{"_id": "50001", "text": "MEASURABLE_CONTINUOUS_COMPOSE |- (!f. (!g. ((((measurable_on f) UNIV) /\\ (((continuous_on g) UNIV) /\\ (!k. ((negligible k) ==> (negligible (GSPEC (\\GEN%PVAR%3702. (?x. (((SETSPEC GEN%PVAR%3702) ((g x) IN k)) x))))))))) ==> ((measurable_on (f o g)) UNIV))))"}
{"_id": "50002", "text": "CONTINUOUS_ON_COMPLEX_RMUL |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\\x. ((complex_mul (f x)) c))) s))))"}
{"_id": "50003", "text": "VECTOR_VARIATION_LIFT_ABS |- (!f. (!s. (((has_bounded_variation_on (\\x. (lift (f x)))) s) ==> ((real_le ((vector_variation s) (\\x. (lift (real_abs (f x)))))) ((vector_variation s) (\\x. (lift (f x))))))))"}
{"_id": "50004", "text": "INCREASING_BOUNDED_REAL_VARIATION |- (!f. (!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))))) ==> ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "50005", "text": "TRIANGULATION_INTER_SIMPLEX |- (!tr. (!t. (!t'. (((triangulation tr) /\\ ((t IN tr) /\\ (t' IN tr))) ==> ((t INTER t') = ((hull convex) ((GSPEC (\\GEN%PVAR%2491. (?x. (((SETSPEC GEN%PVAR%2491) ((extreme_point_of x) t)) x)))) INTER (GSPEC (\\GEN%PVAR%2492. (?x. (((SETSPEC GEN%PVAR%2492) ((extreme_point_of x) t')) x)))))))))))"}
{"_id": "50006", "text": "Functional_equation.functional_halfbump_x1 |- (halfbump_x1 = (promote1_to_6 halfbump_x))"}
{"_id": "50007", "text": "Nonlinear_lemma.vol3f_lm0 |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y1) /\\ (((real_le y2) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\\ ((real_le y6) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ==> ((((((vol3f y1) y2) y6) sqrt2) lmfun) = ((((((vol3f_sqrt2_lmplus y1) y2) y3) y4) y5) y6)))))))))"}
{"_id": "50008", "text": "INT_SUB_LNEG |- (!x. (!y. (((int_sub (int_neg x)) y) = (int_neg ((int_add x) y)))))"}
{"_id": "50009", "text": "Lp_ineqs_proofs.lp_cond_trans |- (!L. (!V. (!E. (!g. (!h. (!ds. (((lp_cond (L , (g , h))) (V , E)) ==> ((lp_cond (((split_list_list L) ds) , (g , h))) (V , ((add_diag_list (g , (V , E))) ds))))))))))"}
{"_id": "50010", "text": "AFFINE_INDEPENDENT_STDBASIS |- (~ (affine_dependent ((vec (NUMERAL _0)) INSERT (GSPEC (\\GEN%PVAR%1517. (?i. (((SETSPEC GEN%PVAR%1517) (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) (basis i))))))))"}
{"_id": "50011", "text": "INTEGER_POW |- (!x. (!n. ((integer x) ==> (integer ((real_pow x) n)))))"}
{"_id": "50012", "text": "RECTIFIABLE_PATH_VECTOR_POLYNOMIAL_FUNCTION |- (!p. ((vector_polynomial_function p) ==> (rectifiable_path p)))"}
{"_id": "50013", "text": "PATH_CONNECTED_NEGATIONS |- (!s. ((path_connected s) ==> (path_connected ((IMAGE vector_neg) s))))"}
{"_id": "50014", "text": "HOMOTOPY_EQUIVALENT |- (!s. (!t. (((homotopy_equivalent s) t) = (?f. (?g. (?h. (((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((continuous_on g) t) /\\ ((((IMAGE g) t) SUBSET s) /\\ (((continuous_on h) t) /\\ ((((IMAGE h) t) SUBSET s) /\\ (((((homotopic_with (\\x. T)) (s , s)) (g o f)) I) /\\ ((((homotopic_with (\\x. T)) (t , t)) (f o h)) I))))))))))))))"}
{"_id": "50015", "text": "MAXIMAL_AFFINE_INDEPENDENT_SUBSET_AFFINE |- (!s. (!b. (((affine s) /\\ ((b SUBSET s) /\\ ((~ (affine_dependent b)) /\\ (!b'. (((b SUBSET b') /\\ ((b' SUBSET s) /\\ (~ (affine_dependent b')))) ==> (b' = b)))))) ==> (((hull affine) b) = s))))"}
{"_id": "50016", "text": "Lp_ineqs_proofs.rhazim3_hi |- (!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 ((rhazim3_fan (V , E)) d)) ((real_add pi) sol0))))))))))"}
{"_id": "50017", "text": "NONNEGATIVE_ABSOLUTELY_REAL_INTEGRABLE |- (!f. (!s. (((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) /\\ ((real_integrable_on f) s)) ==> ((absolutely_real_integrable_on f) s))))"}
{"_id": "50018", "text": "LIM_NULL_COMPLEX_POW_EQ |- (!net. (!f. (!n. ((~ (n = (NUMERAL _0))) ==> ((((--> (\\x. ((complex_pow (f x)) n))) (Cx (real_of_num (NUMERAL _0)))) net) = (((--> f) (Cx (real_of_num (NUMERAL _0)))) net))))))"}
{"_id": "50019", "text": "Trigonometry1.COS_PERIODIC_N2PI |- (!x. (!n. ((cos ((real_add x) ((real_mul (real_of_num n)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) = (cos x))))"}
{"_id": "50020", "text": "ARC_SUBPATH_EQ |- (!g. (!u. (!v. ((arc (((subpath u) v) g)) = ((path (((subpath u) v) g)) /\\ ((~ (u = v)) /\\ (!x. (!y. (((x IN (closed_segment ((CONS (u , v)) NIL))) /\\ ((y IN (closed_segment ((CONS (u , v)) NIL))) /\\ ((g x) = (g y)))) ==> (x = y))))))))))"}
{"_id": "50021", "text": "Tame_lemmas.fully_surrounded_sum_sol |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (((sum (face_set (hypermap_of_fan (V , E)))) (\\f. ((sol (vec (NUMERAL _0))) ((((dartset_leads_into_fan (vec (NUMERAL _0))) V) E) ((IMAGE (ext_dart (V , E))) f))))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))"}
{"_id": "50022", "text": "LIFT_IN_INTERVAL |- ((!a. (!b. (!x. (((lift x) IN (closed_interval ((CONS (a , b)) NIL))) = (x IN (closed_real_interval ((CONS ((drop a) , (drop b))) NIL))))))) /\\ (!a. (!b. (!x. (((lift x) IN (open_interval (a , b))) = (x IN (open_real_interval ((drop a) , (drop b)))))))))"}
{"_id": "50023", "text": "CONTRACTIBLE_CONVEX_TWEAK_BOUNDARY_POINTS |- (!s. (!t. (((convex s) /\\ (((relative_interior s) SUBSET t) /\\ (t SUBSET (closure s)))) ==> (contractible t))))"}
{"_id": "50024", "text": "CONIC_RELATIVE_INTERIOR |- (!s. (((conic s) /\\ ((vec (NUMERAL _0)) IN (relative_interior s))) ==> (conic (relative_interior s))))"}
{"_id": "50025", "text": "Merge_ineq.nonf_mud_135_x |- (!a. (!b. (!c. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_135_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul (((mu_y a) (sqrt x1)) (sqrt x3))) (sqrt (((((((((delta_135_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))"}
{"_id": "50026", "text": "CONTINUOUS_ON_COMPONENTS_FINITE |- (!f. (!s. (((FINITE (components s)) /\\ (!c. ((c IN (components s)) ==> ((continuous_on f) c)))) ==> ((continuous_on f) s))))"}
{"_id": "50027", "text": "Seq2.perm_eq_set_of_list |- (!l. (!s. (((FINITE s) /\\ ((perm_eq l) (list_of_set s))) ==> ((set_of_list l) = s))))"}
{"_id": "50028", "text": "REAL_SUMMABLE_NEG |- (!x. (!s. (((real_summable s) x) ==> ((real_summable s) (\\n. (real_neg (x n)))))))"}
{"_id": "50029", "text": "COMPLEX_SUB_SUB |- (!x. (!y. (((vector_sub ((vector_sub x) y)) x) = (vector_neg y))))"}
{"_id": "50030", "text": "Seq.mem_seq3 |- (!x. (!y1. (!y2. (!y3. (((MEM x) ((CONS y1) ((CONS y2) ((CONS y3) NIL)))) = ((x = y1) \\/ ((x = y2) \\/ (x = y3))))))))"}
{"_id": "50031", "text": "EDELSTEIN_FIX_ITER |- (!f. (!s. (!n. (((compact s) /\\ ((~ (s = EMPTY)) /\\ ((((IMAGE f) s) SUBSET s) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ (~ (x = y)))) ==> ((real_lt (distance ((((ITER n) f) x) , (((ITER n) f) y)))) (distance (x , y))))))))) ==> (?!x. ((x IN s) /\\ ((f x) = x)))))))"}
{"_id": "50032", "text": "ULC_IMP_LOCALLY_CONNECTED |- (!s. ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((real_lt (distance (x , y))) d))) ==> (?c. ((x IN c) /\\ ((y IN c) /\\ ((c SUBSET s) /\\ ((connected c) /\\ ((bounded c) /\\ ((real_le (diameter c)) e)))))))))))))) ==> ((locally connected) s)))"}
{"_id": "50033", "text": "MINIMAL_CONTINUUM |- (!t. (!s. (((t SUBSET s) /\\ ((compact s) /\\ (connected s))) ==> (?u. ((t SUBSET u) /\\ ((u SUBSET s) /\\ ((compact u) /\\ ((connected u) /\\ (!v. (((v SUBSET u) /\\ ((t SUBSET v) /\\ ((compact v) /\\ (connected v)))) ==> (v = u)))))))))))"}
{"_id": "50034", "text": "INSIDE_INSIDE_SUBSET |- (!s. ((inside (inside s)) SUBSET s))"}
{"_id": "50035", "text": "REAL_MEASURABLE_ON_RPOW |- (!f. (!r. (!s. ((((real_measurable_on f) s) /\\ ((real_lt (real_of_num (NUMERAL _0))) r)) ==> ((real_measurable_on (\\x. ((rpow (f x)) r))) s)))))"}
{"_id": "50036", "text": "CONNECTED_COMPONENT_EMPTY |- (!x. (((connected_component EMPTY) x) = EMPTY))"}
{"_id": "50037", "text": "Lp_ineqs_quads.ineq120 |- (!V. (!E. ((lp_fan (V , E)) ==> (quad_nonlinear_v10 ==> ((!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)))))))))) ==> ((!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)))))) ==> ((lp_tau (V , E)) ==> (!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_ge ((real_add ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))) ((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))))))))))))))))) ((azim_dart (V , E)) d))) ((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))))))))))))))) (real_of_num (NUMERAL _0)))))))))))))"}
{"_id": "50038", "text": "POLYTOPE_AFF_DIM_1 |- (!p. (((polytope p) /\\ ((aff_dim p) = (int_of_num (NUMERAL (BIT1 _0))))) = (?a. (?b. ((~ (a = b)) /\\ (p = (closed_segment ((CONS (a , b)) NIL))))))))"}
{"_id": "50039", "text": "EXISTS_VECTOR_3 |- ((?v. (P v)) = (?x. (?y. (?z. (P (vector ((CONS x) ((CONS y) ((CONS z) NIL)))))))))"}
{"_id": "50040", "text": "REAL_LE_TRANS_LE |- (!x. (!y. (((real_le x) y) = (!z. (((real_le y) z) ==> ((real_le x) z))))))"}
{"_id": "50041", "text": "CONIC_HULL_EQ |- (!s. ((((hull conic) s) = s) = (conic s)))"}
{"_id": "50042", "text": "AFF_GE_1_1 |- (!x. (!v. (!w. (((DISJOINT (x INSERT EMPTY)) (v INSERT EMPTY)) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) = (GSPEC (\\GEN%PVAR%5098. (?y. (((SETSPEC GEN%PVAR%5098) (?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": "50043", "text": "IN_RELATIVE_INTERIOR_IN_OPEN_SEGMENT_STRONG |- (!s. (!x. (!a. (((convex s) /\\ ((x IN (relative_interior s)) /\\ ((a IN ((hull affine) s)) /\\ (~ (x = a))))) ==> (?b. ((b IN (relative_interior s)) /\\ (x IN (open_segment (a , b)))))))))"}
{"_id": "50044", "text": "HOLOMORPHIC_ON_LMUL |- (!f. (!c. (!s. (((holomorphic_on f) s) ==> ((holomorphic_on (\\x. ((complex_mul c) (f x)))) s)))))"}
{"_id": "50045", "text": "SUM_BIJECTION |- (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\\ ((p x) = y)))))) ==> (((sum s) f) = ((sum s) (f o p)))))))"}
{"_id": "50046", "text": "RETRACT_FIXPOINT_PROPERTY |- (!s. (!t. ((((retract_of t) s) /\\ (!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": "50047", "text": "DOUBLE_LEBESGUE_MEASURABLE |- (!f. ((!t. ((lebesgue_measurable t) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3690. (?x. (((SETSPEC GEN%PVAR%3690) ((f x) IN t)) x))))))) = (((measurable_on f) UNIV) /\\ (!t. ((negligible t) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3691. (?x. (((SETSPEC GEN%PVAR%3691) ((f x) IN t)) x))))))))))"}
{"_id": "50048", "text": "FRONTIER_SURJECTIVE_LINEAR_IMAGE |- (!f. (((linear f) /\\ (!y. (?x. ((f x) = y)))) ==> ((frontier ((IMAGE f) s)) = ((IMAGE f) (frontier s)))))"}
{"_id": "50049", "text": "Lp_ineqs_proofs.yn_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (yn_fan v)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))"}
{"_id": "50050", "text": "OPEN_EXISTS |- (!Q. ((!a. (open (GSPEC (\\GEN%PVAR%544. (?x. (((SETSPEC GEN%PVAR%544) ((Q a) x)) x)))))) ==> (open (GSPEC (\\GEN%PVAR%545. (?x. (((SETSPEC GEN%PVAR%545) (?a. ((Q a) x))) x)))))))"}
{"_id": "50051", "text": "REAL_INTEGRAL_RESTRICT_INTER |- (!f. (!s. (!t. (((real_integral t) (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) = ((real_integral (s INTER t)) f)))))"}
