drop b)) (drop a))))))"}
{"_id": "3149", "title": "IN_DIFF", "text": "|- (!s. (!t. (!x. ((x IN (s DIFF t)) = ((x IN s) /\\ (~ (x IN t)))))))"}
{"_id": "3150", "title": "Add_triangle.dart_of_fan_eq_image", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((dart_of_fan (V , E)) = ((IMAGE contracted_dart) (d_fan ((vec (NUMERAL _0)) , (V , E))))))))"}
{"_id": "3151", "title": "PATH_CONNECTED_OPEN_DIFF_CARD_LT", "text": "|- (!s. (!t. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ ((open s) /\\ ((connected s) /\\ (t <_c UNIV)))) ==> (path_connected (s DIFF t)))))"}
{"_id": "3152", "title": "MONOIDAL_REAL_ADD", "text": "|- (monoidal real_add)"}
{"_id": "3153", "title": "int_min_th", "text": "|- (!x. (!y. ((real_of_int ((int_min x) y)) = ((real_min (real_of_int x)) (real_of_int y)))))"}
{"_id": "3154", "title": "SUB_SUC", "text": "|- (!m. (!n. (((SUC m) - (SUC n)) = (m - n))))"}
{"_id": "3155", "title": "SUB_0", "text": "|- (!m. ((((NUMERAL _0) - m) = (NUMERAL _0)) /\\ ((m - (NUMERAL _0)) = m)))"}
{"_id": "3156", "title": "CARD_NUMSEG", "text": "|- (!m. (!n. ((CARD (m .. n)) = ((n + (NUMERAL (BIT1 _0))) - m))))"}
{"_id": "3157", "title": "BOUNDED_TRANSLATION_EQ", "text": "|- (!a. (!s. ((bounded ((IMAGE (\\x. ((vector_add a) x))) s)) = (bounded s))))"}
{"_id": "3158", "title": "pairwise", "text": "|- (!s. (!r. (((pairwise r) s) = (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ (~ (x = y)))) ==> ((r x) y)))))))"}
{"_id": "3159", "title": "HAS_BOUNDED_SETVARIATION_ON_EQ", "text": "|- (!f. (!g. (!s. (((!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((f (closed_interval ((CONS (a , b)) NIL))) = (g (closed_interval ((CONS (a , b)) NIL))))))) /\\ ((has_bounded_setvariation_on f) s)) ==> ((has_bounded_setvariation_on g) s)))))"}
{"_id": "3160", "title": "COMPLEX_SUB_0", "text": "|- (!x. (!y. ((((vector_sub x) y) = (Cx (real_of_num (NUMERAL _0)))) = (x = y))))"}
{"_id": "3161", "title": "HAS_INTEGRAL_COMBINE_TAGGED_DIVISION", "text": "|- (!f. (!s. (!p. (!i. ((((tagged_division_of p) s) /\\ (!x. (!k. (((x , k) IN p) ==> (((has_integral f) (i k)) k))))) ==> (((has_integral f) ((vsum p) (GABS (\\f. (!x. (!k. ((GEQ (f (x , k))) (i k)))))))) s))))))"}
{"_id": "3162", "title": "SUBSET_UNION_ABSORPTION", "text": "|- (!s. (!t. ((s SUBSET t) = ((s UNION t) = t))))"}
{"_id": "3163", "title": "RELATIVE_FRONTIER_CLOSURE", "text": "|- (!s. ((convex s) ==> ((relative_frontier (closure s)) = (relative_frontier s))))"}
{"_id": "3164", "title": "COS_NPI", "text": "|- (!n. ((cos ((real_mul (real_of_num n)) pi)) = ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) n)))"}
{"_id": "3165", "title": "REAL_CONTINUOUS_ON_BERNOULLI", "text": "|- (!n. (!s. ((real_continuous_on (bernoulli n)) s)))"}
{"_id": "3166", "title": "CONVEX_CONNECTED_1", "text": "|- (!s. ((convex s) = (connected s)))"}
{"_id": "3167", "title": "CEXP_N", "text": "|- (!n. (!x. ((cexp ((complex_mul (Cx (real_of_num n))) x)) = ((complex_pow (cexp x)) n))))"}
{"_id": "3168", "title": "SUMMABLE_ZETA_INTEGER", "text": "|- (!n. (!m. (((NUMERAL (BIT0 (BIT1 _0))) <= m) ==> ((summable (from n)) (\\k. (complex_inv ((complex_pow (Cx (real_of_num k))) m)))))))"}
{"_id": "3169", "title": "Lp_ineqs_proofs.y9_hi_std", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))"}
{"_id": "3170", "title": "COMPLETE_INJECTIVE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((complete ((IMAGE f) s)) = (complete s)))))"}
{"_id": "3171", "title": "LE_RDIV_EQ", "text": "|- (!a. (!b. (!n. ((~ (a = (NUMERAL _0))) ==> ((n <= (b DIV a)) = ((a * n) <= b))))))"}
{"_id": "3172", "title": "CARD_EQ_IMAGE", "text": "|- (!f. (!s. ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) ==> (((IMAGE f) s) =_c s))))"}
{"_id": "3173", "title": "CONVEX_TRANSLATION_EQ", "text": "|- (!a. (!s. ((convex ((IMAGE (\\x. ((vector_add a) x))) s)) = (convex s))))"}
{"_id": "3174", "title": "ENR_OPEN_IN", "text": "|- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\\ (ENR t)) ==> (ENR s))))"}
{"_id": "3175", "title": "REAL_MUL_RZERO", "text": "|- (!x. (((real_mul x) (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0))))"}
{"_id": "3176", "title": "OPEN_IN_TRANSLATION_EQ", "text": "|- (!a. (!s. (!t. (((open_in ((subtopology euclidean) ((IMAGE (\\x. ((vector_add a) x))) t))) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((open_in ((subtopology euclidean) t)) s)))))"}
{"_id": "3177", "title": "INTERIOR_OPEN", "text": "|- (!s. ((open s) ==> ((interior s) = s)))"}
{"_id": "3178", "title": "DIAMETER_SING", "text": "|- (!a. ((diameter (a INSERT EMPTY)) = (real_of_num (NUMERAL _0))))"}
{"_id": "3179", "title": "URYSOHN_LOCAL_STRONG", "text": "|- (!s. (!t. (!u. (!a. (!b. ((((closed_in ((subtopology euclidean) u)) s) /\\ (((closed_in ((subtopology euclidean) u)) t) /\\ (((s INTER t) = EMPTY) /\\ (~ (a = b))))) ==> (?f. (((continuous_on f) u) /\\ ((!x. ((x IN u) ==> ((f x) IN (closed_segment ((CONS (a , b)) NIL))))) /\\ ((!x. ((x IN u) ==> (((f x) = a) = (x IN s)))) /\\ (!x. ((x IN u) ==> (((f x) = b) = (x IN t))))))))))))))"}
{"_id": "3180", "title": "convex_cone", "text": "|- (!s. ((convex_cone s) = ((~ (s = EMPTY)) /\\ ((convex s) /\\ (conic s)))))"}
{"_id": "3181", "title": "CNJ_CX", "text": "|- (!x. ((cnj (Cx x)) = (Cx x)))"}
{"_id": "3182", "title": "DEPENDENT_EQ_DIM_LT_CARD", "text": "|- (!s. ((dependent s) = ((FINITE s) ==> ((dim s) < (CARD s)))))"}
{"_id": "3183", "title": "DET_MUL", "text": "|- (!A. (!B. ((det ((matrix_mul A) B)) = ((real_mul (det A)) (det B)))))"}
{"_id": "3184", "title": "SIMPLICIAL_COMPLEX_TRANSLATION", "text": "|- (!a. (!tr. ((simplicial_complex ((IMAGE (IMAGE (\\x. ((vector_add a) x)))) tr)) = (simplicial_complex tr))))"}
{"_id": "3185", "title": "HOMOTOPIC_WITH_IMP_PROPERTY", "text": "|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> ((P f) /\\ (P g))))))))"}
{"_id": "3186", "title": "COUNTABLE_IMAGE", "text": "|- (!f. (!s. ((COUNTABLE s) ==> (COUNTABLE ((IMAGE f) s)))))"}
{"_id": "3187", "title": "FINITE_BOOL", "text": "|- (FINITE UNIV)"}
{"_id": "3188", "title": "Nonlin_def.mudLs_126_x", "text": "|- (!d2s. (!y3. (!y4. (!y5. (!d1s. ((((((mudLs_126_x d1s) d2s) y3) y4) y5) = ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y3) y3))) proj_x1) proj_x2) dummy6) dummy6) dummy6)) ((add6 ((mul6 (constant6 ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_add d1s) d2s)))) ((sub6 (((delta_126_x ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5))) (constant6 ((real_mul d1s) d1s))))) (constant6 d1s)))))))))"}
{"_id": "3189", "title": "INT_MUL_LNEG", "text": "|- (!x. (!y. (((int_mul (int_neg x)) y) = (int_neg ((int_mul x) y)))))"}
{"_id": "3190", "title": "ARITH_MULT", "text": "|- ((!m. (!n. (((NUMERAL m) * (NUMERAL n)) = (NUMERAL (m * n))))) /\\ (((_0 * _0) = _0) /\\ ((!n. ((_0 * (BIT0 n)) = _0)) /\\ ((!n. ((_0 * (BIT1 n)) = _0)) /\\ ((!n. (((BIT0 n) * _0) = _0)) /\\ ((!n. (((BIT1 n) * _0) = _0)) /\\ ((!m. (!n. (((BIT0 m) * (BIT0 n)) = (BIT0 (BIT0 (m * n)))))) /\\ ((!m. (!n. (((BIT0 m) * (BIT1 n)) = ((BIT0 m) + (BIT0 (BIT0 (m * n))))))) /\\ ((!m. (!n. (((BIT1 m) * (BIT0 n)) = ((BIT0 n) + (BIT0 (BIT0 (m * n))))))) /\\ (!m. (!n. (((BIT1 m) * (BIT1 n)) = ((BIT1 m) + ((BIT0 n) + (BIT0 (BIT0 (m * n)))))))))))))))))"}
{"_id": "3191", "title": "Ckqowsa_3_points.aff_ge_0_2", "text": "|- (!u. (!v. (((~ (u = (vec (NUMERAL _0)))) /\\ (~ (v = (vec (NUMERAL _0))))) ==> (((aff_ge ((vec (NUMERAL _0)) INSERT EMPTY)) (u INSERT (v INSERT EMPTY))) = (GSPEC (\\GEN%PVAR%7632. (?y. (((SETSPEC GEN%PVAR%7632) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t1) /\\ (((real_le (real_of_num (NUMERAL _0))) t2) /\\ (y = ((vector_add ((% t1) u)) ((% t2) v)))))))) y))))))))"}
{"_id": "3192", "title": "CONVEX_CONE_HULL_LINEAR_IMAGE", "text": "|- (!f. (!s. ((linear f) ==> (((hull convex_cone) ((IMAGE f) s)) = ((IMAGE f) ((hull convex_cone) s))))))"}
{"_id": "3193", "title": "LEBESGUE_MEASURABLE_CLOSED", "text": "|- (!s. ((closed s) ==> (lebesgue_measurable s)))"}
{"_id": "3194", "title": "BALL_LINEAR_IMAGE", "text": "|- (!f. (!x. (!r. (((linear f) /\\ ((!y. (?x. ((f x) = y))) /\\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((ball ((f x) , r)) = ((IMAGE f) (ball (x , r))))))))"}
{"_id": "3195", "title": "EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL", "text": "|- (!s. ((FINITE s) ==> ((GSPEC (\\GEN%PVAR%1789. (?y. (((SETSPEC GEN%PVAR%1789) (?u. ((!x. ((x IN s) ==> (((real_lt (real_of_num (NUMERAL _0))) (u x)) /\\ ((real_lt (u x)) (real_of_num (NUMERAL (BIT1 _0))))))) /\\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((vsum s) (\\x. ((% (u x)) x))) = y))))) y)))) SUBSET (relative_interior ((hull convex) s)))))"}
{"_id": "3196", "title": "AFF_DIM_AFFINE_HULL", "text": "|- (!s. ((aff_dim ((hull affine) s)) = (aff_dim s)))"}
{"_id": "3197", "title": "RAT_LEMMA2", "text": "|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> (((real_add ((real_div x1) y1)) ((real_div x2) y2)) = ((real_mul ((real_add ((real_mul x1) y2)) ((real_mul x2) y1))) ((real_mul (real_inv y1)) (real_inv y2)))))"}
{"_id": "3198", "title": "REAL_ADD_RINV", "text": "|- (!x. (((real_add x) (real_neg x)) = (real_of_num (NUMERAL _0))))"}
{"_id": "3199", "title": "HAS_REAL_INTEGRAL_RESTRICT_UNIV", "text": "|- (!f. (!s. (!i. ((((has_real_integral (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) i) UNIV) = (((has_real_integral f) i) s)))))"}
{"_id": "3200", "title": "REAL_DERIVATIVE_UNIQUE_ATREAL", "text": "|- (!f. (!z. (!f'. (!f''. (((((has_real_derivative f) f') (atreal z)) /\\ (((has_real_derivative f) f'') (atreal z))) ==> (f' = f''))))))"}
{"_id": "3201", "title": "Lp_ineqs_proofs.y2_hi", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (y2_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))"}
{"_id": "3202", "title": "SUBSET_CLOSURE", "text": "|- (!s. (!t. ((s SUBSET t) ==> ((closure s) SUBSET (closure t)))))"}
{"_id": "3203", "title": "DIAMETER_ATTAINED_RELATIVE_FRONTIER", "text": "|- (!s. (((bounded s) /\\ (~ ((diameter s) = (real_of_num (NUMERAL _0))))) ==> (?x. (?y. ((x IN (relative_frontier s)) /\\ ((y IN (relative_frontier s)) /\\ ((vector_norm ((vector_sub x) y)) = (diameter s))))))))"}
{"_id": "3204", "title": "face_of", "text": "|- (!s. (!t. (((face_of t) s) = ((t SUBSET s) /\\ ((convex t) /\\ (!a. (!b. (!x. (((a IN s) /\\ ((b IN s) /\\ ((x IN t) /\\ (x IN (open_segment (a , b)))))) ==> ((a IN t) /\\ (b IN t)))))))))))"}
{"_id": "3205", "title": "FINITE_SEGMENT", "text": "|- ((!a. (!b. ((FINITE (closed_segment ((CONS (a , b)) NIL))) = (a = b)))) /\\ (!a. (!b. ((FINITE (open_segment (a , b))) = (a = b)))))"}
{"_id": "3206", "title": "REAL_SECOND_MEAN_VALUE_THEOREM_BONNET_FULL", "text": "|- (!f. (!g. (!a. (!b. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_le x) y))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ (((has_real_integral (\\x. ((real_mul (g x)) (f x)))) ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f))) (closed_real_interval ((CONS (a , b)) NIL))))))))))"}
{"_id": "3207", "title": "Fan.hypermap_of_fan_rep", "text": "|- (!x. (!V. (!E. (!p. (((FAN (x , (V , E))) /\\ (p = (\\t. ((res (((t x) V) E)) (d1_fan (x , (V , E))))))) ==> (((dart (hypermap1_of_fanx (x , (V , E)))) = (d_fan (x , (V , E)))) /\\ (((edge_map (hypermap1_of_fanx (x , (V , E)))) = (p e_fan)) /\\ (((node_map (hypermap1_of_fanx (x , (V , E)))) = (p n_fan)) /\\ ((face_map (hypermap1_of_fanx (x , (V , E)))) = (p f1_fan))))))))))"}
{"_id": "3208", "title": "Wrgcvdr_cizmrrh.SPEC_HY_ELEMS", "text": "|- (!H. (((tuple_hypermap H) = (D , (e , (n , f)))) = (((dart H) = D) /\\ (((edge_map H) = e) /\\ (((node_map H) = n) /\\ ((face_map H) = f))))))"}
{"_id": "3209", "title": "REAL_LT_LE", "text": "|- (!x. (!y. (((real_lt x) y) = (((real_le x) y) /\\ (~ (x = y))))))"}
{"_id": "3210", "title": "Seq.mem_filter", "text": "|- (!a. (!x. (!s. (((MEM x) ((filter a) s)) = ((a x) /\\ ((MEM x) s))))))"}
{"_id": "3211", "title": "DIMINDEX_GE_1", "text": "|- (!s. ((NUMERAL (BIT1 _0)) <= (dimindex s)))"}
{"_id": "3212", "title": "Ssrnat.half_add", "text": "|- (!m. (!n. ((half (m + n)) = ((((COND ((odd m) /\\ (odd n))) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + ((half m) + (half n))))))"}
{"_id": "3213", "title": "SEPARATING_HYPERPLANE_CLOSED_POINT", "text": "|- (!s. (!z. (((convex s) /\\ ((closed s) /\\ (~ (z IN s)))) ==> (?a. (?b. (((real_lt ((dot a) z)) b) /\\ (!x. ((x IN s) ==> ((real_gt ((dot a) x)) b)))))))))"}
{"_id": "3214", "title": "IN_NUMSEG", "text": "|- (!m. (!n. (!p. ((p IN (m .. n)) = ((m <= p) /\\ (p <= n))))))"}
{"_id": "3215", "title": "Seq.perm_eq_sym", "text": "|- (!s1. (!s2. (((perm_eq s1) s2) = ((perm_eq s2) s1))))"}
{"_id": "3216", "title": "HAUSDIST_NONTRIVIAL", "text": "|- (!s. (!t. (((bounded s) /\\ ((bounded t) /\\ ((~ (s = EMPTY)) /\\ (~ (t = EMPTY))))) ==> ((hausdist (s , t)) = (sup ((GSPEC (\\GEN%PVAR%1186. (?x. (((SETSPEC GEN%PVAR%1186) (x IN s)) (setdist ((x INSERT EMPTY) , t)))))) UNION (GSPEC (\\GEN%PVAR%1187. (?y. (((SETSPEC GEN%PVAR%1187) (y IN t)) (setdist ((y INSERT EMPTY) , s))))))))))))"}
{"_id": "3217", "title": "dependent", "text": "|- (!s. ((dependent s) = (?a. ((a IN s) /\\ (a IN (span (s DELETE a)))))))"}
{"_id": "3218", "title": "DET_I", "text": "|- ((det (mat (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "3219", "title": "Odxlstc.DEFORMATION_DIST_LE_ALL_LE3", "text": "|- (((is_scs_v39 s) /\\ (((MMs_v39 s) w) /\\ (((scs_k_v39 s) = k) /\\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\\ (!i. ((~ ((i MOD k) = (l MOD k))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance ((w l) , (w i)))))))))) ==> (!i. ((~ ((i MOD k) = (l MOD k))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\\ ((real_lt t) e)) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) (w l)) , (w i)))))))))))"}
{"_id": "3220", "title": "Ssrnat.subnK", "text": "|- (!m. (!n. ((m <= n) ==> (((n - m) + m) = n))))"}
{"_id": "3221", "title": "HAS_INTEGRAL_ALT", "text": "|- (!f. (!s. (!i. ((((has_integral f) i) s) = ((!a. (!b. ((integrable_on (\\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) (closed_interval ((CONS (a , b)) NIL))))) /\\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\\ (!a. (!b. (((ball ((vec (NUMERAL _0)) , B)) SUBSET (closed_interval ((CONS (a , b)) NIL))) ==> ((real_lt (vector_norm ((vector_sub ((integral (closed_interval ((CONS (a , b)) NIL))) (\\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))))) i))) e)))))))))))))"}
{"_id": "3222", "title": "NORM_LE_L1", "text": "|- (!x. ((real_le (vector_norm x)) ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. (real_abs (x $ i))))))"}
{"_id": "3223", "title": "REAL_SUB_ARG", "text": "|- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> (((real_sub (Arg w)) (Arg z)) = (((COND ((real_le (Arg z)) (Arg w))) (Arg ((complex_div w) z))) ((real_sub (Arg ((complex_div w) z))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))))))"}
{"_id": "3224", "title": "SPAN_SUB", "text": "|- (!x. (!y. (!s. (((x IN (span s)) /\\ (y IN (span s))) ==> (((vector_sub x) y) IN (span s))))))"}
{"_id": "3225", "title": "EL", "text": "|- ((((EL (NUMERAL _0)) l) = (HD l)) /\\ (((EL (SUC n)) l) = ((EL n) (TL l))))"}
{"_id": "3226", "title": "DISCRETE_BOUNDED_IMP_FINITE", "text": "|- (!s. (!e. ((((real_lt (real_of_num (NUMERAL _0))) e) /\\ ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((real_lt (vector_norm ((vector_sub y) x))) e))) ==> (y = x)))) /\\ (bounded s))) ==> (FINITE s))))"}
{"_id": "3227", "title": "HAS_COMPLEX_DERIVATIVE_WITHIN_SUBSET", "text": "|- (!f. (!s. (!t. (!x. (((((has_complex_derivative f) f') ((within (at x)) s)) /\\ (t SUBSET s)) ==> (((has_complex_derivative f) f') ((within (at x)) t)))))))"}
{"_id": "3228", "title": "RELATIVE_FRONTIER_RETRACT_OF_PUNCTURED_AFFINE_HULL", "text": "|- (!s. (!a. (((convex s) /\\ ((bounded s) /\\ (a IN (relative_interior s)))) ==> ((retract_of (relative_frontier s)) (((hull affine) s) DELETE a)))))"}
{"_id": "3229", "title": "VSUM_DIFF", "text": "|- (!f. (!s. (!t. (((FINITE s) /\\ (t SUBSET s)) ==> (((vsum (s DIFF t)) f) = ((vector_sub ((vsum s) f)) ((vsum t) f)))))))"}
{"_id": "3230", "title": "HAS_BOUNDED_VARIATION_ON_MAX", "text": "|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\\x. (lift ((real_max (drop (f x))) (drop (g x)))))) s)))))"}
{"_id": "3231", "title": "Nonlin_def.constant6", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((((((((constant6 c) x1) x2) x3) x4) x5) x6) = c))))))))"}
{"_id": "3232", "title": "CONNECTED_CLOSURE", "text": "|- (!s. ((connected s) ==> (connected (closure s))))"}
{"_id": "3233", "title": "REAL_LE_SQUARE_ABS", "text": "|- (!x. (!y. (((real_le (real_abs x)) (real_abs y)) = ((real_le ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "3234", "title": "MEASURE_INTERVAL_4", "text": "|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ (((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((real_le (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((real_le (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))) /\\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ (((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((real_le (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((real_le (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 (BIT1 _0))))) (a $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (a $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) (real_of_num (NUMERAL _0)))))))"}
{"_id": "3235", "title": "CARD_SING", "text": "|- (!a. ((CARD (a INSERT EMPTY)) = (NUMERAL (BIT1 _0))))"}
{"_id": "3236", "title": "FRONTIER_STRADDLE", "text": "|- (!a. (!s. ((a IN (frontier s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((?x. ((x IN s) /\\ ((real_lt (distance (a , x))) e))) /\\ (?x. ((~ (x IN s)) /\\ ((real_lt (distance (a , x))) e)))))))))"}
{"_id": "3237", "title": "EVENTUALLY_WITHINREAL_LE", "text": "|- (!s. (!a. (!p. (((eventually p) ((within (atreal a)) s)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (((x IN s) /\\ (((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\\ ((real_le (real_abs ((real_sub x) a))) d))) ==> (p x)))))))))"}
{"_id": "3238", "title": "EXTREME_POINT_NOT_IN_INTERIOR", "text": "|- (!s. (!x. (((extreme_point_of x) s) ==> (~ (x IN (interior s))))))"}
{"_id": "3239", "title": "CBALL_LINEAR_IMAGE", "text": "|- (!f. (!x. (!r. (((linear f) /\\ ((!y. (?x. ((f x) = y))) /\\ (!x. ((vector_norm (f x)) = (vector_norm x))))) ==> ((cball ((f x) , r)) = ((IMAGE f) (cball (x , r))))))))"}
{"_id": "3240", "title": "INT_LT_LE", "text": "|- (!x. (!y. (((int_lt x) y) = (((int_le x) y) /\\ (~ (x = y))))))"}
{"_id": "3241", "title": "Hijqaha.K_SCS_4M5", "text": "|- ((scs_k_v39 scs_4M5') = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))"}
{"_id": "3242", "title": "REAL_OF_NUM_POW", "text": "|- (!x. (!n. (((real_pow (real_of_num x)) n) = (real_of_num (x EXP n)))))"}
{"_id": "3243", "title": "BALL_1", "text": "|- (!x. (!r. (((cball (x , r)) = (closed_interval ((CONS (((vector_sub x) (lift r)) , ((vector_add x) (lift r)))) NIL))) /\\ ((ball (x , r)) = (open_interval (((vector_sub x) (lift r)) , ((vector_add x) (lift r))))))))"}
{"_id": "3244", "title": "COMPLEX_DIFFERENTIABLE_AT_WITHIN", "text": "|- (!f. (!s. (!z. (((complex_differentiable f) (at z)) ==> ((complex_differentiable f) ((within (at z)) s))))))"}
{"_id": "3245", "title": "complex_sub", "text": "|- (((vector_sub w) z) = ((vector_add w) (vector_neg z)))"}