{"_id": "50052", "text": "REAL_LEBESGUE_MEASURABLE_COUNTABLE_INTERS_EXPLICIT |- (!s. ((!n. (real_lebesgue_measurable (s n))) ==> (real_lebesgue_measurable (INTERS (GSPEC (\\GEN%PVAR%5042. (?n. (((SETSPEC GEN%PVAR%5042) (n IN UNIV)) (s n)))))))))"}
{"_id": "50053", "text": "Tactics.WELLDEFINED_FUNCTION_1P |- (!P. ((?f. (!x. ((P x) ==> ((f (s x)) = (t x))))) = (!x. (!x'. (((P x) /\\ ((P x') /\\ ((s x) = (s x')))) ==> ((t x) = (t x')))))))"}
{"_id": "50054", "text": "Merge_ineq.nonf_dih4_x_div_sqrtdelta_posbranch |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih4_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4))) ((((((delta_x4 x4) x2) x6) x1) x5) x3))) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x4) ((((((delta_x x1) x2) x3) x4) x5) x6)))) ((real_pow ((((((delta_x4 x4) x2) x6) x1) x5) x3)) (NUMERAL (BIT0 (BIT1 _0))))))))))))))"}
{"_id": "50055", "text": "REALLIM_POSINFINITY_SEQUENTIALLY |- (!f. (!l. ((((---> f) l) at_posinfinity) ==> (((---> (\\n. (f (real_of_num n)))) l) sequentially))))"}
{"_id": "50056", "text": "COMPLEX_DERIVATIVE_MUL_AT |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\\ ((analytic_on g) (z INSERT EMPTY))) ==> (((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": "50057", "text": "REAL_EXP_NEG_MUL |- (!x. (((real_mul (exp x)) (exp (real_neg x))) = (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "50058", "text": "VECTOR_DERIVATIVE_CONST_AT |- (!c. (!a. (((vector_derivative (\\x. c)) (at a)) = (vec (NUMERAL _0)))))"}
{"_id": "50059", "text": "HAS_BOUNDED_VARIATION_DARBOUX_STRONG |- (!f. (!a. (!b. (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (?h. ((!x. ((f x) = ((vector_sub (g x)) (h 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 (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_le (drop x)) (drop y)))) ==> ((real_le (drop (h x))) (drop (h 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 (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. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((continuous f) ((within (at x)) (closed_interval ((CONS (a , x)) NIL))))) ==> (((continuous g) ((within (at x)) (closed_interval ((CONS (a , x)) NIL)))) /\\ ((continuous h) ((within (at x)) (closed_interval ((CONS (a , x)) NIL))))))) /\\ ((!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((continuous f) ((within (at x)) (closed_interval ((CONS (x , b)) NIL))))) ==> (((continuous g) ((within (at x)) (closed_interval ((CONS (x , b)) NIL)))) /\\ ((continuous h) ((within (at x)) (closed_interval ((CONS (x , b)) NIL))))))) /\\ (!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((continuous f) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))) ==> (((continuous g) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) /\\ ((continuous h) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))))))))))))))"}
{"_id": "50060", "text": "Lp_ineqs_proofs.rho_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (rho_fan v)) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_div sol0) pi)))))))))))"}
{"_id": "50061", "text": "SUMS_OFFSET_REV |- (!f. (!l. (!m. (!n. (((((sums f) l) (from m)) /\\ (((NUMERAL _0) < m) /\\ (n <= m))) ==> (((sums f) ((vector_add l) ((vsum (n .. (m - (NUMERAL (BIT1 _0))))) f))) (from n)))))))"}
{"_id": "50062", "text": "Ssrnat.maxKn |- (!m. (!n. (((minn n) ((maxn m) n)) = n)))"}
{"_id": "50063", "text": "REAL_INFSUM_RMUL |- (!s. (!x. (!c. (((real_summable s) x) ==> (((real_infsum s) (\\n. ((real_mul (x n)) c))) = ((real_mul ((real_infsum s) x)) c))))))"}
{"_id": "50064", "text": "Tactics_jordan.REAL_DOUBLE |- (!x. (((real_add x) x) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)))"}
{"_id": "50065", "text": "Tame_list.HEIGHTS |- ((heights ((((Graph fs) n) f) h)) = h)"}
{"_id": "50066", "text": "Rnsyjxm.f_fan_pair_ext_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((f_fan_pair_ext (V , E)) d) IN (dart1_of_fan (V , E))))))))"}
{"_id": "50067", "text": "Gbycpxs.SIGMA_SY_LE1 |- (!s. ((real_le (sigma_sy s)) (real_of_num (NUMERAL (BIT1 _0)))))"}
{"_id": "50068", "text": "CONNECTED_LIMIT_POINTS |- (!s. ((connected s) ==> (connected (GSPEC (\\GEN%PVAR%585. (?x. (((SETSPEC GEN%PVAR%585) ((limit_point_of x) s)) x)))))))"}
{"_id": "50069", "text": "HOLOMORPHIC_ON_CCOS |- (!s. ((holomorphic_on ccos) s))"}
{"_id": "50070", "text": "SAME_EIGENVECTORS_MATRIX_INV |- (!A. (!c. (!v. (((transp A) = A) ==> ((((matrix_vector_mul (matrix_inv A)) v) = ((% c) v)) = (((matrix_vector_mul A) v) = ((% (real_inv c)) v)))))))"}
{"_id": "50071", "text": "REAL_VARIATION_REFLECT2 |- (!f. (!s. (((real_variation ((IMAGE real_neg) s)) (\\x. (f (real_neg x)))) = ((real_variation s) f))))"}
{"_id": "50072", "text": "CONTINUOUS_MAP_CLOSURES |- (!f. (((continuous_on f) UNIV) = (!s. (((IMAGE f) (closure s)) SUBSET (closure ((IMAGE f) s))))))"}
{"_id": "50073", "text": "Functional_equation.gchi4_x' |- (!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)))))) ==> (((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((((((gchi4_x' x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "50074", "text": "OPEN_MAP_IFF_LOWER_HEMICONTINUOUS_PREIMAGE |- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) (GSPEC (\\GEN%PVAR%885. (?y. (((SETSPEC GEN%PVAR%885) ((y IN t) /\\ ((GSPEC (\\GEN%PVAR%884. (?x. (((SETSPEC GEN%PVAR%884) ((x IN s) /\\ ((f x) = y))) x)))) SUBSET u))) y))))))))))))"}
{"_id": "50075", "text": "Localization.HAFL_CIRCLE_FORM_LOCAL_FAN_ALT |- (!V. (!E. (!FF. (!v. (!w. (!HS. (!fv. ((((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": "50076", "text": "Planarity.RWXUYZZ |- (!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)))))) /\\ (((((dartset_leads_into_fan x) V) E) ds) IN (topological_component_yfan (x , (V , E))))))))))"}
{"_id": "50077", "text": "IN_CBALL_RE |- (!x. (!z. (!e. ((x IN (cball (z , e))) ==> ((real_le (real_abs ((real_sub (Re x)) (Re z)))) e)))))"}
{"_id": "50078", "text": "Lp_ineqs_proofs.f_fan_pair_ext_power_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) IN (dart1_of_fan (V , E)))))))))"}
{"_id": "50079", "text": "Ssrnat.distn_add2l |- (!d. (!m. (!n. (((distn (d + m)) (d + n)) = ((distn m) n)))))"}
{"_id": "50080", "text": "Ssrbool.negbNE |- (!b. ((~ (~ b)) ==> b))"}
{"_id": "50081", "text": "LEBESGUE_DENSITY_THEOREM |- (!s. ((lebesgue_measurable s) ==> (?k. ((negligible k) /\\ (!x. ((~ (x IN k)) ==> (((---> (\\e. ((real_div (measure (s INTER (cball (x , e))))) (measure (cball (x , e)))))) (((COND (x IN s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) ((within (atreal (real_of_num (NUMERAL _0)))) (GSPEC (\\GEN%PVAR%5068. (?e. (((SETSPEC GEN%PVAR%5068) ((real_lt (real_of_num (NUMERAL _0))) e)) e))))))))))))"}
{"_id": "50082", "text": "INT_LT_SUB_LADD |- (!x. (!y. (!z. (((int_lt x) ((int_sub y) z)) = ((int_lt ((int_add x) z)) y)))))"}
{"_id": "50083", "text": "REAL_CONTINUOUS_COMPLEX_COMPONENTS_WITHIN |- (!s. (!z. (((real_continuous Re) ((within (at z)) s)) /\\ ((real_continuous Im) ((within (at z)) s)))))"}
{"_id": "50084", "text": "INTEGRAL_COMPONENT_LE_AE |- (!f. (!g. (!s. (!k. (!t. ((((NUMERAL (BIT1 _0)) <= k) /\\ ((k <= (dimindex UNIV)) /\\ ((negligible t) /\\ (((integrable_on f) s) /\\ (((integrable_on g) s) /\\ (!x. ((x IN (s DIFF t)) ==> ((real_le ((f x) $ k)) ((g x) $ k))))))))) ==> ((real_le (((integral s) f) $ k)) (((integral s) g) $ k))))))))"}
{"_id": "50085", "text": "REAL_INTEGRABLE_COMBINE |- (!f. (!a. (!b. (!c. ((((real_le a) c) /\\ (((real_le c) b) /\\ (((real_integrable_on f) (closed_real_interval ((CONS (a , c)) NIL))) /\\ ((real_integrable_on f) (closed_real_interval ((CONS (c , b)) NIL)))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "50086", "text": "Lp_ineqs_proofs.dart3_sol_fan_eq_sol |- (!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)))) ==> (((sol_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)) = ((((((sol_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": "50087", "text": "AFF_DIM_BALL |- (!a. (!r. ((aff_dim (ball (a , r))) = (((COND ((real_lt (real_of_num (NUMERAL _0))) r)) (int_of_num (dimindex UNIV))) (int_neg (int_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "50088", "text": "INJECTIVE_MAP_OPEN_IFF_CLOSED |- (!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)))))) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!u. (((closed_in ((subtopology euclidean) s)) u) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u)))))))))"}
{"_id": "50089", "text": "CHOICE_UNPAIR_THM |- (!P. ((@ (GABS (\\f. (!x. (!y. ((GEQ (f (x , y))) ((P x) y))))))) = (@p. ((P (FST p)) (SND p)))))"}
{"_id": "50090", "text": "SERIES_DIRICHLET_COMPLEX |- (!f. (!g. (!N. (!k. (!m. (((bounded (GSPEC (\\GEN%PVAR%4782. (?n. (((SETSPEC GEN%PVAR%4782) (n IN UNIV)) ((vsum (m .. n)) f)))))) /\\ ((!n. (real (g n))) /\\ ((!n. ((N <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n))))) /\\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) sequentially)))) ==> ((summable (from k)) (\\n. ((complex_mul (f n)) (g n))))))))))"}
{"_id": "50091", "text": "CONVEX_AFF_LE |- (!s. (!t. (convex ((aff_le s) t))))"}
{"_id": "50092", "text": "IVT_DECREASING_IM |- (!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 b)))) y) /\\ ((real_le y) (Im (f (Cx a))))))) ==> (?x. (((real_le a) x) /\\ (((real_le x) b) /\\ ((Im (f (Cx x))) = y)))))))))"}
{"_id": "50093", "text": "REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_OPEN_INTERVAL |- (!f. (!g. (!a. (!b. ((((real_measurable_on f) UNIV) /\\ ((!x. ((f x) IN (open_real_interval (a , b)))) /\\ ((real_continuous_on g) (open_real_interval (a , b))))) ==> ((real_measurable_on (g o f)) UNIV))))))"}
{"_id": "50094", "text": "LOCALLY_CONVEX |- (!s. (((locally convex) s) = (!x. ((x IN s) ==> (?u. (?v. ((x IN u) /\\ ((u SUBSET v) /\\ ((v SUBSET s) /\\ (((open_in ((subtopology euclidean) s)) u) /\\ (convex v)))))))))))"}
{"_id": "50095", "text": "MEASURABLE_ON_MEASURABLE_PREIMAGE_HALFSPACE_COMPONENT_GT |- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!a. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV))) ==> (measurable (GSPEC (\\GEN%PVAR%3650. (?x. (((SETSPEC GEN%PVAR%3650) ((x IN s) /\\ ((real_gt ((f x) $ k)) a))) x))))))))))))"}
{"_id": "50096", "text": "ONORM_TRIANGLE_LT |- (!f. (!g. (((linear f) /\\ ((linear g) /\\ ((real_lt ((real_add (onorm f)) (onorm g))) e))) ==> ((real_lt (onorm (\\x. ((vector_add (f x)) (g x))))) e))))"}
{"_id": "50097", "text": "NSUM_DEGENERATE |- (!f. (!s. ((~ (FINITE (GSPEC (\\GEN%PVAR%171. (?x. (((SETSPEC GEN%PVAR%171) ((x IN s) /\\ (~ ((f x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) f) = (NUMERAL _0)))))"}
{"_id": "50098", "text": "QUOTIENT_MAP_IMP_CONTINUOUS_CLOSED |- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\\ (!u. ((u SUBSET t) ==> (((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%645. (?x. (((SETSPEC GEN%PVAR%645) ((x IN s) /\\ ((f x) IN u))) x))))) = ((closed_in ((subtopology euclidean) t)) u))))) ==> ((continuous_on f) s)))))"}