{"_id": "3246", "title": "COMMUTING_MATRIX_INV_COVARIANCE", "text": "|- (!A. (((matrix_mul (matrix_inv ((matrix_mul (transp A)) A))) ((matrix_mul (transp A)) A)) = ((matrix_mul ((matrix_mul (transp A)) A)) (matrix_inv ((matrix_mul (transp A)) A)))))"}
{"_id": "3247", "title": "MATRIX_ADD", "text": "|- (!f. (!g. (((linear f) /\\ (linear g)) ==> ((matrix (\\x. ((vector_add (f x)) (g x)))) = ((matrix_add (matrix f)) (matrix g))))))"}
{"_id": "3248", "title": "REAL_LT_POW2", "text": "|- (!n. ((real_lt (real_of_num (NUMERAL _0))) ((real_pow (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)))"}
{"_id": "3249", "title": "COMPLEX_DERIVATIVE_MUL", "text": "|- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\\ ((complex_differentiable g) (at z))) ==> (((complex_derivative (\\w. ((complex_mul (f w)) (g w)))) z) = ((vector_add ((complex_mul (f z)) ((complex_derivative g) z))) ((complex_mul ((complex_derivative f) z)) (g z))))))))"}
{"_id": "3250", "title": "AFF_DIM_LE_DIM", "text": "|- (!s. ((int_le (aff_dim s)) (int_of_num (dim s))))"}
{"_id": "3251", "title": "VECTOR_VARIATION_MINUS_FUNCTION_MONOTONE", "text": "|- (!f. (!a. (!b. (!c. (!d. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (((closed_interval ((CONS (c , d)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL))) /\\ (~ ((closed_interval ((CONS (c , d)) NIL)) = EMPTY)))) ==> ((real_le ((real_sub ((vector_variation (closed_interval ((CONS (c , d)) NIL))) f)) (drop ((vector_sub (f d)) (f c))))) ((real_sub ((vector_variation (closed_interval ((CONS (a , b)) NIL))) f)) (drop ((vector_sub (f b)) (f a)))))))))))"}
{"_id": "3252", "title": "TC_SYM", "text": "|- (!R. ((!x. (!y. (((R x) y) ==> ((R y) x)))) ==> (!x. (!y. ((((TC R) x) y) ==> (((TC R) y) x))))))"}
{"_id": "3253", "title": "NORM_LE", "text": "|- (!x. (!y. (((real_le (vector_norm x)) (vector_norm y)) = ((real_le ((dot x) x)) ((dot y) y)))))"}
{"_id": "3254", "title": "one", "text": "|- (!v. (v = one))"}
{"_id": "3255", "title": "Ssrnat.distnC", "text": "|- (!m. (!n. (((distn m) n) = ((distn n) m))))"}
{"_id": "3256", "title": "MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL", "text": "|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3639. (?x. (((SETSPEC GEN%PVAR%3639) ((x IN s) /\\ ((f x) IN t))) x)))))))))))"}
{"_id": "3257", "title": "SUPPORT_CLAUSES", "text": "|- ((!f. ((((support op) f) EMPTY) = EMPTY)) /\\ ((!f. (!x. (!s. ((((support op) f) (x INSERT s)) = (((COND ((f x) = (neutral op))) (((support op) f) s)) (x INSERT (((support op) f) s))))))) /\\ ((!f. (!x. (!s. ((((support op) f) (s DELETE x)) = ((((support op) f) s) DELETE x))))) /\\ ((!f. (!s. (!t. ((((support op) f) (s UNION t)) = ((((support op) f) s) UNION (((support op) f) t)))))) /\\ ((!f. (!s. (!t. ((((support op) f) (s INTER t)) = ((((support op) f) s) INTER (((support op) f) t)))))) /\\ ((!f. (!s. (!t. ((((support op) f) (s DIFF t)) = ((((support op) f) s) DIFF (((support op) f) t)))))) /\\ (!f. (!g. (!s. ((((support op) g) ((IMAGE f) s)) = ((IMAGE f) (((support op) (g o f)) s))))))))))))"}
{"_id": "3258", "title": "SETDIST_SING_IN_SET", "text": "|- (!x. (!s. ((x IN s) ==> ((setdist ((x INSERT EMPTY) , s)) = (real_of_num (NUMERAL _0))))))"}
{"_id": "3259", "title": "HAS_COMPLEX_DERIVATIVE_VSUM", "text": "|- (!f. (!net. (!s. (((FINITE s) /\\ (!a. ((a IN s) ==> (((has_complex_derivative (f a)) (f' a)) net)))) ==> (((has_complex_derivative (\\x. ((vsum s) (\\a. ((f a) x))))) ((vsum s) f')) net)))))"}
{"_id": "3260", "title": "real_div", "text": "|- (!x. (!y. (((real_div x) y) = ((real_mul x) (real_inv y)))))"}
{"_id": "3261", "title": "REAL_BOUNDS_LT", "text": "|- (!x. (!k. ((((real_lt (real_neg k)) x) /\\ ((real_lt x) k)) = ((real_lt (real_abs x)) k))))"}
{"_id": "3262", "title": "WF_SUBSET", "text": "|- (((!x. (!y. (((<< x) y) ==> ((<<< x) y)))) /\\ (WF <<<)) ==> (WF <<))"}
{"_id": "3263", "title": "LIM_NORM", "text": "|- (!net. (!f. (!l. ((((--> f) l) net) ==> (((--> (\\x. (lift (vector_norm (f x))))) (lift (vector_norm l))) net)))))"}
{"_id": "3264", "title": "HAUSDIST_CLOSURE", "text": "|- ((!s. (!t. ((hausdist ((closure s) , t)) = (hausdist (s , t))))) /\\ (!s. (!t. ((hausdist (s , (closure t))) = (hausdist (s , t))))))"}
{"_id": "3265", "title": "REAL_CONVEX_ON_DERIVATIVES_IMP", "text": "|- (!f. (!f'x. (!f'y. (!s. (!x. (!y. ((((real_convex_on f) s) /\\ (((closed_real_segment ((CONS (x , y)) NIL)) SUBSET s) /\\ ((((has_real_derivative f) f'x) ((within (atreal x)) s)) /\\ (((has_real_derivative f) f'y) ((within (atreal y)) s))))) ==> ((real_le ((real_mul f'x) ((real_sub y) x))) ((real_mul f'y) ((real_sub y) x))))))))))"}
{"_id": "3266", "title": "PCROSS_INTERVAL", "text": "|- (!a. (!b. (!c. (!d. (((closed_interval ((CONS (a , b)) NIL)) PCROSS (closed_interval ((CONS (c , d)) NIL))) = (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL)))))))"}
{"_id": "3267", "title": "FRONTIER_SUBSET_RETRACTION", "text": "|- (!s. (!t. (!r. (((bounded s) /\\ (((frontier s) SUBSET t) /\\ (((continuous_on r) (closure s)) /\\ ((((IMAGE r) s) SUBSET t) /\\ (!x. ((x IN t) ==> ((r x) = x))))))) ==> (s SUBSET t)))))"}
{"_id": "3268", "title": "REALLIM_CONST", "text": "|- (!net. (!a. (((---> (\\x. a)) a) net)))"}
{"_id": "3269", "title": "LOG_MONO_LE", "text": "|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> (((real_le (log x)) (log y)) = ((real_le x) y)))))"}
{"_id": "3270", "title": "MATRIX_INV_MULTIPLE_TRANP_LEFT", "text": "|- (!A. ((matrix_inv A) = ((matrix_mul (transp A)) ((matrix_mul (transp (matrix_inv A))) (matrix_inv A)))))"}
{"_id": "3271", "title": "SUMS_FINITE_UNION", "text": "|- (!f. (!s. (!t. (!l. (((FINITE t) /\\ (((sums f) l) s)) ==> (((sums f) ((vector_add l) ((vsum (t DIFF s)) f))) (s UNION t)))))))"}
{"_id": "3272", "title": "binom", "text": "|- ((!n. ((binom (n , (NUMERAL _0))) = (NUMERAL (BIT1 _0)))) /\\ ((!k. ((binom ((NUMERAL _0) , (SUC k))) = (NUMERAL _0))) /\\ (!n. (!k. ((binom ((SUC n) , (SUC k))) = ((binom (n , (SUC k))) + (binom (n , k))))))))"}
{"_id": "3273", "title": "ROTATE2D_COMPLEX", "text": "|- (!t. (!z. (((rotate2d t) z) = ((complex_mul (cexp ((complex_mul ii) (Cx t)))) z))))"}
{"_id": "3274", "title": "PATH_LINEAR_IMAGE_EQ", "text": "|- (!f. (!g. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((path (f o g)) = (path g)))))"}
{"_id": "3275", "title": "SPAN_INC", "text": "|- (!s. (s SUBSET (span s)))"}
{"_id": "3276", "title": "SELECT_AX", "text": "|- (!P. (!x. ((P x) ==> (P (@ P)))))"}
{"_id": "3277", "title": "HOMEOMORPHIC_TRANSLATION_RIGHT_EQ", "text": "|- (!a. (!s. (!t. (((homeomorphic s) ((IMAGE (\\x. ((vector_add a) x))) t)) = ((homeomorphic s) t)))))"}
{"_id": "3278", "title": "SERIES_TERMS_TOZERO", "text": "|- (!f. (!l. (!n. ((((sums f) l) (from n)) ==> (((--> f) (vec (NUMERAL _0))) sequentially)))))"}
{"_id": "3279", "title": "CONIC_HULL_LINEAR_IMAGE", "text": "|- (!f. (!s. ((linear f) ==> (((hull conic) ((IMAGE f) s)) = ((IMAGE f) ((hull conic) s))))))"}
{"_id": "3280", "title": "AFF_DIM_DIM_0", "text": "|- (!s. (((vec (NUMERAL _0)) IN ((hull affine) s)) ==> ((aff_dim s) = (int_of_num (dim s)))))"}
{"_id": "3281", "title": "LOCALLY_COMPACT_INTER_CBALL", "text": "|- (!s. (((locally compact) s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (closed ((cball (x , e)) INTER s))))))))"}
{"_id": "3282", "title": "REAL_MAX_MAX", "text": "|- (!x. (!y. (((real_le x) ((real_max x) y)) /\\ ((real_le y) ((real_max x) y)))))"}
{"_id": "3283", "title": "POLYHEDRON_IMP_CLOSED", "text": "|- (!s. ((polyhedron s) ==> (closed s)))"}
{"_id": "3284", "title": "IN_INTERS", "text": "|- (!s. (!x. ((x IN (INTERS s)) = (!t. ((t IN s) ==> (x IN t))))))"}
{"_id": "3285", "title": "REAL_INTEGRAL_0", "text": "|- (!s. (((real_integral s) (\\x. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0))))"}
{"_id": "3286", "title": "HAS_MEASURE_TRANSLATION_EQ", "text": "|- (!a. (!s. (!m. (((has_measure ((IMAGE (\\x. ((vector_add a) x))) s)) m) = ((has_measure s) m)))))"}
{"_id": "3287", "title": "CONVEX_INTERIOR", "text": "|- (!s. ((convex s) ==> (convex (interior s))))"}
{"_id": "3288", "title": "DET_3", "text": "|- (!A. ((det A) = ((real_add ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))))) ((real_add ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))))) ((real_sub ((real_sub ((real_sub ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 (BIT1 _0)))))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) ((A $ (NUMERAL (BIT1 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))))))))))"}
{"_id": "3289", "title": "EXTREME_POINT_OF_TRANSLATION_EQ", "text": "|- (!a. (!x. (!s. (((extreme_point_of ((vector_add a) x)) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((extreme_point_of x) s)))))"}
{"_id": "3290", "title": "SPHERE_EMPTY", "text": "|- (!a. (!r. (((real_lt r) (real_of_num (NUMERAL _0))) ==> ((sphere (a , r)) = EMPTY))))"}
{"_id": "3291", "title": "HULL_P", "text": "|- (!P. (!s. ((P s) ==> (((hull P) s) = s))))"}
{"_id": "3292", "title": "COUNTABLE_NON_LIMIT_POINTS", "text": "|- (!s. (COUNTABLE (s DIFF (GSPEC (\\GEN%PVAR%1262. (?x. (((SETSPEC GEN%PVAR%1262) ((limit_point_of x) s)) x)))))))"}
{"_id": "3293", "title": "DOT_RMUL", "text": "|- (!c. (!x. (!y. (((dot x) ((% c) y)) = ((real_mul c) ((dot x) y))))))"}
{"_id": "3294", "title": "Lp_ineqs_proofs.y6_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (y6_fan d))))))))))"}
{"_id": "3295", "title": "SPHERE_SING", "text": "|- (!x. (!e. ((e = (real_of_num (NUMERAL _0))) ==> ((sphere (x , e)) = (x INSERT EMPTY)))))"}
{"_id": "3296", "title": "PATH_COMPONENT_TRANSLATION", "text": "|- (!a. (!s. (!x. (((path_component ((IMAGE (\\x. ((vector_add a) x))) s)) ((vector_add a) x)) = ((IMAGE (\\x. ((vector_add a) x))) ((path_component s) x))))))"}
{"_id": "3297", "title": "COUNTABLE_IMAGE_INJ_EQ", "text": "|- (!f. (!s. ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) ==> ((COUNTABLE ((IMAGE f) s)) = (COUNTABLE s)))))"}
{"_id": "3298", "title": "FINITE_UNIONS", "text": "|- (!s. ((FINITE (UNIONS s)) = ((FINITE s) /\\ (!t. ((t IN s) ==> (FINITE t))))))"}
{"_id": "3299", "title": "Kepler_spec.ZERO_SPEC", "text": "|- ((vec (NUMERAL _0)) = (real3_of_triple ((real_of_num (NUMERAL _0)) , ((real_of_num (NUMERAL _0)) , (real_of_num (NUMERAL _0))))))"}
{"_id": "3300", "title": "real_measurable_on", "text": "|- (!f. (!s. (((real_measurable_on f) s) = ((measurable_on (lift o (f o drop))) ((IMAGE lift) s)))))"}
{"_id": "3301", "title": "REAL_SUMMABLE_CAUCHY", "text": "|- (!f. (!s. (((real_summable s) f) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!m. (!n. ((m >= N) ==> ((real_lt (real_abs ((sum (s INTER (m .. n))) f))) e))))))))))"}
{"_id": "3302", "title": "Ssrnat.addnS", "text": "|- (!m. (!n. ((m + (SUC n)) = (SUC (m + n)))))"}
{"_id": "3303", "title": "COMPLEX_EXPAND", "text": "|- (!z. (z = ((vector_add (Cx (Re z))) ((complex_mul ii) (Cx (Im z))))))"}
{"_id": "3304", "title": "INTEGER_POS", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((integer x) = (?n. (x = (real_of_num n))))))"}
{"_id": "3305", "title": "CARD_EXP_SING", "text": "|- (!s. (!b. (((^_c s) (b INSERT EMPTY)) =_c s)))"}
{"_id": "3306", "title": "Hypermap_and_fan.FINITE_ORBIT_MAP_INVERSE", "text": "|- (!f. (!s. (!x. (!n. (!k. (((FINITE s) /\\ (((permutes f) s) /\\ (((CARD ((orbit_map f) x)) = n) /\\ (k <= n)))) ==> ((((POWER (inverse f)) k) x) = (((POWER f) (n - k)) x))))))))"}
{"_id": "3307", "title": "Tame_defs2.marked_list", "text": "|- (!L. (!N. (!r. (!x. (((((marked_list L) N) r) x) = ((is_restricted (hypermap_of_list L)) /\\ (((normal_list L) N) /\\ (((MEM r) N) /\\ (((MEM x) r) /\\ (((LET (\\qs. (LET_END (((all uniq) qs) /\\ ((all (\\v. ((sizel ((filter (MEM v)) qs)) > (NUMERAL (BIT1 _0))))) (flatten qs)))))) ((quotient_list L) N)) /\\ ((((next_el r) x) = ((f_list L) x)) /\\ (((LET (\\r'. (LET_END ((final_list L) r')))) ((find_list N) (e_list x))) /\\ ((((final_list L) r) ==> ((flag_list L) N)) /\\ ((~ ((final_list L) r)) ==> ((((s_flag_list L) N) r) x)))))))))))))))"}
{"_id": "3308", "title": "CONVERGENT_BOUNDED_MONOTONE_1", "text": "|- (!s. (((bounded ((IMAGE s) UNIV)) /\\ ((!n. ((real_le (drop (s n))) (drop (s (SUC n))))) \\/ (!n. ((real_le (drop (s (SUC n)))) (drop (s n)))))) ==> (?l. (((--> s) l) sequentially))))"}
{"_id": "3309", "title": "Local_lemmas.LOFA_IMP_CARD_FF_V_EQ", "text": "|- ((local_fan (V , (E , FF))) ==> ((CARD FF) = (CARD V)))"}
{"_id": "3310", "title": "REAL_SUMMABLE_COMPLEX", "text": "|- (((real_summable s) f) = ((summable s) (Cx o f)))"}
{"_id": "3311", "title": "NORM_SUB", "text": "|- (!x. (!y. ((vector_norm ((vector_sub x) y)) = (vector_norm ((vector_sub y) x)))))"}
{"_id": "3312", "title": "PRODUCT_CLAUSES_NUMSEG", "text": "|- ((!m. (((product (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))))) /\\ (!m. (!n. (((product (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((real_mul ((product (m .. n)) f)) (f (SUC n)))) ((product (m .. n)) f))))))"}
{"_id": "3313", "title": "EIGENVALUES_CHARACTERISTIC_ALT", "text": "|- (!A. (!c. ((?v. ((~ (v = (vec (NUMERAL _0)))) /\\ (((matrix_vector_mul A) v) = ((% c) v)))) = ((det ((matrix_sub A) ((%% c) (mat (NUMERAL (BIT1 _0)))))) = (real_of_num (NUMERAL _0))))))"}
{"_id": "3314", "title": "ccos", "text": "|- (!z. ((ccos z) = ((complex_div ((vector_add (cexp ((complex_mul ii) z))) (cexp ((complex_mul (vector_neg ii)) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "3315", "title": "Seq.behead", "text": "|- (!h. (!t. (((behead NIL) = NIL) /\\ ((behead ((CONS h) t)) = t))))"}
{"_id": "3316", "title": "MATRIX_INV_COVARIANCE_RMUL", "text": "|- (!A. (((matrix_mul (matrix_inv ((matrix_mul (transp A)) A))) (transp A)) = (matrix_inv A)))"}
{"_id": "3317", "title": "HAS_VECTOR_DERIVATIVE_UNIQUE_AT", "text": "|- (!f. (!f'. (!x. ((((has_vector_derivative f) f') (at x)) ==> (((vector_derivative f) (at x)) = f')))))"}
{"_id": "3318", "title": "CNJ_CEXP", "text": "|- (!z. ((cnj (cexp z)) = (cexp (cnj z))))"}
{"_id": "3319", "title": "LE_EXP", "text": "|- (!x. (!m. (!n. (((x EXP m) <= (x EXP n)) = (((COND (x = (NUMERAL _0))) ((m = (NUMERAL _0)) ==> (n = (NUMERAL _0)))) ((x = (NUMERAL (BIT1 _0))) \\/ (m <= n)))))))"}
{"_id": "3320", "title": "homeomorphism", "text": "|- (!f. (!s. (!g. (!t. (((homeomorphism (s , t)) (f , g)) = ((!x. ((x IN s) ==> ((g (f x)) = x))) /\\ ((((IMAGE f) s) = t) /\\ (((continuous_on f) s) /\\ ((!y. ((y IN t) ==> ((f (g y)) = y))) /\\ ((((IMAGE g) t) = s) /\\ ((continuous_on g) t)))))))))))"}
{"_id": "3321", "title": "tendsto_real", "text": "|- (!f. (!l. (!net. ((((---> f) l) net) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\x. ((real_lt (real_abs ((real_sub (f x)) l))) e))) net)))))))"}
{"_id": "3322", "title": "ADD_ASSOC", "text": "|- (!m. (!n. (!p. ((m + (n + p)) = ((m + n) + p)))))"}
{"_id": "3323", "title": "IS_INTERVAL_CONVEX_1", "text": "|- (!s. ((is_interval s) = (convex s)))"}
{"_id": "3324", "title": "REAL_INV_MUL", "text": "|- (!x. (!y. ((real_inv ((real_mul x) y)) = ((real_mul (real_inv x)) (real_inv y)))))"}
{"_id": "3325", "title": "COMPONENTS_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> ((components ((IMAGE f) s)) = ((IMAGE (IMAGE f)) (components s))))))"}
{"_id": "3326", "title": "Hypermap_iso.bij_contracted_dart1", "text": "|- (!V. (!E. (((BIJ contracted_dart) (d1_fan ((vec (NUMERAL _0)) , (V , E)))) (dart1_of_fan (V , E)))))"}
{"_id": "3327", "title": "INTERVAL_LOWERBOUND_NONEMPTY", "text": "|- (!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) ==> ((interval_lowerbound (closed_interval ((CONS (a , b)) NIL))) = a))))"}
{"_id": "3328", "title": "Fan.set_of_edge", "text": "|- (!v. (!E. (!V. ((((set_of_edge v) V) E) = (GSPEC (\\GEN%PVAR%5448. (?w. (((SETSPEC GEN%PVAR%5448) (((v INSERT (w INSERT EMPTY)) IN E) /\\ (w IN V))) w))))))))"}
{"_id": "3329", "title": "CARD_LE_DIM_SPANNING", "text": "|- (!v. (!b. (((v SUBSET (span b)) /\\ ((FINITE b) /\\ ((CARD b) <= (dim v)))) ==> (independent b))))"}
{"_id": "3330", "title": "Ssrnat.addnC", "text": "|- (!m. (!n. ((m + n) = (n + m))))"}
{"_id": "3331", "title": "COMPLEX_EQ", "text": "|- (!w. (!z. ((w = z) = (((Re w) = (Re z)) /\\ ((Im w) = (Im z))))))"}
{"_id": "3332", "title": "PATH_TRANSLATION_EQ", "text": "|- (!a. (!g. ((path ((\\x. ((vector_add a) x)) o g)) = (path g))))"}
{"_id": "3333", "title": "path", "text": "|- (!g. ((path g) = ((continuous_on g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))"}
{"_id": "3334", "title": "CONNECTED_OUTSIDE", "text": "|- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ (bounded s)) ==> (connected (outside s))))"}
{"_id": "3335", "title": "CARD_LE_CONG", "text": "|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\\ (t =_c t')) ==> ((s <=_c t) = (s' <=_c t')))))))"}
{"_id": "3336", "title": "vec", "text": "|- (!n. ((vec n) = (lambdai. (real_of_num n))))"}
{"_id": "3337", "title": "HAS_REAL_VECTOR_DERIVATIVE_WITHIN", "text": "|- ((((has_real_derivative f) f') ((within (atreal x)) s)) = (((has_vector_derivative (lift o (f o drop))) (lift f')) ((within (at (lift x))) ((IMAGE lift) s))))"}
{"_id": "3338", "title": "HOLOMORPHIC_ON_DIFFERENTIABLE", "text": "|- (!f. (!s. (((holomorphic_on f) s) = (!x. ((x IN s) ==> ((complex_differentiable f) ((within (at x)) s)))))))"}
{"_id": "3339", "title": "CROSS_ORTHOGONAL_MATRIX", "text": "|- (!A. (!x. (!y. ((orthogonal_matrix A) ==> (((cross ((matrix_vector_mul A) x)) ((matrix_vector_mul A) y)) = ((% (det A)) ((matrix_vector_mul A) ((cross x) y))))))))"}
{"_id": "3340", "title": "CNJ_CNJ", "text": "|- (!z. ((cnj (cnj z)) = z))"}
{"_id": "3341", "title": "Seq.size_subseq_leqif", "text": "|- (!s1. (!s2. (((subseq s1) s2) ==> (((leqif (sizel s1)) (sizel s2)) (s1 = s2)))))"}
{"_id": "3342", "title": "LIM_CONTINUOUS", "text": "|- (!net. (!f. (!l. ((((continuous f) net) /\\ ((f (netlimit net)) = l)) ==> (((--> f) l) net)))))"}
{"_id": "3343", "title": "EQ_ADD_LCANCEL", "text": "|- (!m. (!n. (!p. (((m + n) = (m + p)) = (n = p)))))"}
{"_id": "3344", "title": "Hypermap.set_of_orbits", "text": "|- (!D. (!f. (((set_of_orbits D) f) = (GSPEC (\\GEN%PVAR%5395. (?x. (((SETSPEC GEN%PVAR%5395) (x IN D)) ((orbit_map f) x))))))))"}
{"_id": "3345", "title": "HAS_REAL_MEASURE_HAS_MEASURE", "text": "|- (!s. (!m. (((has_real_measure s) m) = ((has_measure ((IMAGE lift) s)) m))))"}
{"_id": "3346", "title": "ANR_IMP_ABSOLUTE_NEIGHBOURHOOD_EXTENSOR", "text": "|- (!f. (!u. (!t. (!s. (((ANR s) /\\ (((continuous_on f) t) /\\ ((((IMAGE f) t) SUBSET s) /\\ ((closed_in ((subtopology euclidean) u)) t)))) ==> (?v. (?g. ((t SUBSET v) /\\ (((open_in ((subtopology euclidean) u)) v) /\\ (((continuous_on g) v) /\\ ((((IMAGE g) v) SUBSET s) /\\ (!x. ((x IN t) ==> ((g x) = (f x)))))))))))))))"}
{"_id": "3347", "title": "Seq2.EL_nth", "text": "|- (!x0. (!s. (!i. ((i < (sizel s)) ==> (((EL i) s) = (((nth x0) s) i))))))"}
{"_id": "3348", "title": "SUBSET_CROSS", "text": "|- (!s. (!t. (!s'. (!t'. (((s CROSS t) SUBSET (s' CROSS t')) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((s SUBSET s') /\\ (t SUBSET t')))))))))"}
{"_id": "3349", "title": "REAL_ADD_LID", "text": "|- (!x. (((real_add (real_of_num (NUMERAL _0))) x) = x))"}
{"_id": "3350", "title": "Fan_misc.EXTENSION_SIGMA_FAN_EQ_RES", "text": "|- (!x. (!V. (!E. (!v. (((((extension_sigma_fan x) V) E) v) = ((res ((((sigma_fan x) V) E) v)) (((set_of_edge v) V) E)))))))"}
{"_id": "3351", "title": "AFFINE_INDEPENDENT_IFF_CARD", "text": "|- (!s. ((~ (affine_dependent s)) = ((FINITE s) /\\ ((aff_dim s) = ((int_sub (int_of_num (CARD s))) (int_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "3352", "title": "OPEN_IMP_FSIGMA", "text": "|- (!s. ((open s) ==> (fsigma s)))"}
{"_id": "3353", "title": "FRONTIER_TRANSLATION", "text": "|- (!a. (!s. ((frontier ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) (frontier s)))))"}
{"_id": "3354", "title": "CX_SIN", "text": "|- (!x. ((Cx (sin x)) = (csin (Cx x))))"}
{"_id": "3355", "title": "CENTRE_IN_BALL", "text": "|- (!x. (!e. ((x IN (ball (x , e))) = ((real_lt (real_of_num (NUMERAL _0))) e))))"}
{"_id": "3356", "title": "path_image", "text": "|- (!g. ((path_image g) = ((IMAGE g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))"}
{"_id": "3357", "title": "SUMS_INFSUM", "text": "|- (!f. (!s. ((((sums f) ((infsum s) f)) s) = ((summable s) f))))"}
{"_id": "3358", "title": "VECTOR_EQ", "text": "|- (!x. (!y. ((x = y) = ((((dot x) x) = ((dot x) y)) /\\ (((dot y) y) = ((dot x) x))))))"}
{"_id": "3359", "title": "IMAGE_AFFINITY_INTERVAL", "text": "|- (!a. (!b. (!m. (!c. (((IMAGE (\\x. ((vector_add ((% m) x)) c))) (closed_interval ((CONS (a , b)) NIL))) = (((COND ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (((COND ((real_le (real_of_num (NUMERAL _0))) m)) (closed_interval ((CONS (((vector_add ((% m) a)) c) , ((vector_add ((% m) b)) c))) NIL))) (closed_interval ((CONS (((vector_add ((% m) b)) c) , ((vector_add ((% m) a)) c))) NIL)))))))))"}
{"_id": "3360", "title": "CONVEX_BALL", "text": "|- (!x. (!e. (convex (ball (x , e)))))"}
{"_id": "3361", "title": "SIN_COS_EQ", "text": "|- (!x. (!y. ((((sin y) = (sin x)) /\\ ((cos y) = (cos x))) = (?n. ((integer n) /\\ (y = ((real_add x) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul n) pi)))))))))"}
{"_id": "3362", "title": "BINOM_LT", "text": "|- (!n. (!k. ((n < k) ==> ((binom (n , k)) = (NUMERAL _0)))))"}
{"_id": "3363", "title": "Hypermap.is_node_going", "text": "|- (!H. (!L. (!x. (!y. (((((is_node_going H) L) x) y) = (?k. ((y = (((POWER (loop_map L)) k) x)) /\\ (!i. ((i <= k) ==> ((((POWER (loop_map L)) i) x) = (((POWER (inverse (node_map H))) i) x)))))))))))"}
{"_id": "3364", "title": "CSQRT_0", "text": "|- ((csqrt (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL _0))))"}
{"_id": "3365", "title": "DIAMETER_BOUNDED_BOUND", "text": "|- (!s. (!x. (!y. (((bounded s) /\\ ((x IN s) /\\ (y IN s))) ==> ((real_le (vector_norm ((vector_sub x) y))) (diameter s))))))"}
{"_id": "3366", "title": "INTEGRABLE_ON_LITTLE_SUBINTERVALS", "text": "|- (!f. (!a. (!b. ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!u. (!v. (((x IN (closed_interval ((CONS (u , v)) NIL))) /\\ (((closed_interval ((CONS (u , v)) NIL)) SUBSET (ball (x , d))) /\\ ((closed_interval ((CONS (u , v)) NIL)) SUBSET (closed_interval ((CONS (a , b)) NIL))))) ==> ((integrable_on f) (closed_interval ((CONS (u , v)) NIL)))))))))) ==> ((integrable_on f) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "3367", "title": "STC_SYM", "text": "|- (!R. (!x. (!y. ((((STC R) x) y) ==> (((STC R) y) x)))))"}
{"_id": "3368", "title": "TENDSTO_REAL", "text": "|- (((---> s) l) = ((--> (lift o s)) (lift l)))"}
{"_id": "3369", "title": "CONTINUOUS_WITHIN_ID", "text": "|- (!a. (!s. ((continuous (\\x. x)) ((within (at a)) s))))"}
{"_id": "3370", "title": "EXTENSION_INTO_AR_LOCAL", "text": "|- (!f. (!c. (!s. (!t. ((((continuous_on f) c) /\\ ((((IMAGE f) c) SUBSET t) /\\ (((retract_of t) UNIV) /\\ ((closed_in ((subtopology euclidean) s)) c)))) ==> (?g. (((continuous_on g) s) /\\ ((((IMAGE g) UNIV) SUBSET t) /\\ (!x. ((x IN c) ==> ((g x) = (f x))))))))))))"}
{"_id": "3371", "title": "EXISTS_PASTECART", "text": "|- ((?p. (P p)) = (?x. (?y. (P ((pastecart x) y)))))"}
{"_id": "3372", "title": "NETLIMIT_AT", "text": "|- (!a. ((netlimit (at a)) = a))"}
{"_id": "3373", "title": "MEASURABLE_ON_PREIMAGE_OPEN", "text": "|- (!f. (((measurable_on f) UNIV) = (!t. ((open t) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3534. (?x. (((SETSPEC GEN%PVAR%3534) ((f x) IN t)) x)))))))))"}
{"_id": "3374", "title": "OUTSIDE_COMPACT_IN_OPEN", "text": "|- (!s. (!t. (((compact s) /\\ ((open t) /\\ ((s SUBSET t) /\\ (~ (t = EMPTY))))) ==> (~ (((outside s) INTER t) = EMPTY)))))"}
{"_id": "3375", "title": "MEM", "text": "|- ((((MEM x) NIL) = F) /\\ (((MEM x) ((CONS h) t)) = ((x = h) \\/ ((MEM x) t))))"}
{"_id": "3376", "title": "VECTOR_ADD_RID", "text": "|- (!x. (((vector_add x) (vec (NUMERAL _0))) = x))"}
{"_id": "3377", "title": "DIST_REFL", "text": "|- (!x. ((distance (x , x)) = (real_of_num (NUMERAL _0))))"}
{"_id": "3378", "title": "CONNECTED_PATH_IMAGE", "text": "|- (!g. ((path g) ==> (connected (path_image g))))"}
{"_id": "3379", "title": "IM_COMPLEX_INV_GE_0", "text": "|- (!z. (((real_le (real_of_num (NUMERAL _0))) (Im (complex_inv z))) = ((real_le (Im z)) (real_of_num (NUMERAL _0)))))"}
{"_id": "3380", "title": "EXP_EQ_0", "text": "|- (!m. (!n. (((m EXP n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) /\\ (~ (n = (NUMERAL _0)))))))"}
{"_id": "3381", "title": "infsum", "text": "|- (!f. (!s. (((infsum s) f) = (@l. (((sums f) l) s)))))"}
{"_id": "3382", "title": "INTEGRABLE_0", "text": "|- (!s. ((integrable_on (\\x. (vec (NUMERAL _0)))) s))"}
{"_id": "3383", "title": "LAMBDA_SKOLEM", "text": "|- ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (?x. ((P i) x)))) = (?x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((P i) (x $ i))))))"}
{"_id": "3384", "title": "AFF_DIM_INJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((aff_dim ((IMAGE f) s)) = (aff_dim s)))))"}
{"_id": "3385", "title": "SURJECTIVE_IMAGE", "text": "|- (!f. ((!t. (?s. (((IMAGE f) s) = t))) = (!y. (?x. ((f x) = y)))))"}
{"_id": "3386", "title": "Real_ext.REAL_SV_SQRT_0", "text": "|- ((sqrt (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))"}
{"_id": "3387", "title": "BILINEAR_COMPLEX_MUL", "text": "|- (bilinear complex_mul)"}
{"_id": "3388", "title": "COMPACT_UNION", "text": "|- (!s. (!t. (((compact s) /\\ (compact t)) ==> (compact (s UNION t)))))"}
{"_id": "3389", "title": "Seq.pred1", "text": "|- (!a. ((pred1 a) = (\\x. (x = a))))"}
{"_id": "3390", "title": "REAL_SUP_UNIQUE", "text": "|- (!s. (!b. (((!x. ((x IN s) ==> ((real_le x) b))) /\\ (!b'. (((real_lt b') b) ==> (?x. ((x IN s) /\\ ((real_lt b') x)))))) ==> ((sup s) = b))))"}
{"_id": "3391", "title": "Misc_defs_and_lemmas.REAL_ADD", "text": "|- (!m. (!n. (((real_add (real_of_num m)) (real_of_num n)) = (real_of_num (m + n)))))"}
{"_id": "3392", "title": "Functional_equation.uni", "text": "|- (!f. (!x. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((uni (f , x)) x1) x2) x3) x4) x5) x6) = (f ((((((x x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "3393", "title": "SYMMETRIC_MATRIX_INV_RMUL", "text": "|- (!A. ((transp ((matrix_mul A) (matrix_inv A))) = ((matrix_mul A) (matrix_inv A))))"}
{"_id": "3394", "title": "AFFINE_HULL_CONVEX_INTER_NONEMPTY_INTERIOR", "text": "|- (!s. (!t. (((convex s) /\\ (~ ((s INTER (interior t)) = EMPTY))) ==> (((hull affine) (s INTER t)) = ((hull affine) s)))))"}
{"_id": "3395", "title": "BETWEEN_MIDPOINT", "text": "|- (!a. (!b. (((between (midpoint (a , b))) (a , b)) /\\ ((between (midpoint (a , b))) (b , a)))))"}
{"_id": "3396", "title": "Seq.uniq_size_uniq", "text": "|- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> ((uniq s2) = ((sizel s2) = (sizel s1)))))))"}
{"_id": "3397", "title": "CLOSED_IN_TRANSLATION_EQ", "text": "|- (!a. (!s. (!t. (((closed_in ((subtopology euclidean) ((IMAGE (\\x. ((vector_add a) x))) t))) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((closed_in ((subtopology euclidean) t)) s)))))"}
{"_id": "3398", "title": "FINITE_IMAGE_INJ_EQ", "text": "|- (!f. (!s. ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) ==> ((FINITE ((IMAGE f) s)) = (FINITE s)))))"}
{"_id": "3399", "title": "INSERT", "text": "|- ((x INSERT s) = (GSPEC (\\GEN%PVAR%5. (?y. (((SETSPEC GEN%PVAR%5) ((y IN s) \\/ (y = x))) y)))))"}
{"_id": "3400", "title": "Nonlinear_lemma.gcy_low", "text": "|- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((gcy y) = (gchi y))))"}
{"_id": "3401", "title": "ANTIDERIVATIVE_CONTINUOUS", "text": "|- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (((has_vector_derivative g) (f x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))))))))))"}
{"_id": "3402", "title": "PATH_COMPONENT_LINEAR_IMAGE", "text": "|- (!f. (!s. (!x. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> (((path_component ((IMAGE f) s)) (f x)) = ((IMAGE f) ((path_component s) x)))))))"}
{"_id": "3403", "title": "VECTOR_MUL_LCANCEL", "text": "|- (!a. (!x. (!y. ((((% a) x) = ((% a) y)) = ((a = (real_of_num (NUMERAL _0))) \\/ (x = y))))))"}
{"_id": "3404", "title": "BETWEEN_DIST_LT", "text": "|- (!r. (!a. (!b. (!c. ((((real_lt (distance (c , a))) r) /\\ (((real_lt (distance (c , b))) r) /\\ ((between x) (a , b)))) ==> ((real_lt (distance (c , x))) r))))))"}
{"_id": "3405", "title": "Sphere.sol0", "text": "|- (sol0 = ((real_sub ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) (acs ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))) pi))"}
{"_id": "3406", "title": "ANALYTIC_AT", "text": "|- (!f. (!z. (((analytic_on f) (z INSERT EMPTY)) = (?s. ((open s) /\\ ((z IN s) /\\ ((holomorphic_on f) s)))))))"}
{"_id": "3407", "title": "ABSOLUTELY_INTEGRABLE_CMUL_EQ", "text": "|- (!f. (!s. (!c. (((absolutely_integrable_on (\\x. ((% c) (f x)))) s) = ((c = (real_of_num (NUMERAL _0))) \\/ ((absolutely_integrable_on f) s))))))"}
{"_id": "3408", "title": "IMAGE_LIFT_UNIV", "text": "|- (((IMAGE lift) UNIV) = UNIV)"}
{"_id": "3409", "title": "CLOSEST_POINT_UNIQUE", "text": "|- (!s. (!a. (!x. (((convex s) /\\ ((closed s) /\\ ((x IN s) /\\ (!z. ((z IN s) ==> ((real_le (distance (a , x))) (distance (a , z)))))))) ==> (x = ((closest_point s) a))))))"}
{"_id": "3410", "title": "FORALL_IN_CLAUSES", "text": "|- ((!P. ((!x. ((x IN EMPTY) ==> (P x))) = T)) /\\ (!P. (!a. (!s. ((!x. ((x IN (a INSERT s)) ==> (P x))) = ((P a) /\\ (!x. ((x IN s) ==> (P x)))))))))"}
{"_id": "3411", "title": "VECTOR_VARIATION_GE_NORM_FUNCTION", "text": "|- (!f. (!s. (!a. (!b. ((((has_bounded_variation_on f) s) /\\ ((closed_segment ((CONS (a , b)) NIL)) SUBSET s)) ==> ((real_le (vector_norm ((vector_sub (f b)) (f a)))) ((vector_variation s) f)))))))"}
{"_id": "3412", "title": "LIM_WITHIN_OPEN", "text": "|- (!f. (!l. (!a. (!s. (((a IN s) /\\ (open s)) ==> ((((--> f) l) ((within (at a)) s)) = (((--> f) l) (at a))))))))"}
{"_id": "3413", "title": "Float.POW_1", "text": "|- (!x. (((real_pow x) (NUMERAL (BIT1 _0))) = x))"}
{"_id": "3414", "title": "NORM_NEG", "text": "|- (!x. ((vector_norm (vector_neg x)) = (vector_norm x)))"}
{"_id": "3415", "title": "REAL_CONTINUOUS_AT_WITHIN", "text": "|- (!f. (!s. (!x. (((real_continuous f) (at x)) ==> ((real_continuous f) ((within (at x)) s))))))"}
{"_id": "3416", "title": "DOT_2", "text": "|- (((dot x) y) = ((real_add ((real_mul (x $ (NUMERAL (BIT1 _0)))) (y $ (NUMERAL (BIT1 _0))))) ((real_mul (x $ (NUMERAL (BIT0 (BIT1 _0))))) (y $ (NUMERAL (BIT0 (BIT1 _0)))))))"}
{"_id": "3417", "title": "Seq.cat1s", "text": "|- (!x. (!s. (((cat ((CONS x) NIL)) s) = ((CONS x) s))))"}
{"_id": "3418", "title": "LEBESGUE_MEASURABLE_ALMOST_FSIGMA", "text": "|- (!s. ((lebesgue_measurable s) ==> (?c. (?t. ((fsigma c) /\\ ((negligible t) /\\ (((c UNION t) = s) /\\ ((DISJOINT c) t))))))))"}
{"_id": "3419", "title": "Trigonometry1.NORM_CAUCHY_SCHWARZ_FRAC", "text": "|- (!u. (!v. (((~ (u = (vec (NUMERAL _0)))) /\\ (~ (v = (vec (NUMERAL _0))))) ==> (((real_le (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((real_div ((dot u) v)) ((real_mul (vector_norm u)) (vector_norm v)))) /\\ ((real_le ((real_div ((dot u) v)) ((real_mul (vector_norm u)) (vector_norm v)))) (real_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "3420", "title": "VECTOR_DIFFERENTIABLE", "text": "|- (!f. (!net. (((differentiable f) net) = (?f'. (((has_vector_derivative f) f') net)))))"}
{"_id": "3421", "title": "Sphere.h0", "text": "|- (h0 = ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))"}
{"_id": "3422", "title": "CARD_EXP_CONG", "text": "|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\\ (t =_c t')) ==> (((^_c s) t) =_c ((^_c s') t')))))))"}
{"_id": "3423", "title": "FORALL_IN_UNIONS", "text": "|- (!P. (!s. ((!x. ((x IN (UNIONS s)) ==> (P x))) = (!t. (!x. (((t IN s) /\\ (x IN t)) ==> (P x)))))))"}
{"_id": "3424", "title": "Bump.MCELL_CELL_PARAMETERS_EXIST", "text": "|- (!V. (!ul. (!k. (!X. (((k <= (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\\ ((packing V) /\\ ((saturated V) /\\ ((X = (((mcell k) V) ul)) /\\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\\ (~ (negligible X))))))) ==> ((FST ((cell_params V) X)) = k))))))"}
{"_id": "3425", "title": "Cuxvzoz.vv_azim_le_alt", "text": "|- (!vv. (!k. (!i. (!j. ((LET (\\V. (LET_END ((LET (\\E. (LET_END ((LET (\\f. (LET_END ((((periodic vv) k) /\\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\\ ((convex_local_fan (V , (E , f))) /\\ (!i. (!j. (((i < k) /\\ ((j < k) /\\ ((vv i) = (vv j)))) ==> (i = j))))))) ==> ((real_le ((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv j))) ((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + (k - (NUMERAL (BIT1 _0))))))))))) ((IMAGE (\\i. ((vv i) , (vv (SUC i))))) UNIV))))) ((IMAGE (\\i. ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY)))) UNIV))))) ((IMAGE vv) UNIV))))))"}
{"_id": "3426", "title": "Ssrnat.minnCA", "text": "|- (!m1. (!m2. (!m3. (((minn m1) ((minn m2) m3)) = ((minn m2) ((minn m1) m3))))))"}
{"_id": "3427", "title": "IN_CBALL_0", "text": "|- (!x. (!e. ((x IN (cball ((vec (NUMERAL _0)) , e))) = ((real_le (vector_norm x)) e))))"}
{"_id": "3428", "title": "Float.ABS_NUM_P", "text": "|- (!m. (!n. ((m <= n) ==> ((real_abs ((real_sub (real_of_num n)) (real_of_num m))) = (real_of_num ((m - n) + (n - m)))))))"}
{"_id": "3429", "title": "NUMERAL", "text": "|- (!n. ((NUMERAL n) = n))"}
{"_id": "3430", "title": "Real_ext.REAL_PROP_NN_DOUBLE", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) ((real_add x) x)) = ((real_le (real_of_num (NUMERAL _0))) x)))"}
{"_id": "3431", "title": "EXP_LOG", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((exp (log x)) = x)))"}
{"_id": "3432", "title": "PATH_CONNECTED_TRANSLATION_EQ", "text": "|- (!a. (!s. ((path_connected ((IMAGE (\\x. ((vector_add a) x))) s)) = (path_connected s))))"}
{"_id": "3433", "title": "REVERSEPATH_TRANSLATION", "text": "|- (!a. (!g. ((reversepath ((\\x. ((vector_add a) x)) o g)) = ((\\x. ((vector_add a) x)) o (reversepath g)))))"}
{"_id": "3434", "title": "MEASURE_LOCALLY_LIPSCHITZ_IMAGE", "text": "|- (!f. (!s. (!B. ((((dimindex UNIV) <= (dimindex UNIV)) /\\ ((measurable s) /\\ (!x. ((x IN s) ==> (?t. ((open t) /\\ ((x IN t) /\\ (!y. ((y IN (s INTER t)) ==> ((real_le (vector_norm ((vector_sub (f y)) (f x)))) ((real_mul B) (vector_norm ((vector_sub y) x))))))))))))) ==> ((real_le (measure ((IMAGE f) s))) ((real_mul ((real_pow B) (dimindex UNIV))) (measure s)))))))"}
{"_id": "3435", "title": "Conforming.conforming_solid_angle_fan", "text": "|- (!x. (!V. (!E. ((conforming_solid_angle_fan (x , (V , E))) = (!f. ((f IN (face_set (hypermap1_of_fanx (x , (V , E))))) ==> ((LET (\\U. (LET_END ((!r. (measurable ((ball (x , r)) INTER U))) /\\ (((eventually_radial x) U) /\\ (((sol x) U) = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) ((sum f) (\\y. ((real_sub (((((azim_fan x) V) E) (pr2 y)) (pr3 y))) pi)))))))))) ((((dartset_leads_into_fan x) V) E) f))))))))"}
{"_id": "3436", "title": "HYPERPLANE_EQ_EMPTY", "text": "|- (!a. (!b. (((GSPEC (\\GEN%PVAR%171. (?x. (((SETSPEC GEN%PVAR%171) (((dot a) x) = b)) x)))) = EMPTY) = ((a = (vec (NUMERAL _0))) /\\ (~ (b = (real_of_num (NUMERAL _0))))))))"}
{"_id": "3437", "title": "IS_REALINTERVAL_CONVEX", "text": "|- (!s. ((is_realinterval s) = (convex ((IMAGE lift) s))))"}
{"_id": "3438", "title": "SUMMABLE_ZETA", "text": "|- (!n. (!z. (((real_lt (real_of_num (NUMERAL (BIT1 _0)))) (Re z)) ==> ((summable (from n)) (\\k. (complex_inv ((cpow (Cx (real_of_num k))) z)))))))"}
{"_id": "3439", "title": "REAL_SUB_TRIANGLE", "text": "|- (!a. (!b. (!c. (((real_add ((real_sub a) b)) ((real_sub b) c)) = ((real_sub a) c)))))"}
{"_id": "3440", "title": "IN_CARD_MUL", "text": "|- (!s. (!t. (!x. (!y. (((x , y) IN ((*_c s) t)) = ((x IN s) /\\ (y IN t)))))))"}
{"_id": "3441", "title": "REAL_ABS_NUM", "text": "|- (!n. ((real_abs (real_of_num n)) = (real_of_num n)))"}
{"_id": "3442", "title": "Tame_defs.restricted_hypermap", "text": "|- (!H. ((restricted_hypermap H) = ((is_no_double_joins H) /\\ ((~ ((dart H) = EMPTY)) /\\ ((planar_hypermap H) /\\ ((connected_hypermap H) /\\ ((plain_hypermap H) /\\ ((simple_hypermap H) /\\ ((is_edge_nondegenerate H) /\\ ((is_node_nondegenerate H) /\\ (!f. ((f IN (face_set H)) ==> ((CARD f) >= (NUMERAL (BIT1 (BIT1 _0))))))))))))))))"}
{"_id": "3443", "title": "APPEND_NIL", "text": "|- (!l. (((APPEND l) NIL) = l))"}
{"_id": "3444", "title": "BILINEAR_SWAP", "text": "|- (!op. ((bilinear (\\x. (\\y. ((op y) x)))) = (bilinear op)))"}
{"_id": "3445", "title": "triangulation", "text": "|- (!tr. ((triangulation tr) = ((FINITE tr) /\\ ((!t. ((t IN tr) ==> (?n. ((simplex n) t)))) /\\ (!t. (!t'. (((t IN tr) /\\ (t' IN tr)) ==> (((face_of (t INTER t')) t) /\\ ((face_of (t INTER t')) t')))))))))"}
{"_id": "3446", "title": "Hypermap_and_fan.DART_EXISTS", "text": "|- (!V. (!E. (!v. ((v IN V) ==> (?w. ((v , w) IN (dart_of_fan (V , E))))))))"}
{"_id": "3447", "title": "HAS_REAL_INTEGRAL_LBOUND", "text": "|- (!f. (!a. (!b. (!i. ((((real_le a) b) /\\ ((((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))) /\\ (!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_le B) (f x)))))) ==> ((real_le ((real_mul B) ((real_sub b) a))) i))))))"}
{"_id": "3448", "title": "FSIGMA_IMP_BOREL", "text": "|- (!s. ((fsigma s) ==> (borel s)))"}
{"_id": "3449", "title": "REAL_LE_01", "text": "|- ((real_le (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "3450", "title": "Appendix.scs_bm_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_bm_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = beta))))))))))"}
{"_id": "3451", "title": "NSUM_CLAUSES_NUMSEG", "text": "|- ((!m. (((nsum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (NUMERAL _0)))) /\\ (!m. (!n. (((nsum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) (((nsum (m .. n)) f) + (f (SUC n)))) ((nsum (m .. n)) f))))))"}
{"_id": "3452", "title": "DEPENDENT_2", "text": "|- (!a. (!b. ((dependent (a INSERT (b INSERT EMPTY))) = (((COND (a = b)) (a = (vec (NUMERAL _0)))) (?x. (?y. ((((vector_add ((% x) a)) ((% y) b)) = (vec (NUMERAL _0))) /\\ (~ ((x = (real_of_num (NUMERAL _0))) /\\ (y = (real_of_num (NUMERAL _0))))))))))))"}
{"_id": "3453", "title": "Seq2.uniq_nthP", "text": "|- (!x0. (!s. ((uniq s) = (!i. (!j. (((i < j) /\\ (j < (sizel s))) ==> (~ ((((nth x0) s) i) = (((nth x0) s) j)))))))))"}
{"_id": "3454", "title": "Oxl_2012.CARD_INSERT", "text": "|- (!x. (!s. ((FINITE s) ==> ((CARD (x INSERT s)) = (((COND (x IN s)) (CARD s)) (SUC (CARD s)))))))"}
{"_id": "3455", "title": "NOT_FORALL_THM", "text": "|- (!P. ((~ (!x. (P x))) = (?x. (~ (P x)))))"}
{"_id": "3456", "title": "COMPLEX_MUL_RNEG", "text": "|- (!x. (!y. (((complex_mul x) (vector_neg y)) = (vector_neg ((complex_mul x) y)))))"}
{"_id": "3457", "title": "CONNECTED_INTERMEDIATE_CLOSURE", "text": "|- (!s. (!t. (((connected s) /\\ ((s SUBSET t) /\\ (t SUBSET (closure s)))) ==> (connected t))))"}
{"_id": "3458", "title": "Ssrnat.leq_trans", "text": "|- (!n. (!m. (!p. ((m <= n) ==> ((n <= p) ==> (m <= p))))))"}