{"_id": "50099", "text": "LEBESGUE_MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ |- (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> ((lebesgue_measurable s) = (lebesgue_measurable t)))))"}
{"_id": "50100", "text": "INT_LE_NEGL |- (!x. (((int_le (int_neg x)) x) = ((int_le (int_of_num (NUMERAL _0))) x)))"}
{"_id": "50101", "text": "EXISTS_SUBARC_OF_ARC_NOENDS |- (!g. (!a. (!b. (((arc g) /\\ ((a IN (path_image g)) /\\ ((b IN (path_image g)) /\\ ((~ (a = b)) /\\ (((a INSERT (b INSERT EMPTY)) INTER ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))) = EMPTY))))) ==> (?h. ((arc h) /\\ (((pathstart h) = a) /\\ (((pathfinish h) = b) /\\ ((path_image h) SUBSET ((path_image g) DIFF ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))))))))))))"}
{"_id": "50102", "text": "Tame_list.AQIUNPP2 |- (!L. (!N. (!r. (!x. (((transform_assumption_v2 L) /\\ ((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": "50103", "text": "URYSOHN |- (!s. (!t. (!a. (!b. (((closed s) /\\ ((closed t) /\\ ((s INTER t) = EMPTY))) ==> (?f. (((continuous_on f) UNIV) /\\ ((!x. ((f x) IN (closed_segment ((CONS (a , b)) NIL)))) /\\ ((!x. ((x IN s) ==> ((f x) = a))) /\\ (!x. ((x IN t) ==> ((f x) = b))))))))))))"}
{"_id": "50104", "text": "CARD_MUL_ASSOC |- (!s. (!t. (!u. (((*_c s) ((*_c t) u)) =_c ((*_c ((*_c s) t)) u)))))"}
{"_id": "50105", "text": "LINEAR_SUBSPACE_GRAPH |- (!f. ((linear f) = (subspace (GSPEC (\\GEN%PVAR%153. (?x. (((SETSPEC GEN%PVAR%153) (x IN UNIV)) ((pastecart x) (f x)))))))))"}
{"_id": "50106", "text": "Lp_ineqs_proofs.rhazim2_hi |- (!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 ((rhazim2_fan (V , E)) d)) ((real_add pi) sol0))))))))))"}
{"_id": "50107", "text": "MEASURABLE_ON_TRANSLATION_EQ |- (!f. (!s. (!a. (((measurable_on (\\x. (f ((vector_add a) x)))) s) = ((measurable_on f) ((IMAGE (\\x. ((vector_add a) x))) s))))))"}
{"_id": "50108", "text": "Lp_gen_theory.sum_face_iso |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!r. (!f. ((f IN (face_set G)) ==> (((sum ((IMAGE g) f)) r) = ((sum f) (r o g))))))))))"}
{"_id": "50109", "text": "Lp_ineqs_proofs.INJ_pair |- (!h1. (!h2. (!g. ((!x. ((g x) = ((h1 (FST x)) , (h2 (SND x))))) ==> (!s1. (!t1. (!s2. (!t2. (((((INJ h1) s1) t1) /\\ (((INJ h2) s2) t2)) ==> (((INJ g) (s1 CROSS s2)) (t1 CROSS t2)))))))))))"}
{"_id": "50110", "text": "IN_COMPONENTS_UNIONS_COMPLEMENT |- (!s. (!c. ((c IN (components s)) ==> ((s DIFF c) = (UNIONS ((components s) DELETE c))))))"}
{"_id": "50111", "text": "Collect_geom2.EQ_IMP_COPLANAR |- (!a. (!b. (!c. (!d. (((a = b) \\/ ((a = c) \\/ (a = d))) ==> (coplanar_alt (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))))))))"}
{"_id": "50112", "text": "HOMOTOPIC_INTO_RETRACT |- (!f. (!g. (!s. (!t. (!u. (((((IMAGE f) s) SUBSET t) /\\ ((((IMAGE g) s) SUBSET t) /\\ (((retract_of t) u) /\\ ((((homotopic_with (\\x. T)) (s , u)) f) g)))) ==> ((((homotopic_with (\\x. T)) (s , t)) f) g)))))))"}
{"_id": "50113", "text": "HAS_BOUNDED_VARIATION_COMPOSE_INJECTIVE |- (!f. (!g. (!s. (((is_interval s) /\\ (((continuous_on f) s) /\\ ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) /\\ ((has_bounded_variation_on g) ((IMAGE f) s))))) ==> ((has_bounded_variation_on (g o f)) s)))))"}
{"_id": "50114", "text": "SUMS_INTERVALS |- ((!a. (!b. (!c. (!d. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (~ ((closed_interval ((CONS (c , d)) NIL)) = EMPTY))) ==> ((GSPEC (\\GEN%PVAR%972. (?x. (?y. (((SETSPEC GEN%PVAR%972) ((x IN (closed_interval ((CONS (a , b)) NIL))) /\\ (y IN (closed_interval ((CONS (c , d)) NIL))))) ((vector_add x) y)))))) = (closed_interval ((CONS (((vector_add a) c) , ((vector_add b) d))) NIL)))))))) /\\ (!a. (!b. (!c. (!d. (((~ ((open_interval (a , b)) = EMPTY)) /\\ (~ ((open_interval (c , d)) = EMPTY))) ==> ((GSPEC (\\GEN%PVAR%973. (?x. (?y. (((SETSPEC GEN%PVAR%973) ((x IN (open_interval (a , b))) /\\ (y IN (open_interval (c , d))))) ((vector_add x) y)))))) = (open_interval (((vector_add a) c) , ((vector_add b) d))))))))))"}
{"_id": "50115", "text": "Functional_equation.functional_acs_sqrt_x2_d4 |- (acs_sqrt_x2_d4 = (uni (acs , ((scalar6 proj_y2) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))"}
{"_id": "50116", "text": "LOCALLY_COMPACT_INTER_CBALLS |- (!s. (((locally compact) s) = (!x. ((x IN s) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (!d. (((real_le d) e) ==> (closed ((cball (x , d)) INTER s))))))))))"}
{"_id": "50117", "text": "REAL_SERIES_DIFFS |- (!f. (!k. ((((---> f) (real_of_num (NUMERAL _0))) sequentially) ==> (((real_sums (\\n. ((real_sub (f n)) (f (n + (NUMERAL (BIT1 _0))))))) (f k)) (from k)))))"}
{"_id": "50118", "text": "REAL_POWER_SERIES_CONV_IMP_ABSCONV |- (!a. (!k. (!w. (!z. ((((real_summable k) (\\n. ((real_mul (a n)) ((real_pow z) n)))) /\\ ((real_lt (real_abs w)) (real_abs z))) ==> ((real_summable k) (\\n. (real_abs ((real_mul (a n)) ((real_pow w) n))))))))))"}
{"_id": "50119", "text": "Tame_list.nth_EL |- (!x. (!xs. (!n. (!k. (((n = (NUMERAL _0)) ==> ((((\\l. (\\n. ((EL n) l))) ((CONS x) xs)) n) = x)) /\\ ((n = (SUC k)) ==> ((((\\l. (\\n. ((EL n) l))) ((CONS x) xs)) n) = (((\\l. (\\n. ((EL n) l))) xs) k))))))))"}
{"_id": "50120", "text": "NPRODUCT_ONE |- (!s. (((nproduct s) (\\n. (NUMERAL (BIT1 _0)))) = (NUMERAL (BIT1 _0))))"}
{"_id": "50121", "text": "SUP_INSERT |- (!x. (!s. ((bounded ((IMAGE lift) s)) ==> ((sup (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_max x) (sup s)))))))"}
{"_id": "50122", "text": "RELPOW_M |- (!m. (!n. (!x. (!y. (((((RELPOW (m + n)) R) x) y) = (?z. (((((RELPOW m) R) x) z) /\\ ((((RELPOW n) R) z) y))))))))"}
{"_id": "50123", "text": "REAL_CONTINUOUS_ON_CASES |- (!P. (!f. (!g. (!s. (!t. (((real_closed s) /\\ ((real_closed t) /\\ (((real_continuous_on f) s) /\\ (((real_continuous_on g) t) /\\ (!x. ((((x IN s) /\\ (~ (P x))) \\/ ((x IN t) /\\ (P x))) ==> ((f x) = (g x)))))))) ==> ((real_continuous_on (\\x. (((COND (P x)) (f x)) (g x)))) (s UNION t))))))))"}
{"_id": "50124", "text": "List_hypermap_iso.fan_list_nodes_iso_inj |- (!L. (!f. (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso f) ((hypermap_of_fan (V , E)) , (hypermap_of_list L))) ==> ((good_list_nodes L) ==> (!x. (!y. (((x IN V) /\\ ((y IN V) /\\ (((fan_list_nodes_iso (f , (V , E))) x) = ((fan_list_nodes_iso (f , (V , E))) y)))) ==> (x = y))))))))))))"}
{"_id": "50125", "text": "List_hypermap_iso.fan_list_nodes_iso_surj |- (!f. (!E. (!V. (!L. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso f) ((hypermap_of_fan (V , E)) , (hypermap_of_list L))) ==> (((SURJ (fan_list_nodes_iso (f , (V , E)))) V) (elements_of_list L)))))))))"}
{"_id": "50126", "text": "REAL_INTEGRABLE_ON_REFL |- (!f. (!a. ((real_integrable_on f) (closed_real_interval ((CONS (a , a)) NIL)))))"}
{"_id": "50127", "text": "REAL_CONTINUOUS_WITHIN_SQRT |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_continuous sqrt) ((within (atreal x)) s)))))"}
{"_id": "50128", "text": "LIM_LOG_OVER_Z |- (((--> (\\z. ((complex_div (clog z)) z))) (Cx (real_of_num (NUMERAL _0)))) at_infinity)"}
{"_id": "50129", "text": "CONIC_PCROSS_EQ |- (!s. (!t. ((conic (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((conic s) /\\ (conic t)))))))"}
{"_id": "50130", "text": "RSTC_TRANS_R |- (!R. (!x. (!y. (!z. ((((R x) y) /\\ (((RSTC R) y) z)) ==> (((RSTC R) x) z))))))"}
{"_id": "50131", "text": "Fatugpd.dot_normalize |- (!v. (((dot v) (normalize v)) = (vector_norm v)))"}
{"_id": "50132", "text": "REAL_CONVEX_ON_LEFT_SECANT |- (!f. (!s. (((real_convex_on f) s) = (!a. (!b. (!x. (((a IN s) /\\ ((b IN s) /\\ (x IN (open_real_segment (a , b))))) ==> ((real_le ((real_div ((real_sub (f x)) (f a))) (real_abs ((real_sub x) a)))) ((real_div ((real_sub (f b)) (f a))) (real_abs ((real_sub b) a)))))))))))"}
{"_id": "50133", "text": "ANALYTIC_AT_ADD |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\\ ((analytic_on g) (z INSERT EMPTY))) ==> ((analytic_on (\\w. ((vector_add (f w)) (g w)))) (z INSERT EMPTY))))))"}
{"_id": "50134", "text": "Lp_gen_theory.hyp_iso_face_set_k |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!k. (((face_set_k k) H) = ((IMAGE (IMAGE g)) ((face_set_k k) G))))))))"}
{"_id": "50135", "text": "INT_SGN_NEG |- (!x. ((int_sgn (int_neg x)) = (int_neg (int_sgn x))))"}
{"_id": "50136", "text": "COMPLEX_EQ_NEG2 |- (!x. (!y. (((vector_neg x) = (vector_neg y)) = (x = y))))"}
{"_id": "50137", "text": "INT_SUB_TRIANGLE |- (!a. (!b. (!c. (((int_add ((int_sub a) b)) ((int_sub b) c)) = ((int_sub a) c)))))"}
{"_id": "50138", "text": "FRONTIER_UNIONS_SUBSET |- (!f. ((FINITE f) ==> ((frontier (UNIONS f)) SUBSET (UNIONS (GSPEC (\\GEN%PVAR%1826. (?t. (((SETSPEC GEN%PVAR%1826) (t IN f)) (frontier t)))))))))"}
{"_id": "50139", "text": "REAL_LOG_CONVEX_IMP_CONVEX |- (!f. (!s. (((real_log_convex_on f) s) ==> ((real_convex_on f) s))))"}
{"_id": "50140", "text": "Lp_main_estimate.ineq_tau3 |- (!V. (!E. ((lp_fan (V , E)) ==> ((!v1. (!v2. (!v3. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v1)) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v2)) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v3)) /\\ (((real_le (vector_norm v1)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\\ (((real_le (vector_norm v2)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\\ (((real_le (vector_norm v3)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v1 , v2))) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v1 , v3))) /\\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (v2 , v3))) /\\ (((real_le (distance (v1 , v2))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\\ (((real_le (distance (v1 , v3))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\\ ((real_le (distance (v2 , v3))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))))))))))) ==> ((real_le (real_of_num (NUMERAL _0))) (((tau3 v1) v2) v3)))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL _0))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))"}
{"_id": "50141", "text": "REAL_INTEGRAL_NEG |- (!f. (!s. (((real_integrable_on f) s) ==> (((real_integral s) (\\x. (real_neg (f x)))) = (real_neg ((real_integral s) f))))))"}
{"_id": "50142", "text": "Lp_ineqs_proofs.RHB |- (!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 ((rhazim_fan (V , E)) d)) ((real_mul ((azim_dart (V , E)) d)) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_div sol0) pi))))))))))))"}
{"_id": "50143", "text": "PATH_LENGTH_EQ_LINE_SEGMENT |- (!g. (((rectifiable_path g) /\\ ((path_length g) = (distance ((pathstart g) , (pathfinish g))))) ==> ((path_image g) = (closed_segment ((CONS ((pathstart g) , (pathfinish g))) NIL)))))"}
{"_id": "50144", "text": "VSUM_CMUL_NUMSEG |- (!f. (!c. (!m. (!n. (((vsum (m .. n)) (\\x. ((% c) (f x)))) = ((% c) ((vsum (m .. n)) f)))))))"}
{"_id": "50145", "text": "CONNECTED_OPEN_VECTOR_POLYNOMIAL_CONNECTED |- (!s. (((open s) /\\ (connected s)) ==> (!x. (!y. (((x IN s) /\\ (y IN s)) ==> (?g. ((vector_polynomial_function g) /\\ (((path_image g) SUBSET s) /\\ (((pathstart g) = x) /\\ ((pathfinish g) = y))))))))))"}