{"_id": "3459", "title": "COUNTABLE_OPEN_COMPONENTS", "text": "|- (!s. ((open s) ==> (COUNTABLE (components s))))"}
{"_id": "3460", "title": "submetric", "text": "|- (!s. (!m. (((submetric m) s) = (metric ((s INTER (mspace m)) , (mdist m))))))"}
{"_id": "3461", "title": "BIJECTIONS_CARD_EQ", "text": "|- (!s. (!t. (!f. (!g. ((((FINITE s) \\/ (FINITE t)) /\\ ((!x. ((x IN s) ==> (((f x) IN t) /\\ ((g (f x)) = x)))) /\\ (!y. ((y IN t) ==> (((g y) IN s) /\\ ((f (g y)) = y)))))) ==> ((CARD s) = (CARD t)))))))"}
{"_id": "3462", "title": "CONTINUOUS_ABS", "text": "|- (!f. (!net. (((continuous f) net) ==> ((continuous (\\x. (lambdai. (real_abs ((f x) $ i))))) net))))"}
{"_id": "3463", "title": "LT_REFL", "text": "|- (!n. (~ (n < n)))"}
{"_id": "3464", "title": "CONTINUOUS_WITHINREAL", "text": "|- (!f. (!x. (!s. (((continuous f) ((within (atreal x)) s)) = (((--> f) (f x)) ((within (atreal x)) s))))))"}
{"_id": "3465", "title": "CRAMER_MATRIX_LEFT", "text": "|- (!A. (!X. (!B. ((~ ((det A) = (real_of_num (NUMERAL _0)))) ==> ((((matrix_mul X) A) = B) = (X = (lambdak. (lambdal. ((real_div (det (lambdai. (lambdaj. (((COND (j = l)) ((B $ k) $ i)) ((A $ j) $ i)))))) (det A))))))))))"}
{"_id": "3466", "title": "lin_combo", "text": "|- (!V. (!f. (((lin_combo V) f) = ((vsum V) (\\v. ((% (f v)) v))))))"}
{"_id": "3467", "title": "SUB_ELIM_THM", "text": "|- ((P (a - b)) = (!d. (((a = (b + d)) \\/ ((a < b) /\\ (d = (NUMERAL _0)))) ==> (P d))))"}
{"_id": "3468", "title": "REAL_MEASURABLE_COUNTABLE_UNIONS_STRONG", "text": "|- (!s. (!B. (((!n. (real_measurable (s n))) /\\ (!n. ((real_le (real_measure (UNIONS (GSPEC (\\GEN%PVAR%4945. (?k. (((SETSPEC GEN%PVAR%4945) (k <= n)) (s k)))))))) B))) ==> (real_measurable (UNIONS (GSPEC (\\GEN%PVAR%4946. (?n. (((SETSPEC GEN%PVAR%4946) (n IN UNIV)) (s n))))))))))"}
{"_id": "3469", "title": "Ssrnat.addn0", "text": "|- (!n. ((n + (NUMERAL _0)) = n))"}
{"_id": "3470", "title": "IM_MUL_II", "text": "|- (!z. (((Im ((complex_mul z) ii)) = (Re z)) /\\ ((Im ((complex_mul ii) z)) = (Re z))))"}
{"_id": "3471", "title": "BROUWER_INESSENTIAL_ANR", "text": "|- (!f. (!s. (((compact s) /\\ ((~ (s = EMPTY)) /\\ ((ANR s) /\\ (((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET s) /\\ (?a. ((((homotopic_with (\\x. T)) (s , s)) f) (\\x. a)))))))) ==> (?x. ((x IN s) /\\ ((f x) = x))))))"}
{"_id": "3472", "title": "HOMEOMORPHIC_SIMPLE_PATH_ARC", "text": "|- (!g. (!h. (((arc g) /\\ ((simple_path h) /\\ ((homeomorphic (path_image g)) (path_image h)))) ==> (arc h))))"}
{"_id": "3473", "title": "List_hypermap.good_list_nodes_condition", "text": "|- (!L. ((good_list L) ==> ((good_list_nodes L) = ((CARD (node_set (hypermap_of_list L))) = (sizel (list_of_elements L))))))"}
{"_id": "3474", "title": "CLOSURE_INTER_SUBSET", "text": "|- (!s. (!t. ((closure (s INTER t)) SUBSET ((closure s) INTER (closure t)))))"}
{"_id": "3475", "title": "aff_gt_def", "text": "|- (aff_gt = (affsign sgn_gt))"}
{"_id": "3476", "title": "AFFINE_EQ_SUBSPACE", "text": "|- (!s. (((vec (NUMERAL _0)) IN s) ==> ((affine s) = (subspace s))))"}
{"_id": "3477", "title": "LINSEG_INSEG", "text": "|- (!l. (!a. ((woset l) ==> ((inseg ((linseg l) a)) l))))"}
{"_id": "3478", "title": "List_hypermap.mem_list_pairs", "text": "|- (!x. (!y. (!l. (((MEM (x , y)) (list_pairs l)) ==> (((MEM x) l) /\\ ((MEM y) l))))))"}
{"_id": "3479", "title": "INT_MUL_LID", "text": "|- (!x. (((int_mul (int_of_num (NUMERAL (BIT1 _0)))) x) = x))"}
{"_id": "3480", "title": "Lp_ineqs_proofs.lp_fan", "text": "|- (!E. (!V. ((lp_fan (V , E)) = ((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((fully_surrounded (V , E)) /\\ ((!v. (!w. (((v IN V) /\\ ((w IN V) /\\ ((~ (v = w)) /\\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (distance (v , w)))))) /\\ ((V SUBSET ball_annulus) /\\ (packing V))))))))"}
{"_id": "3481", "title": "REAL_ABS_REFL", "text": "|- (!x. (((real_abs x) = x) = ((real_le (real_of_num (NUMERAL _0))) x)))"}
{"_id": "3482", "title": "aff_lt_def", "text": "|- (aff_lt = (affsign sgn_lt))"}
{"_id": "3483", "title": "COMPLEX_SUB_POW_R1", "text": "|- (!x. (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (((vector_sub ((complex_pow x) n)) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = ((complex_mul ((vector_sub x) (Cx (real_of_num (NUMERAL (BIT1 _0)))))) ((vsum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\\i. ((complex_pow x) i))))))))"}
{"_id": "3484", "title": "Ssrnat.leqif_imp_eq", "text": "|- (!m. (!n. (!c. ((((leqif m) n) c) ==> ((m = n) = c)))))"}
{"_id": "3485", "title": "LOG_INV", "text": "|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log (real_inv x)) = (real_neg (log x)))))"}
{"_id": "3486", "title": "REAL_POW_ADD", "text": "|- (!x. (!m. (!n. (((real_pow x) (m + n)) = ((real_mul ((real_pow x) m)) ((real_pow x) n))))))"}
{"_id": "3487", "title": "num_gcd", "text": "|- (!a. (!b. ((num_gcd (a , b)) = (num_of_int (int_gcd ((int_of_num a) , (int_of_num b)))))))"}
{"_id": "3488", "title": "Appendix.mk_unadorned_v39", "text": "|- (!k. (!d. (!a. (!b. (((((mk_unadorned_v39 k) d) a) b) = (scs_v39 (k , (d , (a , (a , (b , (b , ((\\i. (\\j. F)) , (EMPTY , (EMPTY , EMPTY)))))))))))))))"}
{"_id": "3489", "title": "PASTECART_FST_SND", "text": "|- (!z. (((pastecart (fstcart z)) (sndcart z)) = z))"}
{"_id": "3490", "title": "CARD_EQ_OPEN_IN", "text": "|- (!u. (!s. ((((locally connected) u) /\\ (((open_in ((subtopology euclidean) u)) s) /\\ (?x. ((x IN s) /\\ ((limit_point_of x) u))))) ==> (s =_c UNIV))))"}
{"_id": "3491", "title": "CLOSURE_SUBSET", "text": "|- (!s. (s SUBSET (closure s)))"}
{"_id": "3492", "title": "HOMEOMORPHISM_INJECTIVE_OPEN_MAP_EQ", "text": "|- (!f. (!s. (!t. ((((continuous_on f) s) /\\ ((((IMAGE f) s) = t) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))))) ==> ((?g. ((homeomorphism (s , t)) (f , g))) = (!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))"}
{"_id": "3493", "title": "HAS_ABSOLUTE_INTEGRAL_CHANGE_OF_VARIABLES", "text": "|- (!f. (!g. (!g'. (!s. (!b. (((lebesgue_measurable s) /\\ ((!x. ((x IN s) ==> (((has_derivative g) (g' x)) ((within (at x)) s)))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((g x) = (g y)))) ==> (x = y)))))) ==> ((((absolutely_integrable_on (\\x. ((% (real_abs (det (matrix (g' x))))) (f (g x))))) s) /\\ (((integral s) (\\x. ((% (real_abs (det (matrix (g' x))))) (f (g x))))) = b)) = (((absolutely_integrable_on f) ((IMAGE g) s)) /\\ (((integral ((IMAGE g) s)) f) = b)))))))))"}
{"_id": "3494", "title": "vector_angle", "text": "|- (!x. (!y. (((vector_angle x) y) = (((COND ((x = (vec (NUMERAL _0))) \\/ (y = (vec (NUMERAL _0))))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (acs ((real_div ((dot x) y)) ((real_mul (vector_norm x)) (vector_norm y))))))))"}
{"_id": "3495", "title": "STARLIKE_IMP_CONTRACTIBLE", "text": "|- (!s. ((starlike s) ==> (contractible s)))"}
{"_id": "3496", "title": "WF_ANTISYM", "text": "|- (!<<. (!x. (!y. ((WF <<) ==> (~ (((<< x) y) /\\ ((<< y) x)))))))"}
{"_id": "3497", "title": "SC_INDUCT", "text": "|- (!R. (!SC'. (((!x. (!y. (((R x) y) ==> ((SC' x) y)))) /\\ (!x. (!y. (((SC' x) y) ==> ((SC' y) x))))) ==> (!a0. (!a1. ((((SC R) a0) a1) ==> ((SC' a0) a1)))))))"}
{"_id": "3498", "title": "SUM_CONST", "text": "|- (!c. (!s. ((FINITE s) ==> (((sum s) (\\n. c)) = ((real_mul (real_of_num (CARD s))) c)))))"}
{"_id": "3499", "title": "IMAGE_AFFINITY_REAL_INTERVAL", "text": "|- (!a. (!b. (!m. (!c. (((IMAGE (\\x. ((real_add ((real_mul m) x)) c))) (closed_real_interval ((CONS (a , b)) NIL))) = (((COND ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (((COND ((real_le (real_of_num (NUMERAL _0))) m)) (closed_real_interval ((CONS (((real_add ((real_mul m) a)) c) , ((real_add ((real_mul m) b)) c))) NIL))) (closed_real_interval ((CONS (((real_add ((real_mul m) b)) c) , ((real_add ((real_mul m) a)) c))) NIL)))))))))"}
{"_id": "3500", "title": "INTERVAL_SUBSET_IS_INTERVAL", "text": "|- (!s. (!a. (!b. ((is_interval s) ==> (((closed_interval ((CONS (a , b)) NIL)) SUBSET s) = (((closed_interval ((CONS (a , b)) NIL)) = EMPTY) \\/ ((a IN s) /\\ (b IN s))))))))"}
{"_id": "3501", "title": "SIMPLE_PATH_LINEAR_IMAGE_EQ", "text": "|- (!f. (!g. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((simple_path (f o g)) = (simple_path g)))))"}
{"_id": "3502", "title": "EXPOSED_POINT_OF_INTER_SUPPORTING_HYPERPLANE_LE", "text": "|- (!s. (!a. (!b. (!c. (((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\\ ((s INTER (GSPEC (\\GEN%PVAR%2202. (?x. (((SETSPEC GEN%PVAR%2202) (((dot a) x) = b)) x))))) = (c INSERT EMPTY))) ==> ((exposed_face_of (c INSERT EMPTY)) s))))))"}
{"_id": "3503", "title": "NOT_COPLANAR_NOT_COLLINEAR", "text": "|- (!v1. (!v2. (!v3. (!w. ((~ (coplanar (v1 INSERT (v2 INSERT (v3 INSERT (w INSERT EMPTY)))))) ==> (~ (collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))))))))"}
{"_id": "3504", "title": "CONNECTED_CHAIN_GEN", "text": "|- (!f. (((!s. ((s IN f) ==> ((closed s) /\\ (connected s)))) /\\ ((?s. ((s IN f) /\\ (compact s))) /\\ (!s. (!t. (((s IN f) /\\ (t IN f)) ==> ((s SUBSET t) \\/ (t SUBSET s))))))) ==> (connected (INTERS f))))"}
{"_id": "3505", "title": "CONTENT_POS_LT", "text": "|- (!a. (!b. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (content (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "3506", "title": "LINEAR_INJECTIVE_ISOMORPHISM", "text": "|- (!f. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (?f'. ((linear f') /\\ ((!x. ((f' (f x)) = x)) /\\ (!x. ((f (f' x)) = x)))))))"}
{"_id": "3507", "title": "MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET_0", "text": "|- (!f. (!g. (!s. (!t. (((closed s) /\\ (((measurable_on f) t) /\\ ((!x. ((f x) IN s)) /\\ (((continuous_on g) s) /\\ (((vec (NUMERAL _0)) IN s) /\\ ((g (vec (NUMERAL _0))) = (vec (NUMERAL _0)))))))) ==> ((measurable_on (g o f)) t))))))"}
{"_id": "3508", "title": "REAL_INTERVAL_TRANSLATION", "text": "|- ((!c. (!a. (!b. ((closed_real_interval ((CONS (((real_add c) a) , ((real_add c) b))) NIL)) = ((IMAGE (\\x. ((real_add c) x))) (closed_real_interval ((CONS (a , b)) NIL))))))) /\\ (!c. (!a. (!b. ((open_real_interval (((real_add c) a) , ((real_add c) b))) = ((IMAGE (\\x. ((real_add c) x))) (open_real_interval (a , b))))))))"}
{"_id": "3509", "title": "BANACH_FIX", "text": "|- (!f. (!s. (!c. (((complete s) /\\ ((~ (s = EMPTY)) /\\ (((real_le (real_of_num (NUMERAL _0))) c) /\\ (((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\\ ((((IMAGE f) s) SUBSET s) /\\ (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (distance ((f x) , (f y)))) ((real_mul c) (distance (x , y)))))))))))) ==> (?!x. ((x IN s) /\\ ((f x) = x)))))))"}
{"_id": "3510", "title": "DIFFERENTIABLE_IMP_CONTINUOUS_ON", "text": "|- (!f. (!s. (((differentiable_on f) s) ==> ((continuous_on f) s))))"}
{"_id": "3511", "title": "CX_NEG", "text": "|- (!x. ((Cx (real_neg x)) = (vector_neg (Cx x))))"}
{"_id": "3512", "title": "NPRODUCT_EQ_1", "text": "|- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (NUMERAL (BIT1 _0))))) ==> (((nproduct s) f) = (NUMERAL (BIT1 _0))))))"}
{"_id": "3513", "title": "COMPLEX_MUL_CSIN_CCOS", "text": "|- (!w. (!z. (((complex_mul (csin w)) (ccos z)) = ((complex_div ((vector_add (csin ((vector_add w) z))) (csin ((vector_sub w) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "3514", "title": "SERIES_DIRICHLET", "text": "|- (!f. (!g. (!N. (!k. (!m. (((bounded (GSPEC (\\GEN%PVAR%1126. (?n. (((SETSPEC GEN%PVAR%1126) (n IN UNIV)) ((vsum (m .. n)) f)))))) /\\ ((!n. ((N <= n) ==> ((real_le (g (n + (NUMERAL (BIT1 _0))))) (g n)))) /\\ (((--> (lift o g)) (vec (NUMERAL _0))) sequentially))) ==> ((summable (from k)) (\\n. ((% (g n)) (f n))))))))))"}
{"_id": "3515", "title": "Nonlinear_lemma.vol3f_palt", "text": "|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!r. (!f. (((y3 = r) /\\ ((y4 = r) /\\ (y5 = r))) ==> ((((((vol3f y1) y2) y6) r) f) = ((real_sub ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_y y1) y2) y3) y4) y5) y6))) ((real_add ((((((dih3_y y1) y2) y3) y4) y5) y6)) ((real_add ((((((dih4_y y1) y2) y3) y4) y5) y6)) ((real_sub ((((((dih5_y y1) y2) y3) y4) y5) y6)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi))))))))) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi))) ((real_add ((real_mul (f ((real_div y1) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (f ((real_div y2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih2_y y1) y2) y3) y4) y5) y6))) ((real_mul (f ((real_div y6) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih6_y y1) y2) y3) y4) y5) y6))))))))))))))))"}
{"_id": "3516", "title": "IS_INTERVAL_INTERVAL", "text": "|- (!a. (!b. ((is_interval (open_interval (a , b))) /\\ (is_interval (closed_interval ((CONS (a , b)) NIL))))))"}
{"_id": "3517", "title": "CLOSURE_BALL", "text": "|- (!x. (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((closure (ball (x , e))) = (cball (x , e))))))"}
{"_id": "3518", "title": "HULL_HULL", "text": "|- (!P. (!s. (((hull P) ((hull P) s)) = ((hull P) s))))"}
{"_id": "3519", "title": "NADD_MUL_SYM", "text": "|- (!x. (!y. ((nadd_eq ((nadd_mul x) y)) ((nadd_mul y) x))))"}
{"_id": "3520", "title": "EXP_ADD", "text": "|- (!m. (!n. (!p. ((m EXP (n + p)) = ((m EXP n) * (m EXP p))))))"}
{"_id": "3521", "title": "IM_MUL_CX", "text": "|- (!x. (!z. (((Im ((complex_mul (Cx x)) z)) = ((real_mul x) (Im z))) /\\ ((Im ((complex_mul z) (Cx x))) = ((real_mul (Im z)) x)))))"}
{"_id": "3522", "title": "MONOIDAL_ADD", "text": "|- (monoidal +)"}
{"_id": "3523", "title": "LIM_COMPLEX_MUL", "text": "|- (!net. (!f. (!g. (!l. (!m. (((((--> f) l) net) /\\ (((--> g) m) net)) ==> (((--> (\\x. ((complex_mul (f x)) (g x)))) ((complex_mul l) m)) net)))))))"}
{"_id": "3524", "title": "rotation_matrix", "text": "|- (!Q. ((rotation_matrix Q) = ((orthogonal_matrix Q) /\\ ((det Q) = (real_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "3525", "title": "Ssrnat.leq_addr", "text": "|- (!m. (!n. (n <= (n + m))))"}
{"_id": "3526", "title": "Ssrnat.halfS", "text": "|- (!n. ((half (SUC n)) = (uphalf n)))"}
{"_id": "3527", "title": "SIMPLEX_FURTHEST_LE", "text": "|- (!a. (!s. (((FINITE s) /\\ (~ (s = EMPTY))) ==> (?y. ((y IN s) /\\ (!x. ((x IN ((hull convex) s)) ==> ((real_le (vector_norm ((vector_sub x) a))) (vector_norm ((vector_sub y) a))))))))))"}
{"_id": "3528", "title": "Packing3.VORONOI_SET_2", "text": "|- (!V. (!u. (!v. (((voronoi_set V) (u INSERT (v INSERT EMPTY))) = (((voronoi_closed V) v) INTER ((voronoi_closed V) u))))))"}
{"_id": "3529", "title": "RESTRICTION_DEFINED", "text": "|- (!s. (!f. (!x. ((x IN s) ==> ((((RESTRICTION s) f) x) = (f x))))))"}
{"_id": "3530", "title": "Trigonometry.JLWZFBH4", "text": "|- (!s. ((coplanar s) = (?u. (?v. (?w. (s SUBSET ((hull affine) (u INSERT (v INSERT (w INSERT EMPTY))))))))))"}
{"_id": "3531", "title": "DIV_LE_EXCLUSION", "text": "|- (!a. (!b. (!c. (!d. (((~ (b = (NUMERAL _0))) /\\ ((b * c) < ((a + (NUMERAL (BIT1 _0))) * d))) ==> ((c DIV d) <= (a DIV b)))))))"}
{"_id": "3532", "title": "ARITH_SUC", "text": "|- ((!n. ((SUC (NUMERAL n)) = (NUMERAL (SUC n)))) /\\ (((SUC _0) = (BIT1 _0)) /\\ ((!n. ((SUC (BIT0 n)) = (BIT1 n))) /\\ (!n. ((SUC (BIT1 n)) = (BIT0 (SUC n)))))))"}
{"_id": "3533", "title": "REAL_NEG_RMUL", "text": "|- (!x. (!y. ((real_neg ((real_mul x) y)) = ((real_mul x) (real_neg y)))))"}
{"_id": "3534", "title": "Conforming.SIGMA_FAN_OF_FANADD_AT_POINT2", "text": "|- (!x. (!V. (!E. (!E1. (!v. (!u. (!w. (((FAN (x , (V , E))) /\\ ((FAN (x , (V , E1))) /\\ (((v INSERT (u INSERT EMPTY)) IN E) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\\ (((((((sigma_fan x) V) E) u) w) = v) /\\ ((fan80 (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))) ==> ((((((sigma_fan x) V) E1) v) u) = w)))))))))"}
{"_id": "3535", "title": "Add_triangle.pair_expand", "text": "|- (!p. (p = ((FST p) , (SND p))))"}
{"_id": "3536", "title": "REAL_LTE_ADD2", "text": "|- (!w. (!x. (!y. (!z. ((((real_lt w) x) /\\ ((real_le y) z)) ==> ((real_lt ((real_add w) y)) ((real_add x) z)))))))"}
{"_id": "3537", "title": "CARD_MUL_FINITE", "text": "|- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> (FINITE ((*_c s) t)))))"}
{"_id": "3538", "title": "PATHFINISH_IN_PATH_IMAGE", "text": "|- (!g. ((pathfinish g) IN (path_image g)))"}
{"_id": "3539", "title": "Pack2.MEASURE_VORONOI_CLOSED_OPEN", "text": "|- (!s. (!v. ((measure ((voronoi_closed s) v)) = (measure ((voronoi_open s) v)))))"}
{"_id": "3540", "title": "csin", "text": "|- (!z. ((csin z) = ((complex_div ((vector_sub (cexp ((complex_mul ii) z))) (cexp ((complex_mul (vector_neg ii)) z)))) ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ii))))"}
{"_id": "3541", "title": "SUBPATH_SCALING_LEMMA", "text": "|- (!u. (!v. (((IMAGE (\\x. ((vector_add u) ((% (drop ((vector_sub v) u))) x)))) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) = (closed_segment ((CONS (u , v)) NIL)))))"}
{"_id": "3542", "title": "AFF_DIM_OPEN_IN", "text": "|- (!s. (!t. (((~ (s = EMPTY)) /\\ (((open_in ((subtopology euclidean) t)) s) /\\ (affine t))) ==> ((aff_dim s) = (aff_dim t)))))"}
{"_id": "3543", "title": "NSUM_ADD_NUMSEG", "text": "|- (!f. (!g. (!m. (!n. (((nsum (m .. n)) (\\i. ((f i) + (g i)))) = (((nsum (m .. n)) f) + ((nsum (m .. n)) g)))))))"}
{"_id": "3544", "title": "EQ_MULT_LCANCEL", "text": "|- (!m. (!n. (!p. (((m * n) = (m * p)) = ((m = (NUMERAL _0)) \\/ (n = p))))))"}
{"_id": "3545", "title": "CONTRACTIBLE_PCROSS_EQ", "text": "|- (!s. (!t. ((contractible (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((contractible s) /\\ (contractible t)))))))"}
{"_id": "3546", "title": "MDIST_SYM", "text": "|- (!m. (!x. (!y. (((x IN (mspace m)) /\\ (y IN (mspace m))) ==> (((mdist m) (x , y)) = ((mdist m) (y , x)))))))"}
{"_id": "3547", "title": "SUM_POS_LE_NUMSEG", "text": "|- (!m. (!n. (!f. ((!p. (((m <= p) /\\ (p <= n)) ==> ((real_le (real_of_num (NUMERAL _0))) (f p)))) ==> ((real_le (real_of_num (NUMERAL _0))) ((sum (m .. n)) f))))))"}
{"_id": "3548", "title": "FRONTIER_BIJECTIVE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> ((frontier ((IMAGE f) s)) = ((IMAGE f) (frontier s))))))"}
{"_id": "3549", "title": "DE_MORGAN_THM", "text": "|- (!t1. (!t2. (((~ (t1 /\\ t2)) = ((~ t1) \\/ (~ t2))) /\\ ((~ (t1 \\/ t2)) = ((~ t1) /\\ (~ t2))))))"}
{"_id": "3550", "title": "TRANSLATION_UNIV", "text": "|- (!a. (((IMAGE (\\x. ((vector_add a) x))) UNIV) = UNIV))"}
{"_id": "3551", "title": "Real_ext.REAL_PROP_ZERO_SQRT", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((sqrt x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))))"}
{"_id": "3552", "title": "Collect_geom.conv", "text": "|- (!S. ((conv S) = (((affsign sgn_ge) EMPTY) S)))"}
{"_id": "3553", "title": "LEFT_FORALL_OR_THM", "text": "|- (!P. (!Q. ((!x. ((P x) \\/ Q)) = ((!x. (P x)) \\/ Q))))"}
{"_id": "3554", "title": "GDELTA_UNION", "text": "|- (!s. (!t. (((gdelta s) /\\ (gdelta t)) ==> (gdelta (s UNION t)))))"}
{"_id": "3555", "title": "INSIDE_TRANSLATION", "text": "|- (!a. (!s. ((inside ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) (inside s)))))"}
{"_id": "3556", "title": "INVERTIBLE_MATRIX_MUL", "text": "|- (!A. (!B. ((invertible ((matrix_mul A) B)) = ((invertible A) /\\ (invertible B)))))"}
{"_id": "3557", "title": "PAIR_EQ", "text": "|- (!x. (!y. (!a. (!b. (((x , y) = (a , b)) = ((x = a) /\\ (y = b)))))))"}
{"_id": "3558", "title": "FSIGMA_INTER", "text": "|- (!s. (!t. (((fsigma s) /\\ (fsigma t)) ==> (fsigma (s INTER t)))))"}
{"_id": "3559", "title": "COMPLEX_MUL_LID", "text": "|- (!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT1 _0))))) x) = x))"}
{"_id": "3560", "title": "HAS_REAL_INTEGRAL_REFLECT_GEN", "text": "|- (!f. (!i. (!s. ((((has_real_integral (\\x. (f (real_neg x)))) i) s) = (((has_real_integral f) i) ((IMAGE real_neg) s))))))"}
{"_id": "3561", "title": "ORDINAL_CHAINED", "text": "|- (!l. (!m. (((ordinal l) /\\ (ordinal m)) ==> (((inseg m) l) \\/ ((inseg l) m)))))"}
{"_id": "3562", "title": "Ssrnat.minn", "text": "|- (!m. (!n. (((minn m) n) = (((COND (m < n)) m) n))))"}
{"_id": "3563", "title": "FORALL_IN_INTERMEDIATE_CLOSURE", "text": "|- (!f. (!s. (!s'. (!t. (((closed t) /\\ (((continuous_on f) s') /\\ ((s SUBSET s') /\\ ((s' SUBSET (closure s)) /\\ (!x. ((x IN s) ==> ((f x) IN t))))))) ==> (!x. ((x IN s') ==> ((f x) IN t))))))))"}
{"_id": "3564", "title": "HAS_BOUNDED_VARIATION_ABSOLUTELY_INTEGRABLE_DERIVATIVE", "text": "|- (!f. (!s. (!a. (!b. (((COUNTABLE s) /\\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> ((differentiable f) (at x)))))) ==> (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = ((absolutely_integrable_on (\\x. ((vector_derivative f) (at x)))) (closed_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "3565", "title": "Appendix.scs_prop_equ_v39", "text": "|- (!s. (!i. (((scs_prop_equ_v39 s) i) = ((LET (\\shift1. (LET_END ((LET (\\shift2. (LET_END ((LET (\\shift2b. (LET_END (scs_v39 ((scs_k_v39 s) , ((scs_d_v39 s) , ((shift2 (scs_a_v39 s)) , ((shift2 (scs_am_v39 s)) , ((shift2 (scs_bm_v39 s)) , ((shift2 (scs_b_v39 s)) , ((shift2b (scs_J_v39 s)) , ((shift1 (scs_lo_v39 s)) , ((shift1 (scs_hi_v39 s)) , (shift1 (scs_str_v39 s))))))))))))))) (\\f. (\\j. (\\j'. ((f (i + j)) (i + j'))))))))) (\\f. (\\j. (\\j'. ((f (i + j)) (i + j'))))))))) (\\f. (\\j. (f (i + j))))))))"}
{"_id": "3566", "title": "COS_PERIODIC", "text": "|- (!x. ((cos ((real_add x) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))) = (cos x)))"}
{"_id": "3567", "title": "Hypermap_and_fan.TRIANGULAR_FACE", "text": "|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (((v , w) IN (dart1_of_fan (V , E))) /\\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0)))))) ==> ((LET (\\w'. (LET_END ((((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , w') INSERT ((w' , v) INSERT EMPTY)))) /\\ (((((((sigma_fan (vec (NUMERAL _0))) V) E) w) w') = v) /\\ ((((((sigma_fan (vec (NUMERAL _0))) V) E) w') v) = w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))"}
{"_id": "3568", "title": "REAL_SERIES_LMUL", "text": "|- (!x. (!x0. (!c. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\\n. ((real_mul c) (x n)))) ((real_mul c) x0)) s))))))"}
{"_id": "3569", "title": "CONVEX_TRANSLATION", "text": "|- (!s. (!a. ((convex s) ==> (convex ((IMAGE (\\x. ((vector_add a) x))) s)))))"}
{"_id": "3570", "title": "COMPLEX_MUL_LZERO", "text": "|- (!x. (((complex_mul (Cx (real_of_num (NUMERAL _0)))) x) = (Cx (real_of_num (NUMERAL _0)))))"}
{"_id": "3571", "title": "FORALL_IN_PCROSS", "text": "|- ((!z. ((z IN (s PCROSS t)) ==> (P z))) = (!x. (!y. (((x IN s) /\\ (y IN t)) ==> (P ((pastecart x) y))))))"}
{"_id": "3572", "title": "NORM_LIFT", "text": "|- (!x. ((vector_norm (lift x)) = (real_abs x)))"}
{"_id": "3573", "title": "MATRIX_MUL_RCANCEL", "text": "|- (!A. (!B. (!C. ((invertible C) ==> ((((matrix_mul A) C) = ((matrix_mul B) C)) = (A = B))))))"}
{"_id": "3574", "title": "SERIES_RATIO", "text": "|- (!c. (!a. (!s. (!N. ((((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\\ (!n. ((n >= N) ==> ((real_le (vector_norm (a (SUC n)))) ((real_mul c) (vector_norm (a n))))))) ==> (?l. (((sums a) l) s)))))))"}
{"_id": "3575", "title": "LINEAR_CONTINUOUS_WITHIN", "text": "|- (!f. (!s. (!x. ((linear f) ==> ((continuous f) ((within (at x)) s))))))"}
{"_id": "3576", "title": "IMAGE_STRETCH_INTERVAL", "text": "|- (!a. (!b. (!m. (((IMAGE (\\x. (lambdak. ((real_mul (m k)) (x $ k))))) (closed_interval ((CONS (a , b)) NIL))) = (((COND ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (closed_interval ((CONS ((lambdak. ((real_min ((real_mul (m k)) (a $ k))) ((real_mul (m k)) (b $ k)))) , (lambdak. ((real_max ((real_mul (m k)) (a $ k))) ((real_mul (m k)) (b $ k)))))) NIL)))))))"}
{"_id": "3577", "title": "Sphere.rotate3", "text": "|- (!f. (!x3. (!x1. (!x2. (!x6. (!x4. (!x5. ((((((((rotate3 f) x1) x2) x3) x4) x5) x6) = ((((((f x3) x1) x2) x6) x4) x5)))))))))"}
{"_id": "3578", "title": "RECTIFIABLE_PATH_TRANSLATION_EQ", "text": "|- (!a. (!g. ((rectifiable_path ((\\x. ((vector_add a) x)) o g)) = (rectifiable_path g))))"}
{"_id": "3579", "title": "COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP_STRONG", "text": "|- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\\ (!b. ((b IN s) ==> ((limit_point_of b) s)))) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\\GEN%PVAR%2080. (?x. (((SETSPEC GEN%PVAR%2080) ((x IN c) /\\ ((p x) = b))) x))))))) = (!t. (((closed_in ((subtopology euclidean) c)) t) ==> ((closed_in ((subtopology euclidean) s)) ((IMAGE p) t)))))))))"}
{"_id": "3580", "title": "SEGMENT_REFL", "text": "|- ((!a. ((closed_segment ((CONS (a , a)) NIL)) = (a INSERT EMPTY))) /\\ (!a. ((open_segment (a , a)) = EMPTY)))"}
{"_id": "3581", "title": "REAL_ABS_1", "text": "|- ((real_abs (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "3582", "title": "HAS_VECTOR_DERIVATIVE_AT_WITHIN", "text": "|- (!f. (!x. (!s. ((((has_vector_derivative f) f') (at x)) ==> (((has_vector_derivative f) f') ((within (at x)) s))))))"}
{"_id": "3583", "title": "REAL_TENDSTO", "text": "|- (((--> s) l) = ((---> (drop o s)) (drop l)))"}
{"_id": "3584", "title": "CLOSED_INTERVAL", "text": "|- (!a. (!b. (closed (closed_interval ((CONS (a , b)) NIL)))))"}
{"_id": "3585", "title": "CARD_LE_REFL", "text": "|- (!s. (s <=_c s))"}
{"_id": "3586", "title": "Lp_ineqs_def.tau_fan", "text": "|- (!V. (!E. (!f. (((tau_fan (V , E)) f) = (real_abs (tauVEF (V , (E , f))))))))"}
{"_id": "3587", "title": "Appendix.scs_lo_v39_explicit", "text": "|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_lo_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = lo))))))))))"}
{"_id": "3588", "title": "Ssrbool.predC", "text": "|- (!p. ((predC p) = (\\x. (~ (p x)))))"}
{"_id": "3589", "title": "REAL_POW_LE", "text": "|- (!x. (!n. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_pow x) n)))))"}
{"_id": "3590", "title": "REAL_LE_BETWEEN", "text": "|- (!a. (!b. (((real_le a) b) = (?x. (((real_le a) x) /\\ ((real_le x) b))))))"}
{"_id": "3591", "title": "LOG_POW", "text": "|- (!n. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log ((real_pow x) n)) = ((real_mul (real_of_num n)) (log x))))))"}
{"_id": "3592", "title": "Lvducxu.FACE_NODE_EDGE_ORBIT_INVERSE", "text": "|- ((((face H) x) = ((orbit_map (inverse (face_map H))) x)) /\\ ((((node H) x) = ((orbit_map (inverse (node_map H))) x)) /\\ (((edge H) x) = ((orbit_map (inverse (edge_map H))) x))))"}
{"_id": "3593", "title": "STEINHAUS_SUMS", "text": "|- (!s. (!t. (((lebesgue_measurable s) /\\ ((~ (negligible s)) /\\ ((lebesgue_measurable t) /\\ (~ (negligible t))))) ==> (~ ((interior (GSPEC (\\GEN%PVAR%4605. (?x. (?y. (((SETSPEC GEN%PVAR%4605) ((x IN s) /\\ (y IN t))) ((vector_add x) y))))))) = EMPTY)))))"}
{"_id": "3594", "title": "Ssrnat.subnn", "text": "|- (!n. ((n - n) = (NUMERAL _0)))"}
{"_id": "3595", "title": "CONNECTED_FROM_OPEN_UNION_AND_INTER_LOCAL", "text": "|- (!u. (!s. (!t. ((((open_in ((subtopology euclidean) u)) s) /\\ (((open_in ((subtopology euclidean) u)) t) /\\ ((connected (s UNION t)) /\\ (connected (s INTER t))))) ==> ((connected s) /\\ (connected t))))))"}
{"_id": "3596", "title": "RETRACT_OF_UNIV", "text": "|- (!s. (((retract_of s) UNIV) = ((AR s) /\\ (closed s))))"}
{"_id": "3597", "title": "EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL", "text": "|- (!s. ((FINITE s) ==> ((GSPEC (\\GEN%PVAR%1790. (?y. (((SETSPEC GEN%PVAR%1790) (?u. ((!x. ((x IN s) ==> ((real_lt (real_of_num (NUMERAL _0))) (u x)))) /\\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((vsum s) (\\x. ((% (u x)) x))) = y))))) y)))) SUBSET (relative_interior ((hull convex) s)))))"}
{"_id": "3598", "title": "LEBESGUE_MEASURABLE_INTERVAL", "text": "|- ((!a. (!b. (lebesgue_measurable (closed_interval ((CONS (a , b)) NIL))))) /\\ (!a. (!b. (lebesgue_measurable (open_interval (a , b))))))"}
{"_id": "3599", "title": "HOMOTOPIC_PATHS_NEARBY_EXPLICIT", "text": "|- (!g. (!s. (!h. (((path g) /\\ ((path h) /\\ (((pathstart h) = (pathstart g)) /\\ (((pathfinish h) = (pathfinish g)) /\\ (!t. (!x. (((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\\ (~ (x IN s))) ==> ((real_lt (vector_norm ((vector_sub (h t)) (g t)))) (vector_norm ((vector_sub (g t)) x)))))))))) ==> (((homotopic_paths s) g) h)))))"}
{"_id": "3600", "title": "HOMOTOPIC_CONSTANT_MAPS", "text": "|- (!s. (!t. (!a. (!b. (((((homotopic_with (\\x. T)) (s , t)) (\\x. a)) (\\x. b)) = ((s = EMPTY) \\/ (((path_component t) a) b)))))))"}
{"_id": "3601", "title": "NORM_LE_INFNORM", "text": "|- (!x. ((real_le (vector_norm x)) ((real_mul (sqrt (real_of_num (dimindex UNIV)))) (infnorm x))))"}
{"_id": "3602", "title": "Tame_defs2.C_EL", "text": "|- (!n. (!l. (((#!# l) n) = ((EL n) l))))"}
{"_id": "3603", "title": "Hypermap_and_fan.INVERSE_F_FAN_PAIR_EXT_EXPLICIT", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((inverse (f_fan_pair_ext (V , E))) = ((res (GABS (\\f. (!v. (!w. ((GEQ (f (v , w))) ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v))))))) (dart1_of_fan (V , E)))))))"}
{"_id": "3604", "title": "Ssrbool.andb_id2l", "text": "|- (!a. (!b. (!c. ((a ==> (b = c)) ==> ((a /\\ b) = (a /\\ c))))))"}
{"_id": "3605", "title": "Seq.uniq_perm_eq", "text": "|- (!s1. (!s2. ((uniq s1) ==> ((uniq s2) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> ((perm_eq s1) s2))))))"}
{"_id": "3606", "title": "FLOOR", "text": "|- (!x. ((integer (floor x)) /\\ (((real_le (floor x)) x) /\\ ((real_lt x) ((real_add (floor x)) (real_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "3607", "title": "HAS_SIZE_UNION", "text": "|- (!s. (!t. (!m. (!n. (((s HAS_SIZE m) /\\ ((t HAS_SIZE n) /\\ ((DISJOINT s) t))) ==> ((s UNION t) HAS_SIZE (m + n)))))))"}
{"_id": "3608", "title": "CARD_EQ_RESTRICTED_POWERSET", "text": "|- (!s. (!t. ((INFINITE s) ==> ((GSPEC (\\GEN%PVAR%48. (?k. (((SETSPEC GEN%PVAR%48) ((k SUBSET s) /\\ (k =_c t))) k)))) =_c (((COND (t <=_c s)) ((^_c s) t)) EMPTY)))))"}
{"_id": "3609", "title": "toset", "text": "|- (!l. ((toset l) = ((poset l) /\\ (!x. (!y. (((x IN (fl l)) /\\ (y IN (fl l))) ==> ((l (x , y)) \\/ (l (y , x)))))))))"}
{"_id": "3610", "title": "ABSOLUTELY_CONTINUOUS_COMPARISON", "text": "|- (!f. (!g. (!s. ((((absolutely_continuous_on f) s) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (distance ((g x) , (g y)))) (distance ((f x) , (f y)))))))) ==> ((absolutely_continuous_on g) s)))))"}
{"_id": "3611", "title": "EQ_ADD_RCANCEL", "text": "|- (!m. (!n. (!p. (((m + p) = (n + p)) = (m = n)))))"}
{"_id": "3612", "title": "POLYHEDRON_LINEAR_IMAGE", "text": "|- (!f. (!s. (((linear f) /\\ (polyhedron s)) ==> (polyhedron ((IMAGE f) s)))))"}
{"_id": "3613", "title": "one_RECURSION", "text": "|- (!e. (?fn. ((fn one) = e)))"}
{"_id": "3614", "title": "ITERATE_CLOSED", "text": "|- (!op. ((monoidal op) ==> (!P. (((P (neutral op)) /\\ (!x. (!y. (((P x) /\\ (P y)) ==> (P ((op x) y)))))) ==> (!f. (!s. ((!x. (((x IN s) /\\ (~ ((f x) = (neutral op)))) ==> (P (f x)))) ==> (P (((iterate op) s) f)))))))))"}
{"_id": "3615", "title": "HOMEOMORPHIC_MINIMAL", "text": "|- (!s. (!t. (((homeomorphic s) t) = (?f. (?g. ((!x. ((x IN s) ==> (((f x) IN t) /\\ ((g (f x)) = x)))) /\\ ((!y. ((y IN t) ==> (((g y) IN s) /\\ ((f (g y)) = y)))) /\\ (((continuous_on f) s) /\\ ((continuous_on g) t)))))))))"}
{"_id": "3616", "title": "Imjxphr.EXISTS_SMALL_LE_CONST", "text": "|- ((((real_lt (real_of_num (NUMERAL _0))) x1) /\\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\\ (((real_lt (real_of_num (NUMERAL _0))) x6) /\\ ((~ (collinear ((vec (NUMERAL _0)) INSERT (v1 INSERT (v2 INSERT EMPTY))))) /\\ ((((real_pow (vector_norm v1)) (NUMERAL (BIT0 (BIT1 _0)))) = x1) /\\ ((((real_pow (vector_norm v2)) (NUMERAL (BIT0 (BIT1 _0)))) = x2) /\\ ((((real_pow (vector_norm ((vector_sub v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))) = x6) /\\ ((a = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) /\\ ((real_lt c) (distance (v2 , w))))))))))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\\ ((real_lt t) e)) ==> ((real_lt c) (distance (((((((((v3_defor_v1 a) v1) v2) x1) x2) x6) x6) ((real_sub x2) t)) , w))))))))"}
{"_id": "3617", "title": "CLOSED_SEQUENTIAL_LIMITS", "text": "|- (!s. ((closed s) = (!x. (!l. (((!n. ((x n) IN s)) /\\ (((--> x) l) sequentially)) ==> (l IN s))))))"}
{"_id": "3618", "title": "Functional_equation.proj_x4", "text": "|- (((((((proj_x4 x1) x2) x3) x4) x5) x6) = x4)"}
{"_id": "3619", "title": "BUTLAST", "text": "|- (((BUTLAST NIL) = NIL) /\\ ((BUTLAST ((CONS h) t)) = (((COND (t = NIL)) NIL) ((CONS h) (BUTLAST t)))))"}
{"_id": "3620", "title": "FINITE_NUMSEG_LT", "text": "|- (!n. (FINITE (GSPEC (\\GEN%PVAR%119. (?m. (((SETSPEC GEN%PVAR%119) (m < n)) m))))))"}
{"_id": "3621", "title": "CONTINUOUS_AT_DIST_CLOSEST_POINT", "text": "|- (!s. (!x. (((closed s) /\\ (~ (s = EMPTY))) ==> ((continuous (\\x. (lift (distance (x , ((closest_point s) x)))))) (at x)))))"}
{"_id": "3622", "title": "NOT_ODD", "text": "|- (!n. ((~ (ODD n)) = (EVEN n)))"}
{"_id": "3623", "title": "NEGLIGIBLE_SING", "text": "|- (!a. (negligible (a INSERT EMPTY)))"}
{"_id": "3624", "title": "Real_ext.REAL_PROP_NN_SQRT", "text": "|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (real_of_num (NUMERAL _0))) (sqrt x))))"}
{"_id": "3625", "title": "REAL_EQ_MUL_LCANCEL", "text": "|- (!x. (!y. (!z. ((((real_mul x) y) = ((real_mul x) z)) = ((x = (real_of_num (NUMERAL _0))) \\/ (y = z))))))"}
{"_id": "3626", "title": "Seq.unzip1", "text": "|- (unzip1 = (map FST))"}
{"_id": "3627", "title": "LAMBDA_UNPAIR_THM", "text": "|- (!f. ((GABS (\\f. (!x. (!y. ((GEQ (f (x , y))) ((f x) y)))))) = (\\p. ((f (FST p)) (SND p)))))"}
{"_id": "3628", "title": "Sphere.asn797k", "text": "|- (!x2. (!x3. (!x4. (!x5. (!x6. (!k. (((((((asn797k k) x2) x3) x4) x5) x6) = ((real_mul k) (asn ((real_mul (cos ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))) (sin ((real_div pi) k))))))))))))"}
{"_id": "3629", "title": "TIETZE_CLOSED_INTERVAL", "text": "|- (!f. (!u. (!s. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (((closed_in ((subtopology euclidean) u)) s) /\\ (((continuous_on f) s) /\\ (!x. ((x IN s) ==> ((f x) IN (closed_interval ((CONS (a , b)) NIL)))))))) ==> (?g. (((continuous_on g) u) /\\ ((!x. ((x IN s) ==> ((g x) = (f x)))) /\\ (!x. ((x IN u) ==> ((g x) IN (closed_interval ((CONS (a , b)) NIL))))))))))))))"}
{"_id": "3630", "title": "FUNDAMENTAL_THEOREM_OF_CALCULUS_STRONG", "text": "|- (!f. (!f'. (!s. (!a. (!b. (((COUNTABLE s) /\\ (((real_le (drop a)) (drop b)) /\\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (f' x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))) ==> (((has_integral f') ((vector_sub (f b)) (f a))) (closed_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "3631", "title": "REAL_CONTINUOUS_WITHINREAL_SUBSET", "text": "|- (!f. (!s. (!t. ((((real_continuous f) ((within (atreal x)) s)) /\\ (t SUBSET s)) ==> ((real_continuous f) ((within (atreal x)) t))))))"}
{"_id": "3632", "title": "SERIES_CMUL", "text": "|- (!x. (!x0. (!c. (!s. ((((sums x) x0) s) ==> (((sums (\\n. ((% c) (x n)))) ((% c) x0)) s))))))"}
{"_id": "3633", "title": "VECTOR_1", "text": "|- (((vector ((CONS x) NIL)) $ (NUMERAL (BIT1 _0))) = x)"}
{"_id": "3634", "title": "List_hypermap.e_list", "text": "|- (!d. ((e_list d) = ((SND d) , (FST d))))"}
{"_id": "3635", "title": "Ssrnat.addnA", "text": "|- (!n. (!m. (!p. ((n + (m + p)) = ((n + m) + p)))))"}
{"_id": "3636", "title": "UNION", "text": "|- (!s. (!t. ((s UNION t) = (GSPEC (\\GEN%PVAR%0. (?x. (((SETSPEC GEN%PVAR%0) ((x IN s) \\/ (x IN t))) x)))))))"}
{"_id": "3637", "title": "DOT_SQUARE_NORM", "text": "|- (!x. (((dot x) x) = ((real_pow (vector_norm x)) (NUMERAL (BIT0 (BIT1 _0))))))"}
{"_id": "3638", "title": "Seq.rcons", "text": "|- (!x. (!t. (!z. ((((rcons NIL) z) = ((CONS z) NIL)) /\\ (((rcons ((CONS x) t)) z) = ((CONS x) ((rcons t) z)))))))"}
{"_id": "3639", "title": "Functional_equation.dih4_x'", "text": "|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\\ (((real_le (real_of_num (NUMERAL _0))) x2) /\\ (((real_le (real_of_num (NUMERAL _0))) x3) /\\ (((real_le (real_of_num (NUMERAL _0))) x4) /\\ (((real_le (real_of_num (NUMERAL _0))) x5) /\\ ((real_le (real_of_num (NUMERAL _0))) x6)))))) ==> (((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4_x' x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "3640", "title": "LINEAR_0", "text": "|- (!f. ((linear f) ==> ((f (vec (NUMERAL _0))) = (vec (NUMERAL _0)))))"}
{"_id": "3641", "title": "INTERIOR_CBALL", "text": "|- (!x. (!e. ((interior (cball (x , e))) = (ball (x , e)))))"}
{"_id": "3642", "title": "DOT_BASIS", "text": "|- (!x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((((dot (basis i)) x) = (x $ i)) /\\ (((dot x) (basis i)) = (x $ i))))))"}
{"_id": "3643", "title": "NSUM_LE", "text": "|- (!f. (!g. (!s. (((FINITE s) /\\ (!x. ((x IN s) ==> ((f x) <= (g x))))) ==> (((nsum s) f) <= ((nsum s) g))))))"}
{"_id": "3644", "title": "CONTINUOUS_GDELTA_PREIMAGE", "text": "|- (!f. (!s. (!t. ((((continuous_on f) s) /\\ ((gdelta s) /\\ (gdelta t))) ==> (gdelta (GSPEC (\\GEN%PVAR%1321. (?x. (((SETSPEC GEN%PVAR%1321) ((x IN s) /\\ ((f x) IN t))) x)))))))))"}
{"_id": "3645", "title": "Seq.mem_undup", "text": "|- (!s. (!x. (((MEM x) (undup s)) = ((MEM x) s))))"}
{"_id": "3646", "title": "REFLECT_UNIV", "text": "|- (((IMAGE vector_neg) UNIV) = UNIV)"}
{"_id": "3647", "title": "SUBSET_INTERVAL_1", "text": "|- (!a. (!b. (!c. (!d. ((((closed_interval ((CONS (a , b)) NIL)) SUBSET (closed_interval ((CONS (c , d)) NIL))) = (((real_lt (drop b)) (drop a)) \\/ (((real_le (drop c)) (drop a)) /\\ (((real_le (drop a)) (drop b)) /\\ ((real_le (drop b)) (drop d)))))) /\\ ((((closed_interval ((CONS (a , b)) NIL)) SUBSET (open_interval (c , d))) = (((real_lt (drop b)) (drop a)) \\/ (((real_lt (drop c)) (drop a)) /\\ (((real_le (drop a)) (drop b)) /\\ ((real_lt (drop b)) (drop d)))))) /\\ ((((open_interval (a , b)) SUBSET (closed_interval ((CONS (c , d)) NIL))) = (((real_le (drop b)) (drop a)) \\/ (((real_le (drop c)) (drop a)) /\\ (((real_lt (drop a)) (drop b)) /\\ ((real_le (drop b)) (drop d)))))) /\\ (((open_interval (a , b)) SUBSET (open_interval (c , d))) = (((real_le (drop b)) (drop a)) \\/ (((real_le (drop c)) (drop a)) /\\ (((real_lt (drop a)) (drop b)) /\\ ((real_le (drop b)) (drop d)))))))))))))"}
{"_id": "3648", "title": "CONTINUOUS_AT_WITHIN", "text": "|- (!f. (!x. (!s. (((continuous f) (at x)) ==> ((continuous f) ((within (at x)) s))))))"}
{"_id": "3649", "title": "STC_CASES_L", "text": "|- (!R. (!x. (!z. ((((STC R) x) z) = (((R x) z) \\/ ((((STC R) z) x) \\/ (?y. ((((STC R) x) y) /\\ ((R y) z)))))))))"}
{"_id": "3650", "title": "RAT_LEMMA5", "text": "|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> ((((real_div x1) y1) = ((real_div x2) y2)) = (((real_mul x1) y2) = ((real_mul x2) y1))))"}
{"_id": "3651", "title": "REAL_MIN_LT", "text": "|- (!x. (!y. (!z. (((real_lt ((real_min x) y)) z) = (((real_lt x) z) \\/ ((real_lt y) z))))))"}
{"_id": "3652", "title": "NORM_CAUCHY_SCHWARZ_ABS", "text": "|- (!x. (!y. ((real_le (real_abs ((dot x) y))) ((real_mul (vector_norm x)) (vector_norm y)))))"}