{"_id": "50146", "text": "INT_MAX_ACI |- ((((int_max x) y) = ((int_max y) x)) /\\ ((((int_max ((int_max x) y)) z) = ((int_max x) ((int_max y) z))) /\\ ((((int_max x) ((int_max y) z)) = ((int_max y) ((int_max x) z))) /\\ ((((int_max x) x) = x) /\\ (((int_max x) ((int_max x) y)) = ((int_max x) y))))))"}
{"_id": "50147", "text": "Functional_equation.functional_proj_y4 |- (proj_y4 = (rotate4 proj_y1))"}
{"_id": "50148", "text": "COMPACT_IN_IFF_COMPACT |- (!s. (((compact_in s) euclidean) = (compact s)))"}
{"_id": "50149", "text": "INFSUM_NEG |- (!s. (!x. (((summable s) x) ==> (((infsum s) (\\n. (vector_neg (x n)))) = (vector_neg ((infsum s) x))))))"}
{"_id": "50150", "text": "KERNEL_MATRIX_INV |- (!A. ((GSPEC (\\GEN%PVAR%227. (?x. (((SETSPEC GEN%PVAR%227) (((matrix_vector_mul (matrix_inv A)) x) = (vec (NUMERAL _0)))) x)))) = (GSPEC (\\GEN%PVAR%228. (?x. (((SETSPEC GEN%PVAR%228) (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))) x))))))"}
{"_id": "50151", "text": "Ssrbool.orbK |- (!a. (!b. (((a \\/ b) /\\ a) = a)))"}
{"_id": "50152", "text": "GATEAUX_DERIVATIVE |- (!f. (!f'. (!x. (!y. ((((has_derivative f) f') (at x)) ==> (((--> (\\t. ((% (real_inv (drop t))) ((vector_sub (f ((vector_add x) ((% (drop t)) y)))) (f x))))) (f' y)) (at (vec (NUMERAL _0)))))))))"}
{"_id": "50153", "text": "CONNECTED_PUNCTURED_CONVEX |- (!s. (!a. (((convex s) /\\ (~ ((aff_dim s) = (int_of_num (NUMERAL (BIT1 _0)))))) ==> (connected (s DELETE a)))))"}
{"_id": "50154", "text": "Hypermap_iso.surj_ext |- (!f. (!g. (!s. (!t. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> ((((SURJ f) s) t) ==> (((SURJ g) s) t)))))))"}
{"_id": "50155", "text": "SYMMETRIX_MATRIX_CONJUGATE |- (!A. (!B. (((transp B) = B) ==> ((transp ((matrix_mul (transp A)) ((matrix_mul B) A))) = ((matrix_mul (transp A)) ((matrix_mul B) A))))))"}
{"_id": "50156", "text": "NSUM_BIJECTION |- (!f. (!p. (!s. (((!x. ((x IN s) ==> ((p x) IN s))) /\\ (!y. ((y IN s) ==> (?!x. ((x IN s) /\\ ((p x) = y)))))) ==> (((nsum s) f) = ((nsum s) (f o p)))))))"}
{"_id": "50157", "text": "CONTINUOUS_ON_CSQRT |- (!s. ((!z. (((z IN s) /\\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z)))) ==> ((continuous_on csqrt) s)))"}
{"_id": "50158", "text": "Tame_list.sub_nth |- (((sub a) n) = ((#!# a) n))"}
{"_id": "50159", "text": "Appendix.scs_unadorned |- (!s. ((is_scs_v39 s) ==> (is_scs_v39 (scs_v39 ((scs_k_v39 s) , ((scs_d_v39 s) , ((scs_a_v39 s) , ((scs_a_v39 s) , ((scs_b_v39 s) , ((scs_b_v39 s) , ((scs_J_v39 s) , (EMPTY , (EMPTY , EMPTY)))))))))))))"}
{"_id": "50160", "text": "EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL |- (!s. (((FINITE s) /\\ (((hull affine) s) = UNIV)) ==> ((GSPEC (\\GEN%PVAR%1794. (?y. (((SETSPEC GEN%PVAR%1794) (?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 (interior ((hull convex) s)))))"}
{"_id": "50161", "text": "CONTINUOUS_INJECTIVE_IMAGE_OPEN_SEGMENT_1 |- (!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) (open_segment (a , b))) = (open_segment ((f a) , (f b))))))))"}
{"_id": "50162", "text": "HAS_MEASURE_ALMOST_EQ |- (!s. (!s'. (!t. (((negligible t) /\\ ((s UNION t) = (s' UNION t))) ==> (((has_measure s) m) = ((has_measure s') m))))))"}
{"_id": "50163", "text": "OPEN_IN_SEGMENT |- (!s. (!a. (!b. ((((open_segment (a , b)) SUBSET s) /\\ (s SUBSET ((hull affine) (open_segment (a , b))))) ==> ((open_in ((subtopology euclidean) s)) (open_segment (a , b)))))))"}
{"_id": "50164", "text": "Functional_equation.functional_vol3f_x_lfun |- (((domain6 (\\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) ((sub6 ((mul6 (constant6 ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi)))) ((add6 ((mul6 two6) (mk_126 dih_x))) ((add6 ((mul6 two6) (mk_126 dih2_x))) ((add6 ((mul6 two6) (mk_126 dih6_x))) ((add6 (mk_126 dih3_x)) ((add6 (mk_126 dih4_x)) ((sub6 (mk_126 dih5_x)) (constant6 ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi)))))))))) ((mul6 (constant6 ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi)))) ((add6 (mk_126 ldih_x)) ((add6 (mk_126 ldih2_x)) (mk_126 ldih6_x))))))"}
{"_id": "50165", "text": "BOUNDED_PARTIAL_REAL_SUMS |- (!f. (!k. ((real_bounded (GSPEC (\\GEN%PVAR%4868. (?n. (((SETSPEC GEN%PVAR%4868) (n IN UNIV)) ((sum (k .. n)) f)))))) ==> (real_bounded (GSPEC (\\GEN%PVAR%4869. (?m. (?n. (((SETSPEC GEN%PVAR%4869) ((m IN UNIV) /\\ (n IN UNIV))) ((sum (m .. n)) f))))))))))"}
{"_id": "50166", "text": "TRANS_ALT |- (!R. (!S. (!U. ((!x. (!z. ((?y. (((R x) y) /\\ ((S y) z))) ==> ((U x) z)))) = (!x. (!y. (!z. ((((R x) y) /\\ ((S y) z)) ==> ((U x) z)))))))))"}
{"_id": "50167", "text": "Ssrnat.maxn_mull |- (!m1. (!m2. (!n. ((((maxn m1) m2) * n) = ((maxn (m1 * n)) (m2 * n))))))"}
{"_id": "50168", "text": "ISO_FUN |- ((((ISO f) f') /\\ ((ISO g) g')) ==> ((ISO (\\h. (\\a'. (g (h (f' a')))))) (\\h. (\\a. (g' (h (f a)))))))"}
{"_id": "50169", "text": "FRECHET_DERIVATIVE_CONST_AT |- (!c. (!a. (((frechet_derivative (\\x. c)) (at a)) = (\\h. (vec (NUMERAL _0))))))"}
{"_id": "50170", "text": "COMPLEX_SUB_POW_L1 |- (!x. (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow x) n)) = ((complex_mul ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) x)) ((vsum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\\i. ((complex_pow x) i))))))))"}
{"_id": "50171", "text": "Trigonometry.LLOYXRK2 |- (!x. ((sin ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x)) = (cos x)))"}
{"_id": "50172", "text": "REAL_INTEGRABLE_REFLECT |- (!f. (!a. (!b. (((real_integrable_on (\\x. (f (real_neg x)))) (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) = ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "50173", "text": "POLYTOPE_PCROSS_EQ |- (!s. (!t. ((polytope (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((polytope s) /\\ (polytope t)))))))"}
{"_id": "50174", "text": "COLLINEAR_CONVEX_HULL_COLLINEAR |- (!s. ((collinear ((hull convex) s)) = (collinear s)))"}
{"_id": "50175", "text": "SETDIST_LE_HAUSDIST |- (!s. (!t. (((bounded s) /\\ (bounded t)) ==> ((real_le (setdist (s , t))) (hausdist (s , t))))))"}
{"_id": "50176", "text": "REAL_DIFFERENTIABLE_ON_COMPOSE |- (!f. (!g. (!s. ((((real_differentiable_on f) s) /\\ ((real_differentiable_on g) ((IMAGE f) s))) ==> ((real_differentiable_on (g o f)) s)))))"}
{"_id": "50177", "text": "Sphere.vol_solid_triangle |- (!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": "50178", "text": "REALLIM_AT_POSINFINITY |- (!f. (!l. ((((---> f) l) at_posinfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge x) b) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))"}
{"_id": "50179", "text": "FRAC_NEG |- (!x. ((frac (real_neg x)) = (((COND (integer x)) (real_of_num (NUMERAL _0))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (frac x)))))"}
{"_id": "50180", "text": "Lp_ineqs_proofs.ln_hi |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (ln_fan v)) (real_of_num (NUMERAL (BIT1 _0))))))))))))"}
{"_id": "50181", "text": "REAL_INTEGRABLE_AFFINITY |- (!f. (!a. (!b. (!m. (!c. ((((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\\ (~ (m = (real_of_num (NUMERAL _0))))) ==> ((real_integrable_on (\\x. (f ((real_add ((real_mul m) x)) c)))) ((IMAGE (\\x. ((real_mul (real_inv m)) ((real_sub x) c)))) (closed_real_interval ((CONS (a , b)) NIL))))))))))"}
{"_id": "50182", "text": "Hypermap_iso.iso_number_of_edges |- (!f. (!G. (!H. (((hyp_iso f) (H , G)) ==> ((number_of_edges G) = (number_of_edges H))))))"}
{"_id": "50183", "text": "Ssrbool.andb_idl |- (!a. (!b. ((b ==> a) ==> ((a /\\ b) = b))))"}
{"_id": "50184", "text": "CONTINUOUS_TRANSFORM_WITHIN_OPEN |- (!f. (!g. (!s. (!a. (((open s) /\\ ((a IN s) /\\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((continuous f) (at a))))) ==> ((continuous g) (at a)))))))"}
{"_id": "50185", "text": "EXISTS_SUBPATH_OF_ARC_NOENDS |- (!g. (!a. (!b. (((arc g) /\\ ((a IN (path_image g)) /\\ ((b IN (path_image g)) /\\ (((a INSERT (b INSERT EMPTY)) INTER ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))) = EMPTY)))) ==> (?h. ((path h) /\\ (((pathstart h) = a) /\\ (((pathfinish h) = b) /\\ ((path_image h) SUBSET ((path_image g) DIFF ((pathstart g) INSERT ((pathfinish g) INSERT EMPTY))))))))))))"}
{"_id": "50186", "text": "REAL_DIFFERENTIABLE_FROM_COMPLEX_AT |- (!f. (!x. ((((complex_differentiable f) (at (Cx x))) /\\ (!z. ((real z) ==> (real (f z))))) ==> ((real_differentiable (Re o (f o Cx))) (atreal x)))))"}
{"_id": "50187", "text": "Lp_ineqs_proofs.dart4_tau_fan_eq_tauq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((tau_fan (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": "50188", "text": "YOUNG_INEQUALITY |- (!a. (!b. (!p. (!q. ((((real_le (real_of_num (NUMERAL _0))) a) /\\ (((real_le (real_of_num (NUMERAL _0))) b) /\\ (((real_lt (real_of_num (NUMERAL _0))) p) /\\ (((real_lt (real_of_num (NUMERAL _0))) q) /\\ (((real_add (real_inv p)) (real_inv q)) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le ((real_mul a) b)) ((real_add ((real_div ((rpow a) p)) p)) ((real_div ((rpow b) q)) q))))))))"}
{"_id": "50189", "text": "FINITE_COLUMNS |- (!A. (FINITE (columns A)))"}
{"_id": "50190", "text": "Lp_ineqs_proofs.yy10_std |- (!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 (ye_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))"}
{"_id": "50191", "text": "Nonlinear_lemma.pathL_bound |- (!y. (!a. ((((real_le (FST (pathL a))) y) /\\ ((real_le y) (SND (pathL a)))) ==> (((real_le (FST a)) y) /\\ ((real_le y) (SND a))))))"}
{"_id": "50192", "text": "Tame_list.list_update_ALT |- (((((list_update NIL) i) v) = NIL) /\\ ((((list_update ((CONS x) xs)) i) v) = (((COND (i = (NUMERAL _0))) ((CONS v) xs)) ((LET (\\j. (LET_END ((CONS x) (((list_update xs) j) v))))) (PRE i)))))"}
{"_id": "50193", "text": "COMPLEX_DIFFERENTIABLE_POW_AT |- (!f. (!n. (!z. (((complex_differentiable f) (at z)) ==> ((complex_differentiable (\\z. ((complex_pow (f z)) n))) (at z))))))"}
{"_id": "50194", "text": "COS_EQ_MINUS1 |- (!x. (((cos x) = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) = (?n. ((integer n) /\\ (x = ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) (real_of_num (NUMERAL (BIT1 _0))))) pi))))))"}
{"_id": "50195", "text": "TAN_ABS_GE_X |- (!x. (((real_lt (real_abs x)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ==> ((real_le (real_abs x)) (real_abs (tan x)))))"}
{"_id": "50196", "text": "REAL_INFSUM_0 |- (((real_infsum s) (\\i. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0)))"}
{"_id": "50197", "text": "INT_POW_EQ_ABS |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\\ (((int_pow x) n) = ((int_pow y) n))) ==> ((int_abs x) = (int_abs y))))))"}
{"_id": "50198", "text": "INT_LE_SQUARE |- (!x. ((int_le (int_of_num (NUMERAL _0))) ((int_mul x) x)))"}
{"_id": "50199", "text": "OPEN_SET_IRRATIONAL_COORDINATES |- (!s. (((open s) /\\ (~ (s = EMPTY))) ==> (?x. ((x IN s) /\\ (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> (~ (rational (x $ i)))))))))"}
{"_id": "50200", "text": "MOD_EXISTS |- (!m. (!n. ((?q. (m = (n * q))) = (((COND (n = (NUMERAL _0))) (m = (NUMERAL _0))) ((m MOD n) = (NUMERAL _0))))))"}