{"_id": "3653", "title": "BOREL_UNIONS", "text": "|- (!u. (((COUNTABLE u) /\\ (!s. ((s IN u) ==> (borel s)))) ==> (borel (UNIONS u))))"}
{"_id": "3654", "title": "HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE", "text": "|- (!f. (!x. ((((has_complex_derivative f) ((complex_derivative f) x)) (at x)) = ((complex_differentiable f) (at x)))))"}
{"_id": "3655", "title": "HOMEOMORPHISM_IMP_CLOSED_MAP", "text": "|- (!f. (!g. (!s. (!t. (!u. ((((homeomorphism (s , t)) (f , g)) /\\ ((closed_in ((subtopology euclidean) s)) u)) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u))))))))"}
{"_id": "3656", "title": "LIFT_SUM", "text": "|- (!k. (!x. ((lift ((sum k) x)) = ((vsum k) (lift o x)))))"}
{"_id": "3657", "title": "CONTINUOUS_AT_IMP_CONTINUOUS_ON", "text": "|- (!f. (!s. ((!x. ((x IN s) ==> ((continuous f) (at x)))) ==> ((continuous_on f) s))))"}
{"_id": "3658", "title": "IMAGE_o", "text": "|- (!f. (!g. (!s. (((IMAGE (f o g)) s) = ((IMAGE f) ((IMAGE g) s))))))"}
{"_id": "3659", "title": "SIN_PI", "text": "|- ((sin pi) = (real_of_num (NUMERAL _0)))"}
{"_id": "3660", "title": "CONTRACTIBLE_EMPTY", "text": "|- (contractible EMPTY)"}
{"_id": "3661", "title": "VECTOR_MUL_LID", "text": "|- (!x. (((% (real_of_num (NUMERAL (BIT1 _0)))) x) = x))"}
{"_id": "3662", "title": "RECTIFIABLE_PATH_JOIN", "text": "|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((rectifiable_path ((++ g1) g2)) = ((rectifiable_path g1) /\\ (rectifiable_path g2))))))"}
{"_id": "3663", "title": "HAS_MEASURE_MEASURABLE_MEASURE", "text": "|- (!s. (!m. (((has_measure s) m) = ((measurable s) /\\ ((measure s) = m)))))"}
{"_id": "3664", "title": "INTEGRAL_NORM_BOUND_INTEGRAL", "text": "|- (!f. (!g. (!s. ((((integrable_on f) s) /\\ (((integrable_on g) s) /\\ (!x. ((x IN s) ==> ((real_le (vector_norm (f x))) (drop (g x))))))) ==> ((real_le (vector_norm ((integral s) f))) (drop ((integral s) g)))))))"}
{"_id": "3665", "title": "Tame_list.IMP_IMP_THM", "text": "|- (!a. (!b. ((a ==> (b ==> c)) = ((a /\\ b) ==> c))))"}
{"_id": "3666", "title": "DIAMETER_CLOSURE", "text": "|- (!s. ((diameter (closure s)) = (diameter s)))"}
{"_id": "3667", "title": "REAL_LT_01", "text": "|- ((real_lt (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0))))"}
{"_id": "3668", "title": "bool_elim", "text": "|- (!P. (((P T) /\\ (P F)) ==> (!x. (P x))))"}
{"_id": "3669", "title": "COLLINEAR_SUBSET", "text": "|- (!s. (!t. (((collinear t) /\\ (s SUBSET t)) ==> (collinear s))))"}
{"_id": "3670", "title": "PERMUTES_INVERSE_EQ", "text": "|- (!p. (!s. (((permutes p) s) ==> (!x. (!y. ((((inverse p) y) = x) = ((p x) = y)))))))"}
{"_id": "3671", "title": "Sphere.packing", "text": "|- (!S. ((packing S) = (!u. (!v. (((S u) /\\ ((S v) /\\ (~ (u = v)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (u , v))))))))"}
{"_id": "3672", "title": "FINE_UNIONS", "text": "|- (!d. (!ps. ((!p. ((p IN ps) ==> ((fine d) p))) ==> ((fine d) (UNIONS ps)))))"}
{"_id": "3673", "title": "Trigonometry2.WHEN_K_POS_ARCV_STABLE", "text": "|- (((real_lt (real_of_num (NUMERAL _0))) k) ==> ((((arcV (vec (NUMERAL _0))) u) v) = (((arcV (vec (NUMERAL _0))) u) ((% k) v))))"}
{"_id": "3674", "title": "HAS_REAL_MEASURE_DIFF_SUBSET", "text": "|- (!s1. (!s2. (!m1. (!m2. ((((has_real_measure s1) m1) /\\ (((has_real_measure s2) m2) /\\ (s2 SUBSET s1))) ==> ((has_real_measure (s1 DIFF s2)) ((real_sub m1) m2)))))))"}
{"_id": "3675", "title": "HAS_REAL_DERIVATIVE_SUM", "text": "|- (!f. (!net. (!s. (((FINITE s) /\\ (!a. ((a IN s) ==> (((has_real_derivative (f a)) (f' a)) net)))) ==> (((has_real_derivative (\\x. ((sum s) (\\a. ((f a) x))))) ((sum s) f')) net)))))"}
{"_id": "3676", "title": "NEGLIGIBLE_CONVEX_FRONTIER", "text": "|- (!s. ((convex s) ==> (negligible (frontier s))))"}
{"_id": "3677", "title": "VECTOR_NEG_EQ_0", "text": "|- (((vector_neg x) = (vec (NUMERAL _0))) = (x = (vec (NUMERAL _0))))"}
{"_id": "3678", "title": "SUM_INTEGRAL_LBOUND_INCREASING", "text": "|- (!f. (!g. (!m. (!n. (((m <= n) /\\ ((!x. ((x IN (closed_segment ((CONS ((Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) , (Cx (real_of_num n)))) NIL))) ==> (((has_complex_derivative g) (f x)) (at x)))) /\\ (!x. (!y. ((((real_le ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) x) /\\ (((real_le x) y) /\\ ((real_le y) (real_of_num n)))) ==> ((real_le (Re (f (Cx x)))) (Re (f (Cx y))))))))) ==> ((real_le (Re ((vector_sub (g (Cx (real_of_num n)))) (g (Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))))) ((sum (m .. n)) (\\k. (Re (f (Cx (real_of_num k))))))))))))"}
{"_id": "3679", "title": "CX_ADD", "text": "|- (!x. (!y. ((Cx ((real_add x) y)) = ((vector_add (Cx x)) (Cx y)))))"}
{"_id": "3680", "title": "VECTOR_EQ_NEG2", "text": "|- (!x. (!y. (((vector_neg x) = (vector_neg y)) = (x = y))))"}
{"_id": "3681", "title": "STD_SIMPLEX", "text": "|- (((hull convex) ((vec (NUMERAL _0)) INSERT (GSPEC (\\GEN%PVAR%1805. (?i. (((SETSPEC GEN%PVAR%1805) (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) (basis i))))))) = (GSPEC (\\GEN%PVAR%1806. (?x. (((SETSPEC GEN%PVAR%1806) ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i)))) /\\ ((real_le ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. (x $ i)))) (real_of_num (NUMERAL (BIT1 _0)))))) x)))))"}
{"_id": "3682", "title": "AFFINE_DEPENDENT_TRANSLATION_EQ", "text": "|- (!a. (!s. ((affine_dependent ((IMAGE (\\x. ((vector_add a) x))) s)) = (affine_dependent s))))"}
{"_id": "3683", "title": "REAL_LT_SUB_LADD", "text": "|- (!x. (!y. (!z. (((real_lt x) ((real_sub y) z)) = ((real_lt ((real_add x) z)) y)))))"}
{"_id": "3684", "title": "fsigma", "text": "|- (!s. ((fsigma s) = (?g. ((COUNTABLE g) /\\ ((!c. ((c IN g) ==> (closed c))) /\\ ((UNIONS g) = s))))))"}
{"_id": "3685", "title": "DROP_ADD", "text": "|- (!x. (!y. ((drop ((vector_add x) y)) = ((real_add (drop x)) (drop y)))))"}
{"_id": "3686", "title": "absolutely_setcontinuous_on", "text": "|- (!s. (!f. (((absolutely_setcontinuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?r. (((real_lt (real_of_num (NUMERAL _0))) r) /\\ (!d. (!t. ((((division_of d) t) /\\ ((t SUBSET s) /\\ ((real_lt ((sum d) content)) r))) ==> ((real_lt ((sum d) (\\k. (vector_norm (f k))))) e)))))))))))"}
{"_id": "3687", "title": "IM", "text": "|- ((Im (complex (x , y))) = y)"}
{"_id": "3688", "title": "NORM_0", "text": "|- ((vector_norm (vec (NUMERAL _0))) = (real_of_num (NUMERAL _0)))"}
{"_id": "3689", "title": "MULT_CLAUSES", "text": "|- ((!n. (((NUMERAL _0) * n) = (NUMERAL _0))) /\\ ((!m. ((m * (NUMERAL _0)) = (NUMERAL _0))) /\\ ((!n. (((NUMERAL (BIT1 _0)) * n) = n)) /\\ ((!m. ((m * (NUMERAL (BIT1 _0))) = m)) /\\ ((!m. (!n. (((SUC m) * n) = ((m * n) + n)))) /\\ (!m. (!n. ((m * (SUC n)) = (m + (m * n))))))))))"}
{"_id": "3690", "title": "IN_INTERVAL_1", "text": "|- (!a. (!b. (!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) = (((real_le (drop a)) (drop x)) /\\ ((real_le (drop x)) (drop b)))) /\\ ((x IN (open_interval (a , b))) = (((real_lt (drop a)) (drop x)) /\\ ((real_lt (drop x)) (drop b))))))))"}
{"_id": "3691", "title": "FUNSPACE", "text": "|- (!s. (!m. (((mspace ((funspace s) m)) = (GSPEC (\\GEN%PVAR%526. (?f. (((SETSPEC GEN%PVAR%526) ((!x. ((x IN s) ==> ((f x) IN (mspace m)))) /\\ ((f IN (EXTENSIONAL s)) /\\ ((mbounded m) ((IMAGE f) s))))) f))))) /\\ (!f. (!g. (((mdist ((funspace s) m)) (f , g)) = (((COND (s = EMPTY)) (real_of_num (NUMERAL _0))) (sup (GSPEC (\\GEN%PVAR%527. (?x. (((SETSPEC GEN%PVAR%527) (x IN s)) ((mdist m) ((f x) , (g x)))))))))))))))"}
{"_id": "3692", "title": "NORM_MUL", "text": "|- (!a. (!x. ((vector_norm ((% a) x)) = ((real_mul (real_abs a)) (vector_norm x)))))"}
{"_id": "3693", "title": "VOLUME_FRUSTT_STRONG", "text": "|- (!v0. (!v1. (!h. (!a. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((bounded ((((frustt v0) v1) h) a)) /\\ ((convex ((((frustt v0) v1) h) a)) /\\ ((measurable ((((frustt v0) v1) h) a)) /\\ ((measure ((((frustt v0) v1) h) a)) = (((COND ((v1 = v0) \\/ (((real_le (real_of_num (NUMERAL (BIT1 _0)))) a) \\/ ((real_lt h) (real_of_num (NUMERAL _0)))))) (real_of_num (NUMERAL _0))) ((real_mul pi) ((real_mul ((real_sub ((real_pow ((real_div h) a)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow h) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_div h) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))))))"}
{"_id": "3694", "title": "MATRIX_ID", "text": "|- ((matrix (\\x. x)) = (mat (NUMERAL (BIT1 _0))))"}
{"_id": "3695", "title": "LOCALLY_PATH_CONNECTED_TRANSLATION_EQ", "text": "|- (!a. (!s. (((locally path_connected) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((locally path_connected) s))))"}
{"_id": "3696", "title": "LIM_AT_ID", "text": "|- (!a. (((--> (\\x. x)) a) (at a)))"}
{"_id": "3697", "title": "CONVEX_CLOSURE_INTERIOR", "text": "|- (!s. (((convex s) /\\ (~ ((interior s) = EMPTY))) ==> ((closure (interior s)) = (closure s))))"}
{"_id": "3698", "title": "MATRIX_VECTOR_MUL_INV_EQ_0", "text": "|- (!A. ((((matrix_vector_mul (matrix_inv A)) x) = (vec (NUMERAL _0))) = (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))))"}
{"_id": "3699", "title": "CLOSURE_MINIMAL", "text": "|- (!s. (!t. (((s SUBSET t) /\\ (closed t)) ==> ((closure s) SUBSET t))))"}
{"_id": "3700", "title": "Hypermap_and_fan.RES_EMPTY", "text": "|- (!f. (((res f) EMPTY) = I))"}
{"_id": "3701", "title": "SUM_ADD_NUMSEG", "text": "|- (!f. (!g. (!m. (!n. (((sum (m .. n)) (\\i. ((real_add (f i)) (g i)))) = ((real_add ((sum (m .. n)) f)) ((sum (m .. n)) g)))))))"}
{"_id": "3702", "title": "Seq.all_count", "text": "|- (!a. (!s. (((all a) s) = (((count a) s) = (sizel s)))))"}
{"_id": "3703", "title": "pathstart", "text": "|- (!g. ((pathstart g) = (g (vec (NUMERAL _0)))))"}
{"_id": "3704", "title": "Lp_ineqs_proofs.face_eq_power", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN ((darts_k k) (hypermap_of_fan (V , E)))) ==> (((face (hypermap_of_fan (V , E))) d) = (GSPEC (\\GEN%PVAR%8730. (?i. (((SETSPEC GEN%PVAR%8730) (i < k)) (((POWER (f_fan_pair_ext (V , E))) i) d))))))))))))"}
{"_id": "3705", "title": "Tame_defs.split_at_rec", "text": "|- (((((split_at_rec c) bs) NIL) = (bs , NIL)) /\\ ((((split_at_rec c) bs) ((CONS a) xs)) = (((COND (a = c)) (bs , xs)) (((split_at_rec c) ((APPEND bs) ((CONS a) NIL))) xs))))"}
{"_id": "3706", "title": "INTERIOR_SEGMENT", "text": "|- ((!a. (!b. ((interior (closed_segment ((CONS (a , b)) NIL))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) EMPTY) (open_segment (a , b)))))) /\\ (!a. (!b. ((interior (open_segment (a , b))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) EMPTY) (open_segment (a , b)))))))"}
{"_id": "3707", "title": "Add_triangle.ext_contracted", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d_fan ((vec (NUMERAL _0)) , (V , E)))) ==> (((ext_dart (V , E)) (contracted_dart d)) = d))))))"}
{"_id": "3708", "title": "HAS_REAL_INTEGRAL_INTEGRAL", "text": "|- (!f. (!s. (((real_integrable_on f) s) = (((has_real_integral f) ((real_integral s) f)) s))))"}
{"_id": "3709", "title": "INDEFINITE_INTEGRAL_UNIFORMLY_CONTINUOUS", "text": "|- (!f. (!a. (!b. (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((uniformly_continuous_on (\\y. ((integral (closed_interval ((CONS ((fstcart y) , (sndcart y))) NIL))) f))) (closed_interval ((CONS (((pastecart a) a) , ((pastecart b) b))) NIL)))))))"}
{"_id": "3710", "title": "ENDS_IN_SEGMENT", "text": "|- (!a. (!b. ((a IN (closed_segment ((CONS (a , b)) NIL))) /\\ (b IN (closed_segment ((CONS (a , b)) NIL))))))"}
{"_id": "3711", "title": "Sphere.rho", "text": "|- (!y. ((rho y) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_sub const1) ((real_mul const1) (ly y))))))"}
{"_id": "3712", "title": "integrable_on", "text": "|- (!f. (!i. (((integrable_on f) i) = (?y. (((has_integral f) y) i)))))"}
{"_id": "3713", "title": "REAL_SERIES_SUB", "text": "|- (!x. (!x0. (!y. (!y0. (!s. (((((real_sums x) x0) s) /\\ (((real_sums y) y0) s)) ==> (((real_sums (\\n. ((real_sub (x n)) (y n)))) ((real_sub x0) y0)) s)))))))"}
{"_id": "3714", "title": "NPRODUCT_EQ", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((nproduct s) f) = ((nproduct s) g))))))"}
{"_id": "3715", "title": "FSTCART_PASTECART", "text": "|- (!x. (!y. ((fstcart ((pastecart x) y)) = x)))"}
{"_id": "3716", "title": "Ssrbool.iffLR", "text": "|- (!P. (!Q. ((P = Q) ==> (P ==> Q))))"}
{"_id": "3717", "title": "Seq2.list_sum_cons", "text": "|- (!h. (!t. (!f. (((list_sum ((CONS h) t)) f) = ((real_add (f h)) ((list_sum t) f))))))"}
{"_id": "3718", "title": "PATH_IMAGE_NONEMPTY", "text": "|- (!g. (~ ((path_image g) = EMPTY)))"}
{"_id": "3719", "title": "Fan_defs.fully_surrounded", "text": "|- (!V. (!E. ((fully_surrounded (V , E)) = (!x. ((x IN (dart_of_fan (V , E))) ==> ((real_lt ((azim_dart (V , E)) x)) pi))))))"}
{"_id": "3720", "title": "HAS_BOUNDED_VARIATION_COMPOSE_DECREASING", "text": "|- (!f. (!g. (!a. (!b. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f y))) (drop (f x)))))) /\\ ((has_bounded_variation_on g) (closed_interval ((CONS ((f b) , (f a))) NIL)))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "3721", "title": "Lp_ineqs_def.y4_fan", "text": "|- (!V. (!E. (!x. (((y4_fan (V , E)) x) = (distance ((f_fan_pair_ext (V , E)) x))))))"}
{"_id": "3722", "title": "EXPOSED_FACE_OF_TRANSLATION_EQ", "text": "|- (!a. (!f. (!s. (((exposed_face_of ((IMAGE (\\x. ((vector_add a) x))) f)) ((IMAGE (\\x. ((vector_add a) x))) s)) = ((exposed_face_of f) s)))))"}
{"_id": "3723", "title": "HAS_REAL_INTEGRAL_EQ", "text": "|- (!f. (!g. (!k. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ (((has_real_integral f) k) s)) ==> (((has_real_integral g) k) s))))))"}
{"_id": "3724", "title": "SUBSET_UNIV", "text": "|- (!s. (s SUBSET UNIV))"}
{"_id": "3725", "title": "TOPSPACE_EUCLIDEANREAL", "text": "|- ((topspace euclideanreal) = UNIV)"}
{"_id": "3726", "title": "COVERING_SPACE_FINITE_SHEETS_EQ_PROPER_MAP", "text": "|- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\\GEN%PVAR%2082. (?x. (((SETSPEC GEN%PVAR%2082) ((x IN c) /\\ ((p x) = b))) x))))))) = (!k. (((k SUBSET s) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%2083. (?x. (((SETSPEC GEN%PVAR%2083) ((x IN c) /\\ ((p x) IN k))) x))))))))))))"}
{"_id": "3727", "title": "Tame_defs.d_tame", "text": "|- (!n. ((d_tame n) = (((COND (n = (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL _0))) (((COND (n = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((COND (n = (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) (((COND (n = (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) tgt))))))"}
{"_id": "3728", "title": "MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_GEN", "text": "|- (!f. (!s. ((((measurable_on (lift o f)) s) /\\ ((convex s) /\\ (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (f (midpoint (x , y)))) ((real_div ((real_add (f x)) (f y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))) ==> (!x. (!y. (!u. (!v. (((x IN s) /\\ ((y IN s) /\\ (((real_le (real_of_num (NUMERAL _0))) u) /\\ (((real_le (real_of_num (NUMERAL _0))) v) /\\ ((((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((closed_segment ((CONS (x , y)) NIL)) SUBSET (frontier s)) ==> (x = y))))))) ==> ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))"}
{"_id": "3729", "title": "Ssrbool.predI", "text": "|- (!p1. (!p2. (((predI p1) p2) = (\\x. ((p1 x) /\\ (p2 x))))))"}
{"_id": "3730", "title": "COMPLEX_NORM_INV", "text": "|- (!z. ((vector_norm (complex_inv z)) = (real_inv (vector_norm z))))"}
{"_id": "3731", "title": "SIMPLY_CONNECTED_TRANSLATION", "text": "|- (!a. (!s. ((simply_connected ((IMAGE (\\x. ((vector_add a) x))) s)) = (simply_connected s))))"}
{"_id": "3732", "title": "REAL_SECOND_MEAN_VALUE_THEOREM_GEN_FULL", "text": "|- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((!x. ((x IN (open_real_interval (a , b))) ==> (((real_le u) (g x)) /\\ ((real_le (g x)) v)))) /\\ (!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_le x) y))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ (((has_real_integral (\\x. ((real_mul (g x)) (f x)))) ((real_add ((real_mul u) ((real_integral (closed_real_interval ((CONS (a , c)) NIL))) f))) ((real_mul v) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f)))) (closed_real_interval ((CONS (a , b)) NIL))))))))))))"}
{"_id": "3733", "title": "IMAGE_UNION", "text": "|- (!f. (!s. (!t. (((IMAGE f) (s UNION t)) = (((IMAGE f) s) UNION ((IMAGE f) t))))))"}
{"_id": "3734", "title": "VECTOR_EQ_ADDR", "text": "|- ((((vector_add x) y) = x) = (y = (vec (NUMERAL _0))))"}
{"_id": "3735", "title": "COMPACT_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON", "text": "|- (!f. (!s. (((!c. (((c SUBSET s) /\\ (compact c)) ==> (compact ((IMAGE f) c)))) /\\ (!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%1929. (?x. (((SETSPEC GEN%PVAR%1929) ((x IN s) /\\ ((f x) = y))) x))))))) ==> ((continuous_on f) s))))"}
{"_id": "3736", "title": "REAL_CNJ", "text": "|- (!z. ((real z) = ((cnj z) = z)))"}
{"_id": "3737", "title": "REAL_LT_TRANS", "text": "|- (!x. (!y. (!z. ((((real_lt x) y) /\\ ((real_lt y) z)) ==> ((real_lt x) z)))))"}
{"_id": "3738", "title": "real_differentiable_on", "text": "|- (!f. (!s. (((real_differentiable_on f) s) = (!x. ((x IN s) ==> (?f'. (((has_real_derivative f) f') ((within (atreal x)) s))))))))"}
{"_id": "3739", "title": "LINEAR_NEG", "text": "|- (!f. (!x. ((linear f) ==> ((f (vector_neg x)) = (vector_neg (f x))))))"}
{"_id": "3740", "title": "HOMOTOPY_EQUIVALENT_CONTRACTIBILITY", "text": "|- (!s. (!t. (((homotopy_equivalent s) t) ==> ((contractible s) = (contractible t)))))"}
{"_id": "3741", "title": "CONTINUOUS_AE_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET", "text": "|- (!f. (!s. (!m. ((((continuous_on f) (s DIFF m)) /\\ ((lebesgue_measurable s) /\\ (negligible m))) ==> ((measurable_on f) s)))))"}
{"_id": "3742", "title": "FLOOR_UNIQUE", "text": "|- (!x. (!a. (((integer a) /\\ (((real_le a) x) /\\ ((real_lt x) ((real_add a) (real_of_num (NUMERAL (BIT1 _0))))))) = ((floor x) = a))))"}
{"_id": "3743", "title": "INTEGRAL_CONST", "text": "|- (!a. (!b. (!c. (((integral (closed_interval ((CONS (a , b)) NIL))) (\\x. c)) = ((% (content (closed_interval ((CONS (a , b)) NIL)))) c)))))"}
{"_id": "3744", "title": "SUBSPACE_IMP_CONVEX", "text": "|- (!s. ((subspace s) ==> (convex s)))"}
{"_id": "3745", "title": "AFF_DIM_SUBSET", "text": "|- (!s. (!t. ((s SUBSET t) ==> ((int_le (aff_dim s)) (aff_dim t)))))"}
{"_id": "3746", "title": "INTEGRABLE_DECREASING_PRODUCT_UNIV", "text": "|- (!f. (!g. (!B. ((((integrable_on f) UNIV) /\\ ((!x. (!y. (((real_le (drop x)) (drop y)) ==> ((real_le (g y)) (g x))))) /\\ (!x. ((real_le (real_abs (g x))) B)))) ==> ((integrable_on (\\x. ((% (g x)) (f x)))) UNIV)))))"}
{"_id": "3747", "title": "ANALYTIC_AT_TWO", "text": "|- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\\ ((analytic_on g) (z INSERT EMPTY))) = (?s. ((open s) /\\ ((z IN s) /\\ (((holomorphic_on f) s) /\\ ((holomorphic_on g) s)))))))))"}
{"_id": "3748", "title": "EMPTY_GSPEC", "text": "|- ((GSPEC (\\GEN%PVAR%14. (?x. (((SETSPEC GEN%PVAR%14) F) x)))) = EMPTY)"}
{"_id": "3749", "title": "DIMINDEX_3", "text": "|- ((dimindex UNIV) = (NUMERAL (BIT1 (BIT1 _0))))"}
{"_id": "3750", "title": "num_cases", "text": "|- (!P. (((P (NUMERAL _0)) /\\ (!n. (P (SUC n)))) ==> (!m. (P m))))"}
{"_id": "3751", "title": "List_hypermap.darts_k", "text": "|- (!H. (!k. (((darts_k k) H) = (GSPEC (\\GEN%PVAR%8595. (?d. (((SETSPEC GEN%PVAR%8595) ((d IN (dart H)) /\\ ((CARD ((face H) d)) = k))) d)))))))"}
{"_id": "3752", "title": "Seq.cat_take_drop", "text": "|- (!n0. (!s. (((cat ((take n0) s)) ((dropl n0) s)) = s)))"}