{"_id": "50201", "text": "COBOUNDED_HAS_BOUNDED_COMPONENT |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ ((bounded (UNIV DIFF s)) /\\ (~ (connected s)))) ==> (?c. ((c IN (components s)) /\\ (bounded c)))))"}
{"_id": "50202", "text": "Nonlinear_lemma.delta_x_eq0 |- ((((((((delta_x (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0))) /\\ (((((((delta_x (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0))))"}
{"_id": "50203", "text": "REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET_0 |- (!f. (!g. (!s. (!t. (((real_closed s) /\\ (((real_measurable_on f) t) /\\ ((!x. ((f x) IN s)) /\\ (((real_continuous_on g) s) /\\ (((real_of_num (NUMERAL _0)) IN s) /\\ ((g (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))))))) ==> ((real_measurable_on (g o f)) t))))))"}
{"_id": "50204", "text": "UNIFORMLY_CONTINUOUS_ON_ID |- (!s. ((uniformly_continuous_on (\\x. x)) s))"}
{"_id": "50205", "text": "ORTHOGONAL_SUBSPACE_DECOMP |- (!s. (!x. (?! (GABS (\\f. (!y. (!z. ((GEQ (f (y , z))) ((y IN (span s)) /\\ ((z IN (GSPEC (\\GEN%PVAR%185. (?z. (((SETSPEC GEN%PVAR%185) (!x. ((x IN (span s)) ==> ((orthogonal z) x)))) z))))) /\\ (x = ((vector_add y) z))))))))))))"}
{"_id": "50206", "text": "Ssrnat.leq_pexp2l |- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> ((n1 <= n2) ==> ((m EXP n1) <= (m EXP n2)))))))"}
{"_id": "50207", "text": "Lp_ineqs_proofs.dart3_rhazim3_fan_eq_rhazim3 |- (!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)))) ==> (((rhazim3_fan (V , E)) d) = ((((((rhazim3 (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": "50208", "text": "ORTHOGONAL_TRANSFORMATION_LOWDIM_HORIZONTAL |- (!s. (((dim s) < (dimindex UNIV)) ==> (?f. ((orthogonal_transformation f) /\\ (((IMAGE f) s) SUBSET (GSPEC (\\GEN%PVAR%457. (?z. (((SETSPEC GEN%PVAR%457) ((z $ (dimindex UNIV)) = (real_of_num (NUMERAL _0)))) z)))))))))"}
{"_id": "50209", "text": "DIV_REFL |- (!n. ((~ (n = (NUMERAL _0))) ==> ((n DIV n) = (NUMERAL (BIT1 _0)))))"}
{"_id": "50210", "text": "Asfutbf.azim_cycle_mono |- (!V. (!E. (!v. (!x. (!x'. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ ((x IN (((set_of_edge v) V) E)) /\\ ((x' IN (((set_of_edge v) V) E)) /\\ (((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) x) = ((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) x'))))) ==> (x = x')))))))"}
{"_id": "50211", "text": "Ssrfun.some_inj |- (injective SOME)"}
{"_id": "50212", "text": "SUM_OVER_PERMUTATIONS_NUMSEG |- (!f. (!m. (!n. ((m <= n) ==> (((sum (GSPEC (\\GEN%PVAR%297. (?p. (((SETSPEC GEN%PVAR%297) ((permutes p) (m .. n))) p))))) f) = ((sum (m .. n)) (\\i. ((sum (GSPEC (\\GEN%PVAR%298. (?p. (((SETSPEC GEN%PVAR%298) ((permutes p) ((m + (NUMERAL (BIT1 _0))) .. n))) p))))) (\\q. (f ((swap (m , i)) o q)))))))))))"}
{"_id": "50213", "text": "SPHERE_EQ_SING |- (!a. (!r. (!x. (((sphere (a , r)) = (x INSERT EMPTY)) = ((x = a) /\\ (r = (real_of_num (NUMERAL _0))))))))"}
{"_id": "50214", "text": "ABSOLUTELY_INTEGRABLE_ON_LINEAR_IMAGE |- (!f. (!g. (!s. ((linear g) ==> (((absolutely_integrable_on f) ((IMAGE g) s)) = (((absolutely_integrable_on (f o g)) s) \\/ ((det (matrix g)) = (real_of_num (NUMERAL _0)))))))))"}
{"_id": "50215", "text": "NON_MEASURABLE_SET |- (?s. (~ (lebesgue_measurable s)))"}
{"_id": "50216", "text": "INT_ABS_SIGN2 |- (!x. (!y. (((int_lt (int_abs ((int_sub x) y))) (int_neg y)) ==> ((int_lt x) (int_of_num (NUMERAL _0))))))"}
{"_id": "50217", "text": "Collect_geom2.ZJEWPAP |- (!v1. (!v2. (!v3. (!v4. ((LET (\\s. (LET_END ((((CARD s) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\\ (~ (coplanar_alt s))) ==> ((real_le (radV (v1 INSERT (v2 INSERT (v3 INSERT EMPTY))))) (radV s)))))) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))))))"}
{"_id": "50218", "text": "URYSOHN_STRONG |- (!s. (!t. (!a. (!b. (((closed s) /\\ ((closed t) /\\ (((s INTER t) = EMPTY) /\\ (~ (a = b))))) ==> (?f. (((continuous_on f) UNIV) /\\ ((!x. ((f x) IN (closed_segment ((CONS (a , b)) NIL)))) /\\ ((!x. (((f x) = a) = (x IN s))) /\\ (!x. (((f x) = b) = (x IN t))))))))))))"}
{"_id": "50219", "text": "INT_BOUNDS_LT |- (!x. (!k. ((((int_lt (int_neg k)) x) /\\ ((int_lt x) k)) = ((int_lt (int_abs x)) k))))"}
{"_id": "50220", "text": "INT_MAX_LT |- (!x. (!y. (!z. (((int_lt ((int_max x) y)) z) = (((int_lt x) z) /\\ ((int_lt y) z))))))"}
{"_id": "50221", "text": "CEXP_MUL_CPOW |- (!w. (!z. ((((real_lt (real_neg pi)) (Im w)) /\\ ((real_le (Im w)) pi)) ==> ((cexp ((complex_mul w) z)) = ((cpow (cexp w)) z)))))"}
{"_id": "50222", "text": "Lp_gen_theory.hyp_iso_comm_special |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((((face_map H) (g d)) = (g ((face_map G) d))) /\\ ((((node_map H) (g d)) = (g ((node_map G) d))) /\\ ((((edge_map H) (g d)) = (g ((edge_map G) d))) /\\ (!k. ((((POWER (face_map H)) k) (g d)) = (g (((POWER (face_map G)) k) d)))))))))))))"}
{"_id": "50223", "text": "COLLINEAR_STANDARD_HYPERPLANE_2 |- (!k. (!b. (((dimindex UNIV) <= (NUMERAL (BIT0 (BIT1 _0)))) ==> (collinear (GSPEC (\\GEN%PVAR%1599. (?x. (((SETSPEC GEN%PVAR%1599) ((x $ k) = b)) x))))))))"}
{"_id": "50224", "text": "List_hypermap_iso.set_of_edge_subset |- (!V. (!E. (!E'. (!x. ((E SUBSET E') ==> ((((set_of_edge x) V) E) SUBSET (((set_of_edge x) V) E')))))))"}
{"_id": "50225", "text": "AFF_DIM_HALFSPACE_LT |- (!a. (!b. ((aff_dim (GSPEC (\\GEN%PVAR%1548. (?x. (((SETSPEC GEN%PVAR%1548) ((real_lt ((dot a) x)) b)) x))))) = (((COND ((a = (vec (NUMERAL _0))) /\\ ((real_le b) (real_of_num (NUMERAL _0))))) (int_neg (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (dimindex UNIV))))))"}
{"_id": "50226", "text": "NSUM_INJECTION |- (!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)))))) ==> (((nsum s) (f o p)) = ((nsum s) f))))))"}
{"_id": "50227", "text": "Nonlinear_lemma.sqrt8_nn |- ((real_le (real_of_num (NUMERAL _0))) sqrt8)"}
{"_id": "50228", "text": "NSUM_LE_GEN |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) <= (g x)))) /\\ (FINITE (GSPEC (\\GEN%PVAR%188. (?x. (((SETSPEC GEN%PVAR%188) ((x IN s) /\\ (~ ((g x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) f) <= ((nsum s) g))))))"}
{"_id": "50229", "text": "INESSENTIAL_ON_COMPONENTS |- (!f. (!s. (!t. ((((locally connected) s) /\\ ((path_connected t) /\\ (!c. ((c IN (components s)) ==> (?a. ((((homotopic_with (\\x. T)) (c , t)) f) (\\x. a))))))) ==> (?a. ((((homotopic_with (\\x. T)) (s , t)) f) (\\x. a)))))))"}
{"_id": "50230", "text": "HAS_REAL_INTEGRAL_ALT |- (!f. (!s. (!i. ((((has_real_integral f) i) s) = ((!a. (!b. ((real_integrable_on (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) (closed_real_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. (((open_real_interval ((real_neg B) , B)) SUBSET (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_lt (real_abs ((real_sub ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0)))))) i))) e)))))))))))))"}
{"_id": "50231", "text": "MEASURABLE_INSERT |- (!x. (!s. ((measurable (x INSERT s)) = (measurable s))))"}
{"_id": "50232", "text": "DIV_EQ_EXCLUSION |- ((((b * c) < ((a + (NUMERAL (BIT1 _0))) * d)) /\\ ((a * d) < ((c + (NUMERAL (BIT1 _0))) * b))) ==> ((a DIV b) = (c DIV d)))"}
{"_id": "50233", "text": "Hypermap_and_fan.FAN_PAIR_FIXED_POINT |- (!V. (!x. ((x IN (GSPEC (\\GEN%PVAR%5632. (?v. (((SETSPEC GEN%PVAR%5632) ((v IN V) /\\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))) ==> ((((n_fan_pair_ext (V , E)) x) = x) /\\ (((f_fan_pair (V , E)) x) = x)))))"}
{"_id": "50234", "text": "INFSUM_CMUL |- (!s. (!x. (!c. (((summable s) x) ==> (((infsum s) (\\n. ((% c) (x n)))) = ((% c) ((infsum s) x)))))))"}
{"_id": "50235", "text": "HOMEOMORPHIC_FIXPOINT_PROPERTY |- (!s. (!t. (((homeomorphic s) t) ==> ((!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": "50236", "text": "LENGTH_MAP2 |- (!f. (!l. (!m. (((LENGTH l) = (LENGTH m)) ==> ((LENGTH (((MAP2 f) l) m)) = (LENGTH m))))))"}
{"_id": "50237", "text": "EVENTUALLY_INV1_LT |- (!e. (((eventually (\\n. ((real_lt (real_inv ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) e))) sequentially) = ((real_lt (real_of_num (NUMERAL _0))) e)))"}
{"_id": "50238", "text": "FRONTIER_SEGMENT |- ((!a. (!b. ((frontier (closed_segment ((CONS (a , b)) NIL))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) (closed_segment ((CONS (a , b)) NIL))) (a INSERT (b INSERT EMPTY)))))) /\\ (!a. (!b. ((frontier (open_segment (a , b))) = (((COND (a = b)) EMPTY) (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) (closed_segment ((CONS (a , b)) NIL))) (a INSERT (b INSERT EMPTY))))))))"}
{"_id": "50239", "text": "ABSOLUTE_RETRACTION_CONVEX_CLOSED_RELATIVE |- (!s. (!t. (((convex s) /\\ ((closed s) /\\ ((~ (s = EMPTY)) /\\ (s SUBSET t)))) ==> (?r. (((retraction (t , s)) r) /\\ (!x. ((x IN (((hull affine) s) DIFF (relative_interior s))) ==> ((r x) IN (relative_frontier s)))))))))"}
{"_id": "50240", "text": "Merge_ineq.nonf_ldih6_x_div_sqrtdelta_posbranch |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((ldih6_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((real_mul (lfun ((real_div (sqrt x6)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6))) ((((((delta_x4 x6) x1) x5) x3) x4) x2))) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x6) ((((((delta_x x1) x2) x3) x4) x5) x6)))) ((real_pow ((((((delta_x4 x6) x1) x5) x3) x4) x2)) (NUMERAL (BIT0 (BIT1 _0)))))))))))))))"}
{"_id": "50241", "text": "MATRIX_CMUL_SUB_LDISTRIB |- (!A. (!B. (!c. (((%% c) ((matrix_sub A) B)) = ((matrix_sub ((%% c) A)) ((%% c) B))))))"}
{"_id": "50242", "text": "REAL_COMPACT_UNION |- (!s. (!t. (((real_compact s) /\\ (real_compact t)) ==> (real_compact (s UNION t)))))"}
{"_id": "50243", "text": "CLOG_MUL_II |- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((clog ((complex_mul ii) z)) = (((COND (((real_le (real_of_num (NUMERAL _0))) (Re z)) \\/ ((real_lt (Im z)) (real_of_num (NUMERAL _0))))) ((vector_add (clog z)) ((complex_mul ii) (Cx ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))) ((vector_sub (clog z)) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))"}
{"_id": "50244", "text": "Lp_ineqs_quads.ineq120_std |- (!V. (!E. ((lp_fan (V , E)) ==> (quad_nonlinear_v10 ==> ((!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)))))))))) ==> ((!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)))))) ==> ((lp_tau (V , E)) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((real_ge ((real_add ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))) ((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))))))))))))))))) ((azim_dart (V , E)) d))) ((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))))))))))))))) (real_of_num (NUMERAL _0)))))))))))))"}
{"_id": "50245", "text": "Functional_equation.functional_proj_x1 |- (proj_x1 = (promote1_to_6 I))"}
{"_id": "50246", "text": "Lp_ineqs_proofs.y8_list_def |- (!V. (!E. (!g. (!L. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , E)))) ==> ((ALL (\\d. (((y8_list (g , (V , E))) d) = ((ye_list g) ((inverse (f_list_ext L)) ((inverse (f_list_ext L)) d)))))) (list_of_darts L)))))))))"}
{"_id": "50247", "text": "CRAMER_MATRIX_LEFT_INVERSE |- (!A. (!A'. ((((matrix_mul A') A) = (mat (NUMERAL (BIT1 _0)))) = ((~ ((det A) = (real_of_num (NUMERAL _0)))) /\\ (A' = (lambdak. (lambdal. ((real_div (det (lambdai. (lambdaj. (((COND (j = l)) (((COND (i = k)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) ((A $ j) $ i)))))) (det A)))))))))"}
{"_id": "50248", "text": "SUBSET_HYPERPLANES |- (!a. (!b. (!a'. (!b'. (((GSPEC (\\GEN%PVAR%442. (?x. (((SETSPEC GEN%PVAR%442) (((dot a) x) = b)) x)))) SUBSET (GSPEC (\\GEN%PVAR%443. (?x. (((SETSPEC GEN%PVAR%443) (((dot a') x) = b')) x))))) = (((GSPEC (\\GEN%PVAR%444. (?x. (((SETSPEC GEN%PVAR%444) (((dot a) x) = b)) x)))) = EMPTY) \\/ (((GSPEC (\\GEN%PVAR%445. (?x. (((SETSPEC GEN%PVAR%445) (((dot a') x) = b')) x)))) = UNIV) \\/ ((GSPEC (\\GEN%PVAR%446. (?x. (((SETSPEC GEN%PVAR%446) (((dot a) x) = b)) x)))) = (GSPEC (\\GEN%PVAR%447. (?x. (((SETSPEC GEN%PVAR%447) (((dot a') x) = b')) x))))))))))))"}
{"_id": "50249", "text": "REAL_SERIES_LE |- (!f. (!g. (!s. (!y. (!z. (((((real_sums f) y) s) /\\ ((((real_sums g) z) s) /\\ (!i. ((i IN s) ==> ((real_le (f i)) (g i)))))) ==> ((real_le y) z)))))))"}
{"_id": "50250", "text": "HAS_BOUNDED_REAL_VARIATION_TRANSLATION_EQ_INTERVAL |- (!a. (!f. (!u. (!v. (((has_bounded_real_variation_on (\\x. (f ((real_add a) x)))) (closed_real_interval ((CONS (u , v)) NIL))) = ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (((real_add a) u) , ((real_add a) v))) NIL))))))))"}
{"_id": "50251", "text": "REAL_LOG_CONVEX_ON_SUBSET |- (!f. (!s. (!t. ((((real_log_convex_on f) t) /\\ (s SUBSET t)) ==> ((real_log_convex_on f) s)))))"}
{"_id": "50252", "text": "Merge_ineq.nonf_edge2_126_x |- (!d. (!x4'. (!x5'. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((edge2_126_x d) x4') x5') x1) x2) x3) x4) x5) x6) = ((((((edge2_flatD_x1 d) x1) x2) x6) x4') x5')))))))))))"}
{"_id": "50253", "text": "INT_ABS_NZ |- (!x. ((~ (x = (int_of_num (NUMERAL _0)))) = ((int_lt (int_of_num (NUMERAL _0))) (int_abs x))))"}
{"_id": "50254", "text": "CONTENT_POS_LT_1 |- (!a. (!b. (((real_lt (drop a)) (drop b)) ==> ((real_lt (real_of_num (NUMERAL _0))) (content (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "50255", "text": "HAS_BOUNDED_VARIATION_ON_COMPLEX_MUL |- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\\ (((has_bounded_variation_on g) s) /\\ (is_interval s))) ==> ((has_bounded_variation_on (\\x. ((complex_mul (f x)) (g x)))) s)))))"}
{"_id": "50256", "text": "NPRODUCT_EQ_NUMSEG |- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\\ (i <= n)) ==> ((f i) = (g i)))) ==> (((nproduct (m .. n)) f) = ((nproduct (m .. n)) g)))))))"}
{"_id": "50257", "text": "Tame_list.INSERT_alt |- ((a INSERT B) = (GSPEC (\\GEN%PVAR%8608. (?x. (((SETSPEC GEN%PVAR%8608) ((x = a) \\/ (x IN B))) x)))))"}
{"_id": "50258", "text": "Hxhytij.HXHYTIJ |- (!s. (!vv. (!ww. (((is_scs_v39 s) /\\ (((BBprime2_v39 s) vv) /\\ ((BBs_v39 s) ww))) ==> (((real_lt ((taustar_v39 s) vv)) ((taustar_v39 s) ww)) \\/ (((BBindex_v39 s) vv) <= ((BBindex_v39 s) ww)))))))"}
{"_id": "50259", "text": "Functional_equation.functional_cayleyR6u |- ((LET (\\x1. (LET_END ((LET (\\x2. (LET_END ((LET (\\x3. (LET_END ((LET (\\x4. (LET_END ((LET (\\x5. (LET_END ((LET (\\x6. (LET_END ((LET (\\four. (LET_END ((LET (\\t01. (LET_END ((LET (\\two. (LET_END ((LET (\\mone. (LET_END (cayleyR6u = ((add6 ((mul6 mone) ((mul6 x1) ((mul6 x1) ((mul6 x4) x4))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) x4))))) ((mul6 x6) ((mul6 x6) ((mul6 x4) x4))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x4) x5))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x4) x5)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) x5)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x6) ((mul6 x6) ((mul6 x4) x5))))) ((mul6 x2) ((mul6 x2) ((mul6 x5) x5))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) x5))))) ((mul6 x6) ((mul6 x6) ((mul6 x5) x5))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x4) four)))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x1) ((mul6 x4) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x4) four))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) four)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x2) ((mul6 x5) four))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x5) four)))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) four)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x5) four))))) ((mul6 x2) ((mul6 x2) ((mul6 four) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 four) four))))) ((mul6 x1) ((mul6 x1) ((mul6 four) four))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 x4) four))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x4) four)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) four)))))) ((add6 ((mul6 two) ((mul6 x6) ((mul6 x6) ((mul6 x4) four))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 x5) four))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x5) four)))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) four)))))) ((add6 ((mul6 two) ((mul6 x6) ((mul6 x6) ((mul6 x5) four))))) ((add6 ((sub6 ((sub6 ((mul6 four) ((mul6 x6) ((mul6 x4) ((mul6 x5) four))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) four)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) four)))))) ((add6 ((mul6 four) ((mul6 x2) ((mul6 x1) ((mul6 four) four))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 x6) ((mul6 four) four))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 four) four)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 four) four)))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 four) four)))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 four) four)))))) ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 four) four)))))) ((mul6 two) ((mul6 x6) ((mul6 x5) ((mul6 four) four)))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 four) ((mul6 four) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 four) ((mul6 four) four))))) ((mul6 x3) ((mul6 x3) ((mul6 four) four))))) ((add6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 four) four))))) ((mul6 x6) ((mul6 x6) ((mul6 four) four))))) ((add6 ((mul6 two) ((mul6 x3) ((mul6 four) ((mul6 four) four))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x6) ((mul6 four) ((mul6 four) four))))) ((mul6 four) ((mul6 four) ((mul6 four) four))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 x1) ((mul6 x5) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((mul6 four) ((mul6 x2) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 x5) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x6) ((mul6 x5) ((mul6 x5) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x1) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 four) ((mul6 x1) ((mul6 x4) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x3) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x6) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x5) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((mul6 two) ((mul6 x5) ((mul6 four) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 x3) ((mul6 x3) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x1) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x3) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x1) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x5) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 x2) ((mul6 x4) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01))))))) ((add6 ((mul6 four) ((mul6 x1) ((mul6 x6) ((mul6 x4) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 x4) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 x4) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 x5) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 x2) ((mul6 x5) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 x5) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x2) ((mul6 x2) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 four) ((mul6 x2) ((mul6 x5) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x3) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x6) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x6) ((mul6 four) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x6) ((mul6 x4) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 four) ((mul6 four) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x4) ((mul6 four) ((mul6 four) ((mul6 t01) t01)))))) ((add6 ((sub6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x3) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((add6 ((sub6 ((sub6 ((sub6 ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x1) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 two) ((mul6 x3) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x1) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x3) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((mul6 two) ((mul6 x2) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01))))))) ((add6 ((mul6 two) ((mul6 x4) ((mul6 x5) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 four) ((mul6 x3) ((mul6 four) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x3) ((mul6 x3) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((sub6 ((mul6 two) ((mul6 x3) ((mul6 x2) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x2) ((mul6 x2) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((add6 ((mul6 two) ((mul6 x3) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((sub6 ((mul6 two) ((mul6 x2) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))) ((mul6 x4) ((mul6 x4) ((mul6 ((mul6 t01) t01)) ((mul6 t01) t01)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) (constant6 (real_neg (real_of_num (NUMERAL (BIT1 _0))))))))) (constant6 (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))) (constant6 ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))) (constant6 (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) proj_x6)))) proj_x5)))) proj_x4)))) proj_x3)))) proj_x2)))) proj_x1)"}
{"_id": "50260", "text": "Seq2.list_sum_undup |- (!s. (!f. (((list_sum s) f) = ((list_sum (undup s)) (\\x. ((real_mul (real_of_num ((count (pred1 x)) s))) (f x)))))))"}
{"_id": "50261", "text": "PRODUCT_CLAUSES_RIGHT |- (!f. (!m. (!n. ((((NUMERAL _0) < n) /\\ (m <= n)) ==> (((product (m .. n)) f) = ((real_mul ((product (m .. (n - (NUMERAL (BIT1 _0))))) f)) (f n)))))))"}
{"_id": "50262", "text": "REAL_CONVEX_ON_EXP |- (!s. ((real_convex_on exp) s))"}
{"_id": "50263", "text": "HOMEOMORPHIC_GRAPH |- (!f. (!s. (((continuous_on f) s) ==> ((homeomorphic (GSPEC (\\GEN%PVAR%1035. (?x. (((SETSPEC GEN%PVAR%1035) (x IN s)) ((pastecart x) (f x))))))) s))))"}
{"_id": "50264", "text": "MEASURABLE_OPEN_IN |- (!s. (!t. ((((open_in ((subtopology euclidean) s)) t) /\\ (measurable s)) ==> (measurable t))))"}
{"_id": "50265", "text": "ORDER_EXISTENCE_FINITE |- (!R. (!f. (!z. (((FINITE (GSPEC (\\GEN%PVAR%77. (?n. (((SETSPEC GEN%PVAR%77) (n IN UNIV)) (R (((ITER n) f) z))))))) /\\ (((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) /\\ (!n. (((R (((ITER n) f) z)) z) = ((num_divides d) n)))))))))"}