{"_id": "3753", "title": "MSPACE", "text": "|- (!s. (!d. ((is_metric_space (s , d)) ==> ((mspace (metric (s , d))) = s))))"}
{"_id": "3754", "title": "Ssrnat.leq_exp2l", "text": "|- (!m. (!n1. (!n2. (((NUMERAL (BIT1 _0)) < m) ==> (((m EXP n1) <= (m EXP n2)) = (n1 <= n2))))))"}
{"_id": "3755", "title": "Fan.th3", "text": "|- (!x. (!v. (!u. ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) ==> ((~ (x = v)) /\\ ((~ (x = u)) /\\ (((DISJOINT (x INSERT (v INSERT EMPTY))) (u INSERT EMPTY)) /\\ (((DISJOINT (x INSERT (u INSERT EMPTY))) (v INSERT EMPTY)) /\\ (((DISJOINT (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY))) /\\ (~ (u IN (aff (x INSERT (v INSERT EMPTY))))))))))))))"}
{"_id": "3756", "title": "COUNTABLE_SUBSET", "text": "|- (!s. (!t. (((COUNTABLE t) /\\ (s SUBSET t)) ==> (COUNTABLE s))))"}
{"_id": "3757", "title": "pair_RECURSION", "text": "|- (!PAIR'. (?fn. (!a0. (!a1. ((fn (a0 , a1)) = ((PAIR' a0) a1))))))"}
{"_id": "3758", "title": "RE_NEG", "text": "|- (!x. ((Re (vector_neg x)) = (real_neg (Re x))))"}
{"_id": "3759", "title": "RIGHT_POLAR_DECOMPOSITION_INVERTIBLE", "text": "|- (!A. ((invertible A) = (?U. (?P. ((orthogonal_matrix U) /\\ ((positive_definite P) /\\ (((matrix_mul U) P) = A)))))))"}
{"_id": "3760", "title": "CX_MUL", "text": "|- (!x. (!y. ((Cx ((real_mul x) y)) = ((complex_mul (Cx x)) (Cx y)))))"}
{"_id": "3761", "title": "POLYHEDRON_UNIV", "text": "|- (polyhedron UNIV)"}
{"_id": "3762", "title": "Topology.IMAGE_SEG", "text": "|- (!n. (!f. (((IMAGE f) (GSPEC (\\GEN%PVAR%5485. (?i. (((SETSPEC GEN%PVAR%5485) (i < n)) i))))) = (GSPEC (\\GEN%PVAR%5486. (?i. (((SETSPEC GEN%PVAR%5486) (i < n)) (f i))))))))"}
{"_id": "3763", "title": "HAS_REAL_INTEGRAL", "text": "|- ((((has_real_integral f) y) (closed_real_interval ((CONS (a , b)) NIL))) = (((has_integral (lift o (f o drop))) (lift y)) (closed_interval ((CONS ((lift a) , (lift b))) NIL))))"}
{"_id": "3764", "title": "Hypermap_and_fan.NODE_SET_AS_IMAGE", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (?f. ((!v. (!w. (((f v) = (f w)) ==> (v = w)))) /\\ ((!v. (!x. ((x IN (f v)) ==> ((FST x) = v)))) /\\ ((!v. ((FST (CHOICE (f v))) = v)) /\\ ((node_set (hypermap_of_fan (V , E))) = ((IMAGE f) V)))))))))"}
{"_id": "3765", "title": "REAL_OPEN_IN", "text": "|- (!s. ((real_open s) = ((open_in euclideanreal) s)))"}
{"_id": "3766", "title": "REAL_NEGNEG", "text": "|- (!x. ((real_neg (real_neg x)) = x))"}
{"_id": "3767", "title": "Localization.convex_local_fan", "text": "|- (!FF. (!V. (!E. ((convex_local_fan (V , (E , FF))) = ((local_fan (V , (E , FF))) /\\ (!x. ((x IN FF) ==> (((real_le ((azim_in_fan x) E)) pi) /\\ (V SUBSET ((wedge_in_fan_ge x) E))))))))))"}
{"_id": "3768", "title": "CARD_ADD_FINITE", "text": "|- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> (FINITE ((+_c s) t)))))"}
{"_id": "3769", "title": "VECTOR_3", "text": "|- ((((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT1 _0))) = x) /\\ ((((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT0 (BIT1 _0)))) = y) /\\ (((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT1 (BIT1 _0)))) = z)))"}
{"_id": "3770", "title": "SYMMETRIC_MATRIX_EQ_DIAGONALIZABLE_ALT", "text": "|- (!A. (((transp A) = A) = (?P. (?D. ((orthogonal_matrix P) /\\ ((diagonal_matrix D) /\\ (A = ((matrix_mul (transp P)) ((matrix_mul D) P)))))))))"}
{"_id": "3771", "title": "LEBESGUE_MEASURABLE_OPEN_IN", "text": "|- (!s. (!t. ((((open_in ((subtopology euclidean) s)) t) /\\ (lebesgue_measurable s)) ==> (lebesgue_measurable t))))"}
{"_id": "3772", "title": "INTER_ASSOC", "text": "|- (!s. (!t. (!u. (((s INTER t) INTER u) = (s INTER (t INTER u))))))"}
{"_id": "3773", "title": "CONVEX_ON_EQ", "text": "|- (!f. (!g. (!s. (((convex s) /\\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((convex_on f) s))) ==> ((convex_on g) s)))))"}
{"_id": "3774", "title": "NUMSEG_EMPTY", "text": "|- (!m. (!n. (((m .. n) = EMPTY) = (n < m))))"}
{"_id": "3775", "title": "MATRIX_VECTOR_MUL_ASSOC", "text": "|- (!A. (!B. (!x. (((matrix_vector_mul A) ((matrix_vector_mul B) x)) = ((matrix_vector_mul ((matrix_mul A) B)) x)))))"}
{"_id": "3776", "title": "Tame_defs.tame_5a", "text": "|- (!H. ((tame_5a H) = (is_no_double_joins H)))"}
{"_id": "3777", "title": "UNIV_NOT_EMPTY", "text": "|- (~ (UNIV = EMPTY))"}
{"_id": "3778", "title": "Nonlin_def.ldih6_x_div_sqrtdelta_posbranch", "text": "|- (ldih6_x_div_sqrtdelta_posbranch = (rotate6 ldih_x_div_sqrtdelta_posbranch))"}
{"_id": "3779", "title": "AFFINE_HYPERPLANE", "text": "|- (!a. (!b. (affine (GSPEC (\\GEN%PVAR%1401. (?x. (((SETSPEC GEN%PVAR%1401) (((dot a) x) = b)) x)))))))"}
{"_id": "3780", "title": "CLOSED_HALFSPACE_IM_LE", "text": "|- (!b. (closed (GSPEC (\\GEN%PVAR%4757. (?z. (((SETSPEC GEN%PVAR%4757) ((real_le (Im z)) b)) z))))))"}
{"_id": "3781", "title": "Hypermap_and_fan.FACE_SUBSET_DART1_OF_FAN", "text": "|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (x IN (dart1_of_fan (V , E)))) ==> (((face (hypermap_of_fan (V , E))) x) SUBSET (dart1_of_fan (V , E)))))))"}
{"_id": "3782", "title": "POSITIVE_SEMIDEFINITE_MUL_EQ", "text": "|- (!A. (!B. (((positive_semidefinite A) /\\ (positive_semidefinite B)) ==> ((positive_semidefinite ((matrix_mul A) B)) = (((matrix_mul A) B) = ((matrix_mul B) A))))))"}
{"_id": "3783", "title": "real_compact", "text": "|- (!s. ((real_compact s) = (compact ((IMAGE lift) s))))"}
{"_id": "3784", "title": "matrix_cmul", "text": "|- (!c. (!A. (((%% c) A) = (lambdai. (lambdaj. ((real_mul c) ((A $ i) $ j)))))))"}
{"_id": "3785", "title": "Wrgcvdr_cizmrrh.X_IN_ITS_ORBIT", "text": "|- (x IN ((orbit_map f) x))"}
{"_id": "3786", "title": "DIAGONAL_MATRIX_MUL", "text": "|- (!A. (!B. (((diagonal_matrix A) /\\ (diagonal_matrix B)) ==> (diagonal_matrix ((matrix_mul A) B)))))"}
{"_id": "3787", "title": "INSEG_WOSET", "text": "|- (!l. (!m. ((((inseg m) l) /\\ (woset l)) ==> (woset m))))"}
{"_id": "3788", "title": "INTEGRABLE_ON_EMPTY", "text": "|- (!f. ((integrable_on f) EMPTY))"}
{"_id": "3789", "title": "Functional_equation.mk_126", "text": "|- (!f. ((mk_126 f) = (((((((compose6 f) proj_x1) proj_x2) two6) two6) two6) proj_x6)))"}
{"_id": "3790", "title": "SUM_RESTRICT_SET", "text": "|- (!P. (!s. (!f. (((sum (GSPEC (\\GEN%PVAR%199. (?x. (((SETSPEC GEN%PVAR%199) ((x IN s) /\\ (P x))) x))))) f) = ((sum s) (\\x. (((COND (P x)) (f x)) (real_of_num (NUMERAL _0)))))))))"}
{"_id": "3791", "title": "MONOIDAL_MUL", "text": "|- (monoidal *)"}
{"_id": "3792", "title": "IMAGE_LIFT_REAL_INTERVAL", "text": "|- ((((IMAGE lift) (closed_real_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS ((lift a) , (lift b))) NIL))) /\\ (((IMAGE lift) (open_real_interval (a , b))) = (open_interval ((lift a) , (lift b)))))"}
{"_id": "3793", "title": "RIGHT_IMP_EXISTS_THM", "text": "|- (!P. (!Q. ((P ==> (?x. (Q x))) = (?x. (P ==> (Q x))))))"}
{"_id": "3794", "title": "COMPLEX_NORM_GE_RE_IM", "text": "|- (!z. (((real_le (real_abs (Re z))) (vector_norm z)) /\\ ((real_le (real_abs (Im z))) (vector_norm z))))"}
{"_id": "3795", "title": "MEASURABLE_NONNEGLIGIBLE_IMP_LARGE", "text": "|- (!s. (((measurable s) /\\ ((real_lt (real_of_num (NUMERAL _0))) (measure s))) ==> (s =_c UNIV)))"}
{"_id": "3796", "title": "SUP_EQ", "text": "|- (!s. (!t. ((!b. ((!x. ((x IN s) ==> ((real_le x) b))) = (!x. ((x IN t) ==> ((real_le x) b))))) ==> ((sup s) = (sup t)))))"}
{"_id": "3797", "title": "UNION_EMPTY", "text": "|- ((!s. ((EMPTY UNION s) = s)) /\\ (!s. ((s UNION EMPTY) = s)))"}
{"_id": "3798", "title": "absolutely_continuous_on", "text": "|- (!f. (!s. (((absolutely_continuous_on f) s) = ((absolutely_setcontinuous_on (\\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))) s))))"}
{"_id": "3799", "title": "CFUNSPACE", "text": "|- ((!top. (!m. ((mspace ((cfunspace top) m)) = (GSPEC (\\GEN%PVAR%534. (?f. (((SETSPEC GEN%PVAR%534) ((!x. ((x IN (topspace top)) ==> ((f x) IN (mspace m)))) /\\ ((f IN (EXTENSIONAL (topspace top))) /\\ (((mbounded m) ((IMAGE f) (topspace top))) /\\ (((topcontinuous top) (mtopology m)) f))))) f))))))) /\\ (!f. (!g. (((mdist ((cfunspace top) m)) (f , g)) = (((COND ((topspace top) = EMPTY)) (real_of_num (NUMERAL _0))) (sup (GSPEC (\\GEN%PVAR%535. (?x. (((SETSPEC GEN%PVAR%535) (x IN (topspace top))) ((mdist m) ((f x) , (g x)))))))))))))"}
{"_id": "3800", "title": "ABSOLUTELY_INTEGRABLE_REFLECT_GEN", "text": "|- (!f. (!s. (((absolutely_integrable_on (\\x. (f (vector_neg x)))) s) = ((absolutely_integrable_on f) ((IMAGE vector_neg) s)))))"}
{"_id": "3801", "title": "CONTINUOUS_AT_ID", "text": "|- (!a. ((continuous (\\x. x)) (at a)))"}
{"_id": "3802", "title": "Real_ext.pow", "text": "|- ((((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (!n. (((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n)))))"}
{"_id": "3803", "title": "Functional_equation.proj_y2", "text": "|- (!x1. (!x3. (!x4. (!x5. (!x6. (!x2. (((((((proj_y2 x1) x2) x3) x4) x5) x6) = (sqrt x2))))))))"}
{"_id": "3804", "title": "SELECT_UNIQUE", "text": "|- (!P. (!x. ((!y. ((P y) = (y = x))) ==> ((@ P) = x))))"}
{"_id": "3805", "title": "Sphere.voronoi_closed", "text": "|- (!S. (!v. (((voronoi_closed S) v) = (GSPEC (\\GEN%PVAR%5242. (?x. (((SETSPEC GEN%PVAR%5242) (!w. ((S w) ==> ((real_le (distance (x , v))) (distance (x , w)))))) x)))))))"}
{"_id": "3806", "title": "ANALYTIC_ON_INV", "text": "|- (!f. (!s. ((((analytic_on f) s) /\\ (!z. ((z IN s) ==> (~ ((f z) = (Cx (real_of_num (NUMERAL _0)))))))) ==> ((analytic_on (\\z. (complex_inv (f z)))) s))))"}
{"_id": "3807", "title": "FRONTIER_INTER_SUBSET", "text": "|- (!s. (!t. ((frontier (s INTER t)) SUBSET ((frontier s) UNION (frontier t)))))"}
{"_id": "3808", "title": "LIM_1_OVER_LOG", "text": "|- (((--> (\\n. ((complex_div (Cx (real_of_num (NUMERAL (BIT1 _0))))) (clog (Cx (real_of_num n)))))) (Cx (real_of_num (NUMERAL _0)))) sequentially)"}
{"_id": "3809", "title": "REAL_OF_INT_OF_REAL", "text": "|- (!x. ((integer x) ==> ((real_of_int (int_of_real x)) = x)))"}
{"_id": "3810", "title": "SETDIST_EQ_0_COMPACT_CLOSED", "text": "|- (!s. (!t. (((compact s) /\\ (closed t)) ==> (((setdist (s , t)) = (real_of_num (NUMERAL _0))) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ (~ ((s INTER t) = EMPTY))))))))"}
{"_id": "3811", "title": "SETDIST_FRONTIER", "text": "|- ((!s. (!t. (((DISJOINT s) t) ==> ((setdist ((frontier s) , t)) = (setdist (s , t)))))) /\\ (!s. (!t. (((DISJOINT s) t) ==> ((setdist (s , (frontier t))) = (setdist (s , t)))))))"}
{"_id": "3812", "title": "OPEN_IMP_ANR", "text": "|- (!s. ((open s) ==> (ANR s)))"}
{"_id": "3813", "title": "SPAN_VSUM", "text": "|- (!s. (!f. (!t. (((FINITE t) /\\ (!x. ((x IN t) ==> ((f x) IN (span s))))) ==> (((vsum t) f) IN (span s))))))"}
{"_id": "3814", "title": "o_THM", "text": "|- (!f. (!g. (!x. (((f o g) x) = (f (g x))))))"}
{"_id": "3815", "title": "ANR_PCROSS_EQ", "text": "|- (!s. (!t. ((ANR (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((ANR s) /\\ (ANR t)))))))"}
{"_id": "3816", "title": "REAL_POW_MONO", "text": "|- (!m. (!n. (!x. ((((real_le (real_of_num (NUMERAL (BIT1 _0)))) x) /\\ (m <= n)) ==> ((real_le ((real_pow x) m)) ((real_pow x) n))))))"}
{"_id": "3817", "title": "HAS_BOUNDED_REAL_VARIATION_ON_LMUL", "text": "|- (!f. (!c. (!s. (((has_bounded_real_variation_on f) s) ==> ((has_bounded_real_variation_on (\\x. ((real_mul c) (f x)))) s)))))"}
{"_id": "3818", "title": "CLOSED_IN_UNIONS", "text": "|- (!top. (!s. (((FINITE s) /\\ (!t. ((t IN s) ==> ((closed_in top) t)))) ==> ((closed_in top) (UNIONS s)))))"}
{"_id": "3819", "title": "Lp_main_estimate.tau_fun_local", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((((tau_fun ((v_prime V) f)) ((e_prime E) f)) f) = (((tau_fun V) E) f)))))))"}
{"_id": "3820", "title": "COMPLEX_DIFF_CHAIN_WITHIN", "text": "|- (!f. (!g. (!f'. (!g'. (!x. (!s. (((((has_complex_derivative f) f') ((within (at x)) s)) /\\ (((has_complex_derivative g) g') ((within (at (f x))) ((IMAGE f) s)))) ==> (((has_complex_derivative (g o f)) ((complex_mul g') f')) ((within (at x)) s)))))))))"}
{"_id": "3821", "title": "NEGLIGIBLE_AFFINE_HULL", "text": "|- (!s. (((FINITE s) /\\ ((CARD s) <= (dimindex UNIV))) ==> (negligible ((hull affine) s))))"}
{"_id": "3822", "title": "Functional_equation.proj_x3", "text": "|- (((((((proj_x3 x1) x2) x3) x4) x5) x6) = x3)"}
{"_id": "3823", "title": "has_real_integral", "text": "|- (!f. (!y. (!s. ((((has_real_integral f) y) s) = (((has_integral (lift o (f o drop))) (lift y)) ((IMAGE lift) s))))))"}
{"_id": "3824", "title": "ONORM", "text": "|- (!f. ((linear f) ==> ((!x. ((real_le (vector_norm (f x))) ((real_mul (onorm f)) (vector_norm x)))) /\\ (!b. ((!x. ((real_le (vector_norm (f x))) ((real_mul b) (vector_norm x)))) ==> ((real_le (onorm f)) b))))))"}
{"_id": "3825", "title": "Tame_general.CONTRAVENING_DART_DIST", "text": "|- (!V. (!x. (((contravening V) /\\ (x IN (dart_of_fan (V , (ESTD V))))) ==> (((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (distance x)) /\\ ((real_le (distance x)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))"}
{"_id": "3826", "title": "REAL_NORM", "text": "|- (!z. ((real z) ==> ((vector_norm z) = (real_abs (Re z)))))"}
{"_id": "3827", "title": "INTERIOR_TRANSLATION", "text": "|- (!a. (!s. ((interior ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (\\x. ((vector_add a) x))) (interior s)))))"}
{"_id": "3828", "title": "INSIDE_OUTSIDE", "text": "|- (!s. ((inside s) = (UNIV DIFF (s UNION (outside s)))))"}
{"_id": "3829", "title": "rowvector", "text": "|- (!v. ((rowvector v) = (lambdai. (lambdaj. (v $ j)))))"}
{"_id": "3830", "title": "FRONTIER_COMPLEMENT", "text": "|- (!s. ((frontier (UNIV DIFF s)) = (frontier s)))"}
{"_id": "3831", "title": "CONTINUOUS_ON_CONST", "text": "|- (!s. (!c. ((continuous_on (\\x. c)) s)))"}
{"_id": "3832", "title": "Topology.set_of_orbits_points_fan", "text": "|- (!x. (!V. (!E. (!v. (!u. ((((((set_of_orbits_points_fan x) V) E) v) u) = (GSPEC (\\GEN%PVAR%5483. (?i. (((SETSPEC GEN%PVAR%5483) ((NUMERAL _0) <= i)) (((((((power_map_points sigma_fan) x) V) E) v) u) i)))))))))))"}
{"_id": "3833", "title": "Tame_defs.split_at", "text": "|- (!c. (!xs. (((split_at c) xs) = (((split_at_rec c) NIL) xs))))"}
{"_id": "3834", "title": "BOREL_IMP_LEBESGUE_MEASURABLE", "text": "|- (!s. ((borel s) ==> (lebesgue_measurable s)))"}
{"_id": "3835", "title": "Seq.rev_cons", "text": "|- (!x. (!s. ((rev ((CONS x) s)) = ((rcons (rev s)) x))))"}
{"_id": "3836", "title": "REAL_MUL", "text": "|- (!w. (!z. (((real w) /\\ (real z)) ==> (real ((complex_mul w) z)))))"}
{"_id": "3837", "title": "POSITIVE_DEFINITE_MUL_EQ", "text": "|- (!A. (!B. (((positive_definite A) /\\ (positive_definite B)) ==> ((positive_definite ((matrix_mul A) B)) = (((matrix_mul A) B) = ((matrix_mul B) A))))))"}
{"_id": "3838", "title": "REAL_NEG_MINUS1", "text": "|- (!x. ((real_neg x) = ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x)))"}
{"_id": "3839", "title": "Add_triangle.f_fan_pair_power", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. (!i. ((d IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair (V , E))) i) d) = (((POWER (f_fan_pair_ext (V , E))) i) d))))))))"}
{"_id": "3840", "title": "AFFINE_LINEAR_IMAGE", "text": "|- (!f. (!s. (((affine s) /\\ (linear f)) ==> (affine ((IMAGE f) s)))))"}
{"_id": "3841", "title": "Sphere.acs_sqrt_x2_d4", "text": "|- (!x1. (!x3. (!x4. (!x5. (!x6. (!x2. (((((((acs_sqrt_x2_d4 x1) x2) x3) x4) x5) x6) = (acs ((real_div (sqrt x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))"}
{"_id": "3842", "title": "CNJ_EQ_CX", "text": "|- (!x. (!z. (((cnj z) = (Cx x)) = (z = (Cx x)))))"}
{"_id": "3843", "title": "polytope", "text": "|- (!s. ((polytope s) = (?v. ((FINITE v) /\\ (s = ((hull convex) v))))))"}
{"_id": "3844", "title": "PATH_CONTINUOUS_IMAGE", "text": "|- (!f. (!g. (((path g) /\\ ((continuous_on f) (path_image g))) ==> (path (f o g)))))"}
{"_id": "3845", "title": "HAS_REAL_INTEGRAL_SUBSTITUTION_SIMPLE", "text": "|- (!f. (!g. (!g'. (!a. (!b. (!c. (!d. ((((real_continuous_on f) (closed_real_interval ((CONS (c , d)) NIL))) /\\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> (((has_real_derivative g) (g' x)) ((within (atreal x)) (closed_real_interval ((CONS (a , b)) NIL)))))) /\\ ((((IMAGE g) (closed_real_interval ((CONS (a , b)) NIL))) SUBSET (closed_real_interval ((CONS (c , d)) NIL))) /\\ (((real_le a) b) /\\ (((real_le c) d) /\\ ((real_le (g a)) (g b))))))) ==> (((has_real_integral (\\x. ((real_mul (f (g x))) (g' x)))) ((real_integral (closed_real_interval ((CONS ((g a) , (g b))) NIL))) f)) (closed_real_interval ((CONS (a , b)) NIL)))))))))))"}
{"_id": "3846", "title": "REALLIM_CONG_AT", "text": "|- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((---> (\\x. (f x))) l) (at a)) = (((---> g) l) (at a))))"}
{"_id": "3847", "title": "relative_frontier", "text": "|- (!s. ((relative_frontier s) = ((closure s) DIFF (relative_interior s))))"}
{"_id": "3848", "title": "REAL_INV_EQ_0", "text": "|- (!x. (((real_inv x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0)))))"}
{"_id": "3849", "title": "FUN_EQ_THM", "text": "|- (!f. (!g. ((f = g) = (!x. ((f x) = (g x))))))"}
{"_id": "3850", "title": "Ssrnat.distnEr", "text": "|- (!m. (!n. ((m <= n) ==> (((distn m) n) = (n - m)))))"}
{"_id": "3851", "title": "Lp_ineqs_def.azim2_fan", "text": "|- (!V. (!E. (!x. (((azim2_fan (V , E)) x) = ((azim_dart (V , E)) ((f_fan_pair_ext (V , E)) x))))))"}
{"_id": "3852", "title": "Lp_ineqs_proofs.y7_lo", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y7_fan (V , E)) d))))))))))"}
{"_id": "3853", "title": "Nonlinear_lemma.dih_y_sym", "text": "|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((dih_y y1) y2) y3) y4) y5) y6) = ((((((dih_y y1) y3) y2) y4) y6) y5))))))))"}
{"_id": "3854", "title": "SUM_OFFSET", "text": "|- (!p. (!f. (!m. (!n. (((sum ((m + p) .. (n + p))) f) = ((sum (m .. n)) (\\i. (f (i + p)))))))))"}
{"_id": "3855", "title": "Real_ext.REAL_PROP_POS_LADD", "text": "|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))"}
{"_id": "3856", "title": "Float.POW_2", "text": "|- (!x. (((real_pow x) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_mul x) x)))"}
{"_id": "3857", "title": "LINEAR_ADD", "text": "|- (!f. (!x. (!y. ((linear f) ==> ((f ((vector_add x) y)) = ((vector_add (f x)) (f y)))))))"}
{"_id": "3858", "title": "REAL_NEG_LMUL", "text": "|- (!x. (!y. ((real_neg ((real_mul x) y)) = ((real_mul (real_neg x)) y))))"}
{"_id": "3859", "title": "COLLINEAR_3_EXPLICIT", "text": "|- (!x. (!y. (!z. ((collinear (x INSERT (y INSERT (z INSERT EMPTY)))) = (?a. (?b. (?c. ((((vector_add ((% a) x)) ((vector_add ((% b) y)) ((% c) z))) = (vec (NUMERAL _0))) /\\ ((((real_add a) ((real_add b) c)) = (real_of_num (NUMERAL _0))) /\\ (~ ((a = (real_of_num (NUMERAL _0))) /\\ ((b = (real_of_num (NUMERAL _0))) /\\ (c = (real_of_num (NUMERAL _0)))))))))))))))"}
{"_id": "3860", "title": "Real_ext.REAL_PROP_NN_ADD2", "text": "|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_add x) y)))))"}
{"_id": "3861", "title": "IM_CLOG_POS_LE", "text": "|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((real_le (real_of_num (NUMERAL _0))) (Im (clog z))) = ((real_le (real_of_num (NUMERAL _0))) (Im z)))))"}
{"_id": "3862", "title": "Hypermap.lemma_in_face_set", "text": "|- (!H. (!x. ((x IN (dart H)) = (((face H) x) IN (face_set H)))))"}
{"_id": "3863", "title": "Lp_ineqs_proofs.y9_eq", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> ((LET (\\w'. (LET_END ((LET (\\u. (LET_END (((y9_fan (V , E)) (v , w)) = (distance (w , u)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))))"}