{"_id": "50266", "text": "RSTC_CASES_L |- (!R. (!x. (!z. ((((RSTC R) x) z) = ((x = z) \\/ (((R x) z) \\/ ((((RSTC R) z) x) \\/ (?y. ((((RSTC R) x) y) /\\ ((R y) z))))))))))"}
{"_id": "50267", "text": "RSTC_INC_STC |- (!R. (!x. (!y. ((((STC R) x) y) ==> (((RSTC R) x) y)))))"}
{"_id": "50268", "text": "NSUM_CONST_NUMSEG |- (!c. (!m. (!n. (((nsum (m .. n)) (\\n. c)) = (((n + (NUMERAL (BIT1 _0))) - m) * c)))))"}
{"_id": "50269", "text": "Deformation.COMPACT_SPHERE_0 |- (!a. (compact (GSPEC (\\GEN%PVAR%7989. (?x. (((SETSPEC GEN%PVAR%7989) ((vector_norm x) = a)) x))))))"}
{"_id": "50270", "text": "CONTINUOUS_ON_CPOW_RIGHT |- (!w. (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\\z. ((cpow w) (f z)))) s)))))"}
{"_id": "50271", "text": "CONTINUOUS_WITHIN_CEXP |- (!s. (!z. ((continuous cexp) ((within (at z)) s))))"}
{"_id": "50272", "text": "ORTHOGONAL_LVSUM |- (!f. (!s. (!y. (((FINITE s) /\\ (!x. ((x IN s) ==> ((orthogonal (f x)) y)))) ==> ((orthogonal ((vsum s) f)) y)))))"}
{"_id": "50273", "text": "INJECTIVE_PREIMAGE |- (!f. ((!t. (!t'. (((GSPEC (\\GEN%PVAR%145. (?x. (((SETSPEC GEN%PVAR%145) ((f x) IN t)) x)))) = (GSPEC (\\GEN%PVAR%146. (?x. (((SETSPEC GEN%PVAR%146) ((f x) IN t')) x))))) ==> (t = t')))) = (((IMAGE f) UNIV) = UNIV)))"}
{"_id": "50274", "text": "ISO_REFL |- ((ISO (\\x. x)) (\\x. x))"}
{"_id": "50275", "text": "HOMOTOPIC_LINEAR_POSITIVE_SEMIDEFINITE_MAPS |- (!f. (!g. (((((homotopic_with (\\f. ((linear f) /\\ (positive_semidefinite (matrix f))))) (UNIV , UNIV)) f) g) = ((linear f) /\\ ((linear g) /\\ ((positive_semidefinite (matrix f)) /\\ (positive_semidefinite (matrix g))))))))"}
{"_id": "50276", "text": "CANTOR_BENDIXSON |- (!s. ((closed s) ==> (?t. (?u. ((closed t) /\\ ((!x. ((x IN t) ==> ((limit_point_of x) t))) /\\ ((COUNTABLE u) /\\ (s = (t UNION u)))))))))"}
{"_id": "50277", "text": "INT_EQ_ADD_RCANCEL |- (!x. (!y. (!z. ((((int_add x) z) = ((int_add y) z)) = (x = y)))))"}
{"_id": "50278", "text": "COVERING_SPACE_LIFT |- (!p. (!c. (!s. (!f. (!u. ((((covering_space (c , p)) s) /\\ ((simply_connected u) /\\ (((locally path_connected) u) /\\ (((continuous_on f) u) /\\ (((IMAGE f) u) SUBSET s))))) ==> (?g. (((continuous_on g) u) /\\ ((((IMAGE g) u) SUBSET c) /\\ (!y. ((y IN u) ==> ((p (g y)) = (f y)))))))))))))"}
{"_id": "50279", "text": "TENDSTO_LIM |- (!net. (!f. (!l. (((~ (trivial_limit net)) /\\ (((--> f) l) net)) ==> (((lim net) f) = l)))))"}
{"_id": "50280", "text": "CEXP_VSUM |- (!f. (!s. ((FINITE s) ==> ((cexp ((vsum s) f)) = ((cproduct s) (\\x. (cexp (f x))))))))"}
{"_id": "50281", "text": "AFF_LE_EQ_AFFINE_HULL |- (!s. (!t. (((aff_le s) EMPTY) = ((hull affine) s))))"}
{"_id": "50282", "text": "INT_LNEG_UNIQ |- (!x. (!y. ((((int_add x) y) = (int_of_num (NUMERAL _0))) = (x = (int_neg y)))))"}
{"_id": "50283", "text": "Topology.FINITE_SERIES |- (!n. (!f. (FINITE (GSPEC (\\GEN%PVAR%5487. (?i. (((SETSPEC GEN%PVAR%5487) (i < n)) (f i))))))))"}
{"_id": "50284", "text": "FORALL_INTEGER |- (!P. (((!n. (P (real_of_num n))) /\\ (!x. ((P x) ==> (P (real_neg x))))) ==> (!x. ((integer x) ==> (P x)))))"}
{"_id": "50285", "text": "MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED_EQ |- (!f. (!s. ((((measurable_on f) s) /\\ (measurable s)) = (!t. ((closed t) ==> (measurable (GSPEC (\\GEN%PVAR%3659. (?x. (((SETSPEC GEN%PVAR%3659) ((x IN s) /\\ ((f x) IN t))) x))))))))))"}
{"_id": "50286", "text": "SIMPLEX_FURTHEST_LE_EXISTS |- (!a. (!s. ((FINITE s) ==> (!x. ((x IN ((hull convex) s)) ==> (?y. ((y IN s) /\\ ((real_le (vector_norm ((vector_sub x) a))) (vector_norm ((vector_sub y) a))))))))))"}
{"_id": "50287", "text": "HOMEOMORPHIC_STANDARD_HYPERPLANE_HYPERPLANE |- (!a. (!b. (!k. (!c. (((~ (a = (vec (NUMERAL _0)))) /\\ (((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV)))) ==> ((homeomorphic (GSPEC (\\GEN%PVAR%1060. (?x. (((SETSPEC GEN%PVAR%1060) ((x $ k) = c)) x))))) (GSPEC (\\GEN%PVAR%1061. (?x. (((SETSPEC GEN%PVAR%1061) (((dot a) x) = b)) x))))))))))"}
{"_id": "50288", "text": "CARD_FACES_OF_SIMPLEX |- (!n. (!s. (((simplex n) s) ==> ((CARD (GSPEC (\\GEN%PVAR%2481. (?f. (((SETSPEC GEN%PVAR%2481) ((face_of f) s)) f))))) = ((NUMERAL (BIT0 (BIT1 _0))) EXP (num_of_int ((int_add n) (int_of_num (NUMERAL (BIT1 _0))))))))))"}
{"_id": "50289", "text": "REAL_SERIES_GOESTOZERO |- (!s. (!x. (((real_summable s) x) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\\n. ((n IN s) ==> ((real_lt (real_abs (x n))) e)))) sequentially))))))"}
{"_id": "50290", "text": "Localization.slicef |- (!E. (!FF. (!w. (!v. (((((slicef E) FF) v) w) = ((GSPEC (\\GEN%PVAR%7541. (?f. (((SETSPEC GEN%PVAR%7541) (?u. ((u IN (((((slicev E) FF) v) w) DELETE w)) /\\ (f = (u , ((rho_node1 FF) u)))))) f)))) UNION ((w , v) INSERT EMPTY)))))))"}
{"_id": "50291", "text": "AFF_DIM_EQ_INTER_HYPERPLANE |- (!s. (!t. (((affine s) /\\ ((affine t) /\\ ((t SUBSET s) /\\ (((int_add (aff_dim t)) (int_of_num (NUMERAL (BIT1 _0)))) = (aff_dim s))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\\ (((GSPEC (\\GEN%PVAR%1574. (?x. (((SETSPEC GEN%PVAR%1574) (((dot a) x) = b)) x)))) INTER s) = t)))))))"}
{"_id": "50292", "text": "INFSUM_LINEAR |- (!f. (!h. (!s. ((((summable s) f) /\\ (linear h)) ==> (((infsum s) (\\n. (h (f n)))) = (h ((infsum s) f)))))))"}
{"_id": "50293", "text": "Hypermap_iso.disjoint_diff |- (!s. (!t. ((DISJOINT s) (t DIFF s))))"}
{"_id": "50294", "text": "ORTHOGONAL_MATRIX_ORTHONORMAL_COLUMNS_PAIRWISE |- (!A. ((orthogonal_matrix A) = (((CARD (columns A)) = (dimindex UNIV)) /\\ ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((vector_norm ((column i) A)) = (real_of_num (NUMERAL (BIT1 _0)))))) /\\ ((pairwise orthogonal) (columns A))))))"}
{"_id": "50295", "text": "LIMPT_PCROSS |- (!s. (!t. (!x. (!y. ((((limit_point_of x) s) /\\ ((limit_point_of y) t)) ==> ((limit_point_of ((pastecart x) y)) (s PCROSS t)))))))"}
{"_id": "50296", "text": "Pack_defs.fcc_compatible |- (!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": "50297", "text": "COMPLEX_NEG_ADD |- (!x. (!y. ((vector_neg ((vector_add x) y)) = ((vector_add (vector_neg x)) (vector_neg y)))))"}
{"_id": "50298", "text": "REAL_DIV_SQRT |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_div x) (sqrt x)) = (sqrt x))))"}
{"_id": "50299", "text": "REAL_CARD_INTSEG_INT |- (!a. (!b. (((integer a) /\\ (integer b)) ==> ((real_of_num (CARD (GSPEC (\\GEN%PVAR%86. (?x. (((SETSPEC GEN%PVAR%86) ((integer x) /\\ (((real_le a) x) /\\ ((real_le x) b)))) x)))))) = (((COND ((real_lt b) a)) (real_of_num (NUMERAL _0))) ((real_add ((real_sub b) a)) (real_of_num (NUMERAL (BIT1 _0)))))))))"}
{"_id": "50300", "text": "AFF_DIM_CONVEX_INTER_NONEMPTY_INTERIOR |- (!s. (!t. (((convex s) /\\ (~ ((s INTER (interior t)) = EMPTY))) ==> ((aff_dim (s INTER t)) = (aff_dim s)))))"}
{"_id": "50301", "text": "SYMMETRIC_MATRIX_ORTHOGONAL_EIGENVECTORS |- (!A. (!v. (!w. (!a. (!b. ((((transp A) = A) /\\ ((((matrix_vector_mul A) v) = ((% a) v)) /\\ ((((matrix_vector_mul A) w) = ((% b) w)) /\\ (~ (a = b))))) ==> ((orthogonal v) w)))))))"}
{"_id": "50302", "text": "REAL_INTEGRABLE_ON_EMPTY |- (!f. ((real_integrable_on f) EMPTY))"}
{"_id": "50303", "text": "REAL_UNIFORMLY_CONTINUOUS_ON_SUM |- (!t. (!f. (!s. (((FINITE s) /\\ (!a. ((a IN s) ==> ((real_uniformly_continuous_on (f a)) t)))) ==> ((real_uniformly_continuous_on (\\x. ((sum s) (\\a. ((f a) x))))) t)))))"}
{"_id": "50304", "text": "AR_IMP_PATH_CONNECTED |- (!s. ((AR s) ==> (path_connected s)))"}
{"_id": "50305", "text": "REAL_SUM_INTEGRAL_BOUNDS_INCREASING |- (!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_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_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))))) 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_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))))) ==> ((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))))) /\\ ((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": "50306", "text": "SUMMABLE_IMP_SUMS_BOUNDED |- (!f. (!k. (((summable (from k)) f) ==> (bounded (GSPEC (\\GEN%PVAR%1120. (?n. (((SETSPEC GEN%PVAR%1120) (n IN UNIV)) ((vsum (k .. n)) f)))))))))"}
{"_id": "50307", "text": "RETRACT_OF_HOMOTOPICALLY_TRIVIAL_NULL |- (!s. (!t. (!u. ((((retract_of t) s) /\\ (!f. ((((continuous_on f) u) /\\ (((IMAGE f) u) SUBSET s)) ==> (?c. ((((homotopic_with (\\x. T)) (u , s)) f) (\\x. c)))))) ==> (!f. ((((continuous_on f) u) /\\ (((IMAGE f) u) SUBSET t)) ==> (?c. ((((homotopic_with (\\x. T)) (u , t)) f) (\\x. c)))))))))"}
{"_id": "50308", "text": "RELATIVE_FRONTIER_RELATIVE_INTERIOR |- (!s. ((convex s) ==> ((relative_frontier (relative_interior s)) = (relative_frontier s))))"}
{"_id": "50309", "text": "DIST_TRIANGLE_ADD_HALF |- (!x. (!x'. (!y. (!y'. ((((real_lt (distance (x , x'))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\\ ((real_lt (distance (y , y'))) ((real_div e) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((real_lt (distance (((vector_add x) y) , ((vector_add x') y')))) e))))))"}
{"_id": "50310", "text": "ABSOLUTELY_CONTINUOUS_INDEFINITE_INTEGRAL_LEFT |- (!f. (!a. (!b. (((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((absolutely_continuous_on (\\x. ((integral (closed_interval ((CONS (x , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "50311", "text": "ORTHOGONAL_MATRIX_ORTHOGONAL_EIGENVECTORS |- (!A. (!v. (!w. (!a. (!b. (((orthogonal_matrix A) /\\ ((((matrix_vector_mul A) v) = ((% a) v)) /\\ ((((matrix_vector_mul A) w) = ((% b) w)) /\\ (~ (a = b))))) ==> ((orthogonal v) w)))))))"}
{"_id": "50312", "text": "REAL_CONTINUOUS_ON_ACS |- (!s. ((!x. ((x IN s) ==> ((real_le (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((real_continuous_on acs) s)))"}
{"_id": "50313", "text": "COMPLEX_DERIVATIVE_SUB_AT |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\\ ((analytic_on g) (z INSERT EMPTY))) ==> (((complex_derivative (\\w. ((vector_sub (f w)) (g w)))) z) = ((vector_sub ((complex_derivative f) z)) ((complex_derivative g) z)))))))"}
{"_id": "50314", "text": "MEASURE_INTERVAL_4_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ (((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((real_lt (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((real_lt (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_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ (((real_lt (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((real_lt (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((real_lt (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": "50315", "text": "HAS_REAL_MEASURE_UNION_REAL_NEGLIGIBLE_EQ |- (!s. (!t. (!m. ((real_negligible t) ==> (((has_real_measure (s UNION t)) m) = ((has_real_measure s) m))))))"}
{"_id": "50316", "text": "INDEPENDENT_EQ_DIM_EQ_CARD |- (!s. ((independent s) = ((FINITE s) /\\ ((dim s) = (CARD s)))))"}
{"_id": "50317", "text": "NONNEGATIVE_INTEGER |- (!x. (((integer x) /\\ ((real_le (real_of_num (NUMERAL _0))) x)) = (?n. (x = (real_of_num n)))))"}