{"_id": "3864", "title": "CONTINUOUS_ON_COMPONENTWISE_LIFT", "text": "|- (!f. (!s. (((continuous_on f) s) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((continuous_on (\\x. (lift ((f x) $ i)))) s))))))"}
{"_id": "3865", "title": "Seq.size_rot", "text": "|- (!n0. (!s. ((sizel ((rot n0) s)) = (sizel s))))"}
{"_id": "3866", "title": "HAS_COMPLEX_DERIVATIVE_ID", "text": "|- (!net. (((has_complex_derivative (\\x. x)) (Cx (real_of_num (NUMERAL (BIT1 _0))))) net))"}
{"_id": "3867", "title": "HOMEOMORPHIC_SYM", "text": "|- (!s. (!t. (((homeomorphic s) t) = ((homeomorphic t) s))))"}
{"_id": "3868", "title": "conic", "text": "|- (!s. ((conic s) = (!x. (!c. (((x IN s) /\\ ((real_le (real_of_num (NUMERAL _0))) c)) ==> (((% c) x) IN s))))))"}
{"_id": "3869", "title": "FRAC_FLOOR", "text": "|- (!x. ((frac x) = ((real_sub x) (floor x))))"}
{"_id": "3870", "title": "Hypermap_iso.fan_hypermaps_iso_explicit", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((hyp_iso contracted_dart) ((hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E))) , (hypermap_of_fan (V , E)))))))"}
{"_id": "3871", "title": "Sphere.INITIAL_SUBLIST", "text": "|- (!zl. (!xl. (((initial_sublist xl) zl) = (?yl. (zl = ((APPEND xl) yl))))))"}
{"_id": "3872", "title": "INT_OF_NUM_OF_INT", "text": "|- (!x. (((int_le (int_of_num (NUMERAL _0))) x) ==> ((int_of_num (num_of_int x)) = x)))"}
{"_id": "3873", "title": "COMPLEX_POW_1", "text": "|- (!x. (((complex_pow x) (NUMERAL (BIT1 _0))) = x))"}
{"_id": "3874", "title": "HULL_SUBSET", "text": "|- (!P. (!s. (s SUBSET ((hull P) s))))"}
{"_id": "3875", "title": "CONNECTED_SEGMENT", "text": "|- ((!a. (!b. (connected (closed_segment ((CONS (a , b)) NIL))))) /\\ (!a. (!b. (connected (open_segment (a , b))))))"}
{"_id": "3876", "title": "SUM_CLAUSES_NUMSEG", "text": "|- ((!m. (((sum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (real_of_num (NUMERAL _0))))) /\\ (!m. (!n. (((sum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((real_add ((sum (m .. n)) f)) (f (SUC n)))) ((sum (m .. n)) f))))))"}
{"_id": "3877", "title": "COMPACT_CONVEX_COLLINEAR_SEGMENT", "text": "|- (!s. (((~ (s = EMPTY)) /\\ ((compact s) /\\ ((convex s) /\\ (collinear s)))) ==> (?a. (?b. (s = (closed_segment ((CONS (a , b)) NIL)))))))"}
{"_id": "3878", "title": "DOT_EQ_0", "text": "|- (!x. ((((dot x) x) = (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))"}
{"_id": "3879", "title": "LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((lebesgue_measurable ((IMAGE f) s)) = (lebesgue_measurable s)))))"}
{"_id": "3880", "title": "LT", "text": "|- ((!m. ((m < (NUMERAL _0)) = F)) /\\ (!m. (!n. ((m < (SUC n)) = ((m = n) \\/ (m < n))))))"}
{"_id": "3881", "title": "IN_CLOSURE_OF", "text": "|- (!top. (!s. (!x. ((x IN ((closure_of top) s)) = ((x IN (topspace top)) /\\ (!t. (((x IN t) /\\ ((open_in top) t)) ==> (?y. ((y IN s) /\\ (y IN t))))))))))"}
{"_id": "3882", "title": "HAUSDIST_LINEAR_IMAGE", "text": "|- (!f. (!s. (!t. (((linear f) /\\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((hausdist (((IMAGE f) s) , ((IMAGE f) t))) = (hausdist (s , t)))))))"}
{"_id": "3883", "title": "FACE_OF_SING", "text": "|- (!x. (!s. (((face_of (x INSERT EMPTY)) s) = ((extreme_point_of x) s))))"}
{"_id": "3884", "title": "Functional_equation.div6", "text": "|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((div6 f) g) x1) x2) x3) x4) x5) x6) = ((real_div ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))"}
{"_id": "3885", "title": "RE_VSUM", "text": "|- (!f. (!s. ((FINITE s) ==> ((Re ((vsum s) f)) = ((sum s) (\\x. (Re (f x))))))))"}
{"_id": "3886", "title": "real_integrable_on", "text": "|- (!f. (!i. (((real_integrable_on f) i) = (?y. (((has_real_integral f) y) i)))))"}
{"_id": "3887", "title": "FUBINI_INTEGRAL_INTERVAL", "text": "|- (!f. (!a. (!b. (!c. (!d. (((absolutely_integrable_on f) (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) ==> (((integral (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) f) = ((integral (closed_interval ((CONS (a , b)) NIL))) (\\x. ((integral (closed_interval ((CONS (c , d)) NIL))) (\\y. (f ((pastecart x) y)))))))))))))"}
{"_id": "3888", "title": "Seq.head", "text": "|- (!x0. (!h. (!t. ((((headl x0) NIL) = x0) /\\ (((headl x0) ((CONS h) t)) = h)))))"}
{"_id": "3889", "title": "HAS_REAL_INTEGRAL_AFFINITY", "text": "|- (!f. (!i. (!a. (!b. (!m. (!c. (((((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))) /\\ (~ (m = (real_of_num (NUMERAL _0))))) ==> (((has_real_integral (\\x. (f ((real_add ((real_mul m) x)) c)))) ((real_mul (real_inv (real_abs m))) i)) ((IMAGE (\\x. ((real_mul (real_inv m)) ((real_sub x) c)))) (closed_real_interval ((CONS (a , b)) NIL)))))))))))"}
{"_id": "3890", "title": "REAL_SGN_POW_2", "text": "|- (!x. ((real_sgn ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = (real_sgn (real_abs x))))"}
{"_id": "3891", "title": "LINEPATH_LINEAR_IMAGE", "text": "|- (!f. ((linear f) ==> (!b. (!c. ((linepath ((f b) , (f c))) = (f o (linepath (b , c))))))))"}
{"_id": "3892", "title": "VECTOR_SUB_RDISTRIB", "text": "|- (((% ((real_sub a) b)) x) = ((vector_sub ((% a) x)) ((% b) x)))"}
{"_id": "3893", "title": "Packing3.SING_UNION_EQ_INSERT", "text": "|- (!s. (!x. (((x INSERT EMPTY) UNION s) = (x INSERT s))))"}
{"_id": "3894", "title": "REAL_EXP_POS_LE", "text": "|- (!x. ((real_le (real_of_num (NUMERAL _0))) (exp x)))"}
{"_id": "3895", "title": "HAS_BOUNDED_VARIATION_AFFINITY_EQ", "text": "|- (!m. (!c. (!f. (!s. (((has_bounded_variation_on (\\x. (f ((vector_add ((% m) x)) c)))) s) = ((m = (real_of_num (NUMERAL _0))) \\/ ((has_bounded_variation_on f) ((IMAGE (\\x. ((vector_add ((% m) x)) c))) s))))))))"}
{"_id": "3896", "title": "Fan.fan5", "text": "|- (!V. (!E. (!x. ((fan5 (x , (V , E))) = (!e. (!f. (((e IN E) /\\ ((f IN E) /\\ (~ (e = f)))) ==> ((((aff_gt (x INSERT EMPTY)) e) INTER ((aff_gt (x INSERT EMPTY)) f)) = EMPTY))))))))"}
{"_id": "3897", "title": "CARD_EXP_ADD", "text": "|- (!s. (!t. (!u. (((^_c s) ((+_c t) u)) =_c ((*_c ((^_c s) t)) ((^_c s) u))))))"}
{"_id": "3898", "title": "RELATIVE_INTERIOR_SEGMENT", "text": "|- ((!a. (!b. ((relative_interior (closed_segment ((CONS (a , b)) NIL))) = (((COND (a = b)) (a INSERT EMPTY)) (open_segment (a , b)))))) /\\ (!a. (!b. ((relative_interior (open_segment (a , b))) = (open_segment (a , b))))))"}
{"_id": "3899", "title": "Hypermap_iso.surj_res_inv_right", "text": "|- (!f. (!s. (!t. (!y. (((((SURJ f) s) t) /\\ (y IN t)) ==> (((f (((res_inv f) s) y)) = y) /\\ ((((res_inv f) s) y) IN s)))))))"}
{"_id": "3900", "title": "REAL_DIFFERENTIABLE_ATREAL_WITHIN", "text": "|- (!f. (!s. (!z. (((real_differentiable f) (atreal z)) ==> ((real_differentiable f) ((within (atreal z)) s))))))"}
{"_id": "3901", "title": "COMPLEX_INV_EQ_0", "text": "|- (!x. (((complex_inv x) = (Cx (real_of_num (NUMERAL _0)))) = (x = (Cx (real_of_num (NUMERAL _0))))))"}
{"_id": "3902", "title": "HOMOTOPIC_WITH", "text": "|- ((!h. (!k. ((!x. ((x IN X) ==> ((h x) = (k x)))) ==> ((P h) = (P k))))) ==> (((((homotopic_with P) (X , Y)) p) q) = (?h. (((continuous_on h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) /\\ ((((IMAGE h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) SUBSET Y) /\\ ((!x. ((x IN X) ==> ((h ((pastecart (vec (NUMERAL _0))) x)) = (p x)))) /\\ ((!x. ((x IN X) ==> ((h ((pastecart (vec (NUMERAL (BIT1 _0)))) x)) = (q x)))) /\\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> (P (\\x. (h ((pastecart t) x)))))))))))))"}
{"_id": "3903", "title": "LINEAR_SURJECTIVE_IMP_INJECTIVE", "text": "|- (!f. (((linear f) /\\ (!y. (?x. ((f x) = y)))) ==> (!x. (!y. (((f x) = (f y)) ==> (x = y))))))"}
{"_id": "3904", "title": "Fan_defs.e_fan_pair", "text": "|- (!V. (!E. (!w. (!v. (((e_fan_pair (V , E)) (v , w)) = (w , v))))))"}
{"_id": "3905", "title": "CBALL_SING", "text": "|- (!x. (!e. ((e = (real_of_num (NUMERAL _0))) ==> ((cball (x , e)) = (x INSERT EMPTY)))))"}
{"_id": "3906", "title": "DIST_EQ_0", "text": "|- (!x. (!y. (((distance (x , y)) = (real_of_num (NUMERAL _0))) = (x = y))))"}
{"_id": "3907", "title": "Kepler_spec.dist_spec", "text": "|- (!x. (!y. (!z. (!x'. (!y'. (!z'. ((distance ((real3_of_triple (x , (y , z))) , (real3_of_triple (x' , (y' , z'))))) = (sqrt ((real_add ((real_pow ((real_sub x) x')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_add ((real_pow ((real_sub y) y')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub z) z')) (NUMERAL (BIT0 (BIT1 _0))))))))))))))"}
{"_id": "3908", "title": "Lp_ineqs_proofs.y6_hi_std2", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))"}
{"_id": "3909", "title": "Hypermap.THREE", "text": "|- ((NUMERAL (BIT1 (BIT1 _0))) = (SUC (NUMERAL (BIT0 (BIT1 _0)))))"}
{"_id": "3910", "title": "REAL_COMPACT_IMP_BOUNDED", "text": "|- (!s. ((real_compact s) ==> (real_bounded s)))"}
{"_id": "3911", "title": "Real_ext.REAL_SV_NEG_0", "text": "|- ((real_neg (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))"}
{"_id": "3912", "title": "PATH_CONNECTED_LINEAR_IMAGE_EQ", "text": "|- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((path_connected ((IMAGE f) s)) = (path_connected s)))))"}
{"_id": "3913", "title": "BASIS_ORTHOGONAL", "text": "|- (!b. ((FINITE b) ==> (?c. ((FINITE c) /\\ (((CARD c) <= (CARD b)) /\\ (((span c) = (span b)) /\\ ((pairwise orthogonal) c)))))))"}
{"_id": "3914", "title": "ABSOLUTELY_INTEGRABLE_CONTINUOUS", "text": "|- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "3915", "title": "REFLECT_ALONG_LINEAR_IMAGE", "text": "|- (!f. (!v. (!x. (((linear f) /\\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((reflect_along (f v)) (f x)) = (f ((reflect_along v) x)))))))"}
{"_id": "3916", "title": "CEXP_ADD", "text": "|- (!w. (!z. ((cexp ((vector_add w) z)) = ((complex_mul (cexp w)) (cexp z)))))"}
{"_id": "3917", "title": "PSUBSET", "text": "|- (!s. (!t. ((s PSUBSET t) = ((s SUBSET t) /\\ (~ (s = t))))))"}
{"_id": "3918", "title": "IN", "text": "|- (!P. (!x. ((x IN P) = (P x))))"}
{"_id": "3919", "title": "STC_CLOSED", "text": "|- (!R. (((STC R) = R) = ((!x. (!y. (((R x) y) ==> ((R y) x)))) /\\ (!x. (!y. (!z. ((((R x) y) /\\ ((R y) z)) ==> ((R x) z))))))))"}
{"_id": "3920", "title": "PRODUCT_EQ", "text": "|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((product s) f) = ((product s) g))))))"}
{"_id": "3921", "title": "MEASURABLE_CONVEX", "text": "|- (!s. (((convex s) /\\ (bounded s)) ==> (measurable s)))"}
{"_id": "3922", "title": "Add_triangle.sigma_in_dart1", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> ((v , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) IN (dart1_of_fan (V , E)))))))))"}
{"_id": "3923", "title": "Seq.rot", "text": "|- (!n. (!s. (((rot n) s) = ((cat ((dropl n) s)) ((take n) s)))))"}
{"_id": "3924", "title": "ENR_ANR", "text": "|- (!s. ((ENR s) = ((ANR s) /\\ ((locally compact) s))))"}
{"_id": "3925", "title": "REAL_CONTINUOUS_ON", "text": "|- (!f. (!s. (((real_continuous_on f) s) = ((continuous_on (lift o (f o drop))) ((IMAGE lift) s)))))"}
{"_id": "3926", "title": "Appendix.BBindex_min_v39", "text": "|- (!s. ((BBindex_min_v39 s) = (min_num ((IMAGE (BBindex_v39 s)) (BBprime_v39 s)))))"}
{"_id": "3927", "title": "SUBSET", "text": "|- (!s. (!t. ((s SUBSET t) = (!x. ((x IN s) ==> (x IN t))))))"}
{"_id": "3928", "title": "Tame_general.ly_EQ_lmfun", "text": "|- (!x. (((real_le (vector_norm (FST x))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((lmfun (h_dart x)) = (ly (vector_norm (FST x))))))"}
{"_id": "3929", "title": "LE_ADD", "text": "|- (!m. (!n. (m <= (m + n))))"}
{"_id": "3930", "title": "LEFT_OR_FORALL_THM", "text": "|- (!P. (!Q. (((!x. (P x)) \\/ Q) = (!x. ((P x) \\/ Q)))))"}
{"_id": "3931", "title": "FINITE_IMAGE", "text": "|- (!f. (!s. ((FINITE s) ==> (FINITE ((IMAGE f) s)))))"}
{"_id": "3932", "title": "ANR_IMP_ABSOLUTE_CLOSED_NEIGHBOURHOOD_RETRACT", "text": "|- (!s. (!u. (!s'. (((ANR s) /\\ (((homeomorphic s) s') /\\ ((closed_in ((subtopology euclidean) u)) s'))) ==> (?v. (?w. (((open_in ((subtopology euclidean) u)) v) /\\ (((closed_in ((subtopology euclidean) u)) w) /\\ ((s' SUBSET v) /\\ ((v SUBSET w) /\\ ((retract_of s') w)))))))))))"}
{"_id": "3933", "title": "LIM_AT_INFINITY_COMPLEX_0", "text": "|- (!f. (!l. ((((--> f) l) at_infinity) = (((--> (f o complex_inv)) l) (at (Cx (real_of_num (NUMERAL _0))))))))"}
{"_id": "3934", "title": "RIGHT_ADD_DISTRIB", "text": "|- (!m. (!n. (!p. (((m + n) * p) = ((m * p) + (n * p))))))"}
{"_id": "3935", "title": "IN_INSERT", "text": "|- (!x. (!y. (!s. ((x IN (y INSERT s)) = ((x = y) \\/ (x IN s))))))"}
{"_id": "3936", "title": "SPAN_CONIC_HULL", "text": "|- (!s. ((span ((hull conic) s)) = (span s)))"}
{"_id": "3937", "title": "VECTOR_POLYNOMIAL_FUNCTION_CONST", "text": "|- (!c. (vector_polynomial_function (\\x. c)))"}
{"_id": "3938", "title": "SERIES_COMPONENT", "text": "|- (!f. (!s. (!l. (!k. (((((sums f) l) s) /\\ (((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV)))) ==> (((sums (\\i. (lift ((f i) $ k)))) (lift (l $ k))) s))))))"}
{"_id": "3939", "title": "COMPLEX_DERIVATIVE_SUB", "text": "|- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\\ ((complex_differentiable g) (at z))) ==> (((complex_derivative (\\w. ((vector_sub (f w)) (g w)))) z) = ((vector_sub ((complex_derivative f) z)) ((complex_derivative g) z)))))))"}
{"_id": "3940", "title": "Ssrnat.succnK", "text": "|- (!n. (((SUC n) - (NUMERAL (BIT1 _0))) = n))"}
{"_id": "3941", "title": "ISOMETRY_LINEAR", "text": "|- (!f. ((((f (vec (NUMERAL _0))) = (vec (NUMERAL _0))) /\\ (!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y)))))) ==> (linear f)))"}
{"_id": "3942", "title": "INDEPENDENT_IMP_FINITE", "text": "|- (!s. ((independent s) ==> (FINITE s)))"}
{"_id": "3943", "title": "Sphere.tame_table_d", "text": "|- (!r. (!s. (((tame_table_d r) s) = (((COND ((r + ((NUMERAL (BIT0 (BIT1 _0))) * s)) > (NUMERAL (BIT1 (BIT1 _0))))) ((real_add ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_sub (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num s)))) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((real_add (real_of_num r)) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num s))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))))))"}
{"_id": "3944", "title": "REAL_LE_HAUSDIST", "text": "|- (!s. (!t. (!a. (!b. (!c. (!z. (((~ (s = EMPTY)) /\\ ((~ (t = EMPTY)) /\\ ((!x. ((x IN s) ==> ((real_le (setdist ((x INSERT EMPTY) , t))) b))) /\\ ((!y. ((y IN t) ==> ((real_le (setdist ((y INSERT EMPTY) , s))) c))) /\\ (((z IN s) /\\ ((real_le a) (setdist ((z INSERT EMPTY) , t)))) \\/ ((z IN t) /\\ ((real_le a) (setdist ((z INSERT EMPTY) , s))))))))) ==> ((real_le a) (hausdist (s , t))))))))))"}
{"_id": "3945", "title": "Seq.map", "text": "|- ((((map f) ((CONS x) s)) = ((CONS (f x)) ((map f) s))) /\\ (((map f) NIL) = NIL))"}
{"_id": "3946", "title": "DIFF_CHAIN_AT", "text": "|- (!f. (!g. (!f'. (!g'. (!x. (((((has_derivative f) f') (at x)) /\\ (((has_derivative g) g') (at (f x)))) ==> (((has_derivative (g o f)) (g' o f')) (at x))))))))"}
{"_id": "3947", "title": "Hexagons.sqrt8_LE_CSTAB", "text": "|- ((real_le sqrt8) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))"}
{"_id": "3948", "title": "PASTECART_INJ", "text": "|- (!x. (!y. (!w. (!z. ((((pastecart x) y) = ((pastecart w) z)) = ((x = w) /\\ (y = z)))))))"}
{"_id": "3949", "title": "ORTHOGONAL_TRANSFORMATION_LINEAR", "text": "|- (!f. ((orthogonal_transformation f) ==> (linear f)))"}
{"_id": "3950", "title": "LINEAR_CX_RE", "text": "|- (linear (Cx o Re))"}
{"_id": "3951", "title": "ARITH_LT", "text": "|- ((!m. (!n. (((NUMERAL m) < (NUMERAL n)) = (m < n)))) /\\ (((_0 < _0) = F) /\\ ((!n. (((BIT0 n) < _0) = F)) /\\ ((!n. (((BIT1 n) < _0) = F)) /\\ ((!n. ((_0 < (BIT0 n)) = (_0 < n))) /\\ ((!n. ((_0 < (BIT1 n)) = T)) /\\ ((!m. (!n. (((BIT0 m) < (BIT0 n)) = (m < n)))) /\\ ((!m. (!n. (((BIT0 m) < (BIT1 n)) = (m <= n)))) /\\ ((!m. (!n. (((BIT1 m) < (BIT0 n)) = (m < n)))) /\\ (!m. (!n. (((BIT1 m) < (BIT1 n)) = (m < n)))))))))))))"}
{"_id": "3952", "title": "VECTOR_VARIATION_TRIANGLE", "text": "|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\\ ((has_bounded_variation_on g) s)) ==> ((real_le ((vector_variation s) (\\x. ((vector_add (f x)) (g x))))) ((real_add ((vector_variation s) f)) ((vector_variation s) g)))))))"}
{"_id": "3953", "title": "SUBSET_LE_DIM", "text": "|- (!s. (!t. ((s SUBSET (span t)) ==> ((dim s) <= (dim t)))))"}
{"_id": "3954", "title": "IN_ELIM_PASTECART_THM", "text": "|- (!P. (!a. (!b. ((((pastecart a) b) IN (GSPEC (\\GEN%PVAR%238. (?x. (?y. (((SETSPEC GEN%PVAR%238) ((P x) y)) ((pastecart x) y))))))) = ((P a) b)))))"}
{"_id": "3955", "title": "CONTINUOUS_AT_COMPOSE", "text": "|- (!f. (!g. (!x. ((((continuous f) (at x)) /\\ ((continuous g) (at (f x)))) ==> ((continuous (g o f)) (at x))))))"}
{"_id": "3956", "title": "Lp_ineqs_proofs.tau_fan_eq_tauVEF", "text": "|- (!V. (!E. (!f. (((lp_tau (V , E)) /\\ (f IN (face_set (hypermap_of_fan (V , E))))) ==> (((tau_fan (V , E)) f) = (tauVEF (V , (E , f))))))))"}
{"_id": "3957", "title": "Hypermap_iso.hyp_iso_imp_iso", "text": "|- (!H. (!G. (!f. (((hyp_iso f) (H , G)) ==> ((iso H) G)))))"}
{"_id": "3958", "title": "CONNECTED_COMPONENT_TRANSLATION", "text": "|- (!a. (!s. (!x. (((connected_component ((IMAGE (\\x. ((vector_add a) x))) s)) ((vector_add a) x)) = ((IMAGE (\\x. ((vector_add a) x))) ((connected_component s) x))))))"}
{"_id": "3959", "title": "REAL_ADD_RDISTRIB", "text": "|- (!x. (!y. (!z. (((real_mul ((real_add x) y)) z) = ((real_add ((real_mul x) z)) ((real_mul y) z))))))"}
{"_id": "3960", "title": "IM_ADD", "text": "|- (!x. (!y. ((Im ((vector_add x) y)) = ((real_add (Im x)) (Im y)))))"}
{"_id": "3961", "title": "List_hypermap.good_list_nodes", "text": "|- (!L. ((good_list_nodes L) = ((node_set (hypermap_of_list L)) = (set_of_list (nodes_of_list L)))))"}
{"_id": "3962", "title": "LIM_POSINFINITY_SEQUENTIALLY", "text": "|- (!f. (!l. ((((--> f) l) at_posinfinity) ==> (((--> (\\n. (f (real_of_num n)))) l) sequentially))))"}
{"_id": "3963", "title": "Dih2k_hypermap.cstab", "text": "|- (cstab = ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))"}
{"_id": "3964", "title": "Nonlin_def.sqrt_x4", "text": "|- (((((((sqrt_x4 x1) x2) x3) x4) x5) x6) = (sqrt x4))"}
{"_id": "3965", "title": "REAL_MUL_LINV", "text": "|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (((real_mul (real_inv x)) x) = (real_of_num (NUMERAL (BIT1 _0))))))"}
{"_id": "3966", "title": "differentiable_on", "text": "|- (!f. (!s. (((differentiable_on f) s) = (!x. ((x IN s) ==> ((differentiable f) ((within (at x)) s)))))))"}
{"_id": "3967", "title": "Lp_ineqs_proofs.dart4_cross_eq_y4'", "text": "|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y4'_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))"}
{"_id": "3968", "title": "CONTINUOUS_AT_CACS", "text": "|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((continuous cacs) (at z))))"}
{"_id": "3969", "title": "PERMUTATION_SWAP", "text": "|- (!a. (!b. (permutation (swap (a , b)))))"}
{"_id": "3970", "title": "REAL_SGN_MUL", "text": "|- (!x. (!y. ((real_sgn ((real_mul x) y)) = ((real_mul (real_sgn x)) (real_sgn y)))))"}
{"_id": "3971", "title": "COMPONENTS_TRANSLATION", "text": "|- (!a. (!s. ((components ((IMAGE (\\x. ((vector_add a) x))) s)) = ((IMAGE (IMAGE (\\x. ((vector_add a) x)))) (components s)))))"}
{"_id": "3972", "title": "Fan.POWER", "text": "|- ((!f. (((POWER f) (NUMERAL _0)) = I)) /\\ (!f. (!n. (((POWER f) (SUC n)) = (((POWER f) n) o f)))))"}