{"_id": "50318", "text": "ANR_INTERIOR |- (!s. (ANR (interior s)))"}
{"_id": "50319", "text": "CLOSED_INTERVAL_RIGHT |- (!a. (closed (GSPEC (\\GEN%PVAR%762. (?x. (((SETSPEC GEN%PVAR%762) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (a $ i)) (x $ i))))) x))))))"}
{"_id": "50320", "text": "COMMUTING_WITH_SQUARE_ROOT_MATRIX |- (!A. (!S. (!B. (((positive_semidefinite S) /\\ ((((matrix_mul S) S) = A) /\\ (((matrix_mul A) B) = ((matrix_mul B) A)))) ==> (((matrix_mul S) B) = ((matrix_mul B) S))))))"}
{"_id": "50321", "text": "RESTRICTION_EQ |- (!s. (!f. (!x. (!y. (((x IN s) /\\ ((f x) = y)) ==> ((((RESTRICTION s) f) x) = y))))))"}
{"_id": "50322", "text": "Functional_equation.functional_halfbump_x4 |- (halfbump_x4 = (rotate4 halfbump_x1))"}
{"_id": "50323", "text": "POSITIVE_SEMIDEFINITE_2 |- (!A. ((positive_semidefinite A) = ((((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))) = ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((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_pow ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))))))))"}
{"_id": "50324", "text": "Lp_ineqs_proofs.tau_sum3 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT1 (BIT1 _0)))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_add pi) sol0)))))))))))))"}
{"_id": "50325", "text": "GDELTA_UNIONS |- (!g. (((FINITE g) /\\ (!u. ((u IN g) ==> (gdelta u)))) ==> (gdelta (UNIONS g))))"}
{"_id": "50326", "text": "HOMEOMORPHISM_IMP_QUOTIENT_MAP |- (!f. (!g. (!s. (!t. (((homeomorphism (s , t)) (f , g)) ==> (!u. ((u SUBSET t) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%1030. (?x. (((SETSPEC GEN%PVAR%1030) ((x IN s) /\\ ((f x) IN u))) x))))) = ((open_in ((subtopology euclidean) t)) u)))))))))"}
{"_id": "50327", "text": "Dih2k_hypermap.VECMAT_VEC |- (!i. (!n. ((i < (dimindex UNIV)) ==> (((vecmat i) (vec n)) = (vec n)))))"}
{"_id": "50328", "text": "Vasyyau.A_EQ2h0_IMP_B |- (((!i. ((((((scs_a_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((((scs_b_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) \\/ (((((scs_a_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) /\\ ((((scs_b_v39 s) i) (i + (NUMERAL (BIT1 _0)))) = cstab)))) /\\ ((((scs_a_v39 s) p) (p + (NUMERAL (BIT1 _0)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> ((((scs_b_v39 s) p) (p + (NUMERAL (BIT1 _0)))) = cstab))"}
{"_id": "50329", "text": "SIN_VECTOR_ANGLE_RMUL |- (!a. (!x. (!y. ((sin ((vector_angle x) ((% a) y))) = (((COND (a = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (sin ((vector_angle x) y)))))))"}
{"_id": "50330", "text": "CONIC_HULLS_EQ_IMP_SPANS_EQ |- (!s. (!t. ((((hull conic) s) = ((hull conic) t)) ==> ((span s) = (span t)))))"}
{"_id": "50331", "text": "COUNTABLE_ISOLATED_SET |- (!s. ((!a. ((a IN s) ==> (~ ((limit_point_of a) s)))) ==> (COUNTABLE s)))"}
{"_id": "50332", "text": "UNIFORM_LIM_COMPLEX_DIV |- (!net. (!P. (!f. (!g. (!l. (!m. (!b1. (!b2. ((((eventually (\\x. (!n. ((P n) ==> ((real_le (vector_norm (l n))) b1))))) net) /\\ (((real_lt (real_of_num (NUMERAL _0))) b2) /\\ (((eventually (\\x. (!n. ((P n) ==> ((real_le b2) (vector_norm (m n))))))) 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_div ((f n) x)) ((g n) x))) ((complex_div (l n)) (m n))))) e))))) net))))))))))))"}
{"_id": "50333", "text": "Functional_equation.functional_dih2_x |- (dih2_x = (rotate2 dih_x))"}
{"_id": "50334", "text": "MATRIX_CMUL_RZERO |- (!c. (((%% c) (mat (NUMERAL _0))) = (mat (NUMERAL _0))))"}
{"_id": "50335", "text": "Tactics_jordan.switch_lemma_lt |- (!A. (!B. (!C. ((((real_lt (real_of_num (NUMERAL _0))) A) ==> (((real_lt (real_of_num (NUMERAL _0))) B) ==> C)) = (((real_lt (real_of_num (NUMERAL _0))) B) ==> (((real_lt (real_of_num (NUMERAL _0))) A) ==> C))))))"}
{"_id": "50336", "text": "INT_LT_ADDNEG |- (!x. (!y. (!z. (((int_lt y) ((int_add x) (int_neg z))) = ((int_lt ((int_add y) z)) x)))))"}
{"_id": "50337", "text": "INT_EQ_MUL_RCANCEL |- (!x. (!y. (!z. ((((int_mul x) z) = ((int_mul y) z)) = ((x = y) \\/ (z = (int_of_num (NUMERAL _0))))))))"}
{"_id": "50338", "text": "CARD_EQ_OPEN_IN_AFFINE |- (!u. (!s. (((affine u) /\\ ((~ ((aff_dim u) = (int_of_num (NUMERAL _0)))) /\\ (((open_in ((subtopology euclidean) u)) s) /\\ (~ (s = EMPTY))))) ==> (s =_c UNIV))))"}
{"_id": "50339", "text": "More_list_hypermap.contravening_dist_not_dart |- (!V. (!h. (!L. (!g. ((contravening V) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , (ESTD V))))) ==> ((((BIJ h) (elements_of_list L)) V) ==> ((g = (\\d. ((h (FST d)) , (h (SND d))))) ==> (!s. ((((ALL (\\d. (((MEM (FST d)) (list_of_elements L)) /\\ (((MEM (SND d)) (list_of_elements L)) /\\ (((FST d) = (SND d)) = F))))) s) /\\ ((ALL (\\d. (((MEM d) (list_of_darts L)) = F))) s)) ==> ((ALL (\\d. ((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (distance (g d))))) s))))))))))))"}
{"_id": "50340", "text": "RETRACTION_ARC |- (!p. ((arc p) ==> (?f. (((continuous_on f) UNIV) /\\ ((((IMAGE f) UNIV) SUBSET (path_image p)) /\\ (!x. ((x IN (path_image p)) ==> ((f x) = x))))))))"}
{"_id": "50341", "text": "INVERTIBLE_MAT |- (!a. ((invertible (mat a)) = (~ (a = (NUMERAL _0)))))"}
{"_id": "50342", "text": "OPEN_MAP_CLOSED_SUPERSET_PREIMAGE_EQ |- (!f. (!s. (!t. ((((IMAGE f) s) SUBSET t) ==> ((!k. (((open_in ((subtopology euclidean) s)) k) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) k)))) = (!u. (!w. ((((closed_in ((subtopology euclidean) s)) u) /\\ ((w SUBSET t) /\\ ((GSPEC (\\GEN%PVAR%699. (?x. (((SETSPEC GEN%PVAR%699) ((x IN s) /\\ ((f x) IN w))) x)))) SUBSET u))) ==> (?v. (((closed_in ((subtopology euclidean) t)) v) /\\ ((w SUBSET v) /\\ ((GSPEC (\\GEN%PVAR%700. (?x. (((SETSPEC GEN%PVAR%700) ((x IN s) /\\ ((f x) IN v))) x)))) SUBSET u))))))))))))"}
{"_id": "50343", "text": "TC_SC |- (!R. (!x. (!y. ((((SC (TC R)) x) y) ==> (((TC (SC R)) x) y)))))"}
{"_id": "50344", "text": "Optimize.EXPAND_1var_lemma |- (!f. (!a. (!b. ((!y. ((ineq ((CONS (a , (y , b))) NIL)) (f y))) = (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((ineq ((CONS (a , (x1 , b))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x2 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x3 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x4 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x5 , (real_of_num (NUMERAL (BIT1 _0)))))) ((CONS ((real_of_num (NUMERAL (BIT1 _0))) , (x6 , (real_of_num (NUMERAL (BIT1 _0)))))) NIL))))))) (f x1))))))))))))"}
{"_id": "50345", "text": "INVERTIBLE_EIGENVALUES |- (!A. ((invertible A) = (!c. (!v. (((((matrix_vector_mul A) v) = ((% c) v)) /\\ (~ (v = (vec (NUMERAL _0))))) ==> (~ (c = (real_of_num (NUMERAL _0)))))))))"}
{"_id": "50346", "text": "AR_IMP_ABSOLUTE_RETRACT_UNIV |- (!s. (!s'. (((AR s) /\\ (((homeomorphic s) s') /\\ (closed s'))) ==> ((retract_of s') UNIV))))"}
{"_id": "50347", "text": "INT_POW_NZ |- (!x. (!n. ((~ (x = (int_of_num (NUMERAL _0)))) ==> (~ (((int_pow x) n) = (int_of_num (NUMERAL _0)))))))"}
{"_id": "50348", "text": "COLUMN_0 |- (!i. (((column i) (mat (NUMERAL _0))) = (vec (NUMERAL _0))))"}
{"_id": "50349", "text": "SKOLEM_THM_GEN |- (!P. (!s. ((!x. ((P x) ==> (?y. ((R x) y)))) = (?f. (!x. ((P x) ==> ((R x) (f x))))))))"}
{"_id": "50350", "text": "CONTINUOUS_AT_CX_RE |- (!z. ((continuous (Cx o Re)) (at z)))"}
{"_id": "50351", "text": "Local_lemmas.AFF_GE_EQ |- ((((real_lt (real_of_num (NUMERAL _0))) t) /\\ (((vector_sub a) x) = ((% t) ((vector_sub b) x)))) ==> (((aff_ge (x INSERT EMPTY)) (a INSERT EMPTY)) = ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))"}
{"_id": "50352", "text": "COMPLEX_POW_EQ_1 |- (!z. (!n. ((((complex_pow z) n) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) ==> (((vector_norm z) = (real_of_num (NUMERAL (BIT1 _0)))) \\/ (n = (NUMERAL _0))))))"}
{"_id": "50353", "text": "INT_LTE_ADD2 |- (!w. (!x. (!y. (!z. ((((int_lt w) x) /\\ ((int_le y) z)) ==> ((int_lt ((int_add w) y)) ((int_add x) z)))))))"}
{"_id": "50354", "text": "INT_ABS_BETWEEN1 |- (!x. (!y. (!z. ((((int_lt x) z) /\\ ((int_lt (int_abs ((int_sub y) x))) ((int_sub z) x))) ==> ((int_lt y) z)))))"}
{"_id": "50355", "text": "INTEGRABLE_COMPLEX_RMUL_EQ |- (!f. (!s. (!c. (((integrable_on (\\x. ((complex_mul (f x)) c))) s) = ((c = (Cx (real_of_num (NUMERAL _0)))) \\/ ((integrable_on f) s))))))"}
{"_id": "50356", "text": "CNJ_CPRODUCT |- (!f. (!s. ((FINITE s) ==> ((cnj ((cproduct s) f)) = ((cproduct s) (\\i. (cnj (f i))))))))"}
{"_id": "50357", "text": "REAL_CONTINUOUS_WITHIN_ACS |- (!s. (!x. (((real_lt (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))) ==> ((real_continuous acs) ((within (atreal x)) s)))))"}
{"_id": "50358", "text": "REALLIM_1_OVER_N_OFFSET |- (!a. (((---> (\\n. (real_inv ((real_add (real_of_num n)) a)))) (real_of_num (NUMERAL _0))) sequentially))"}
{"_id": "50359", "text": "BETWEEN_NORM_LE |- (!r. (!a. (!b. (!x. ((((real_le (vector_norm a)) r) /\\ (((real_le (vector_norm b)) r) /\\ ((between x) (a , b)))) ==> ((real_le (vector_norm x)) r))))))"}
{"_id": "50360", "text": "CONIC_CAP_DEGENERATE |- (!v0. (!r. (!a. (((((conic_cap v0) v0) r) a) = EMPTY))))"}
{"_id": "50361", "text": "HOLOMORPHIC_ON_EMPTY |- (!f. ((holomorphic_on f) EMPTY))"}
{"_id": "50362", "text": "REAL_COMPLEX_INTEGRAL |- (!f. (!s. ((((integrable_on f) s) /\\ (!x. ((x IN s) ==> (real (f x))))) ==> (real ((integral s) f)))))"}
{"_id": "50363", "text": "CONVEX_HULL_SUBSET_SPAN |- (!s. (((hull convex) s) SUBSET (span s)))"}
{"_id": "50364", "text": "PSUBSET_SUBSET_TRANS |- (!s. (!t. (!u. (((s PSUBSET t) /\\ (t SUBSET u)) ==> (s PSUBSET u)))))"}
{"_id": "50365", "text": "Geomdetail.CONV_EM |- ((conv EMPTY) = EMPTY)"}
{"_id": "50366", "text": "INT_ABS_BETWEEN2 |- (!x0. (!x. (!y0. (!y. ((((int_lt x0) y0) /\\ (((int_lt ((int_mul (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) (int_abs ((int_sub x) x0)))) ((int_sub y0) x0)) /\\ ((int_lt ((int_mul (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) (int_abs ((int_sub y) y0)))) ((int_sub y0) x0)))) ==> ((int_lt x) y))))))"}
{"_id": "50367", "text": "Pqcsxwg.PQCSXWG2 |- (!v0. (!v1. (!v2. (!v3. (!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) /\\ ((~ (collinear (v0 INSERT (v1 INSERT (v2 INSERT EMPTY))))) /\\ ((x1 = ((real_pow (distance (v1 , v0))) (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((x2 = ((real_pow (distance (v2 , v0))) (NUMERAL (BIT0 (BIT1 _0))))) /\\ ((x6 = ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\\ (v3 = (((((((((mk_simplex1 v0) v1) v2) x1) x2) x3) x4) x5) x6))))))))))))) ==> ((continuous (\\q. (((((((((mk_simplex1 v0) v1) v2) x1) x2) x3) x4) q) x6))) (atreal x5)))))))))))))"}
{"_id": "50368", "text": "COMPLEX_DIFFERENTIABLE_WITHIN_CASN |- (!s. (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((complex_differentiable casn) ((within (at z)) s)))))"}
{"_id": "50369", "text": "SEPARATING_HYPERPLANE_POLYHEDRA |- (!s. (!t. (((polyhedron s) /\\ ((polyhedron t) /\\ ((~ (s = EMPTY)) /\\ ((~ (t = EMPTY)) /\\ ((DISJOINT s) t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\\ ((!x. ((x IN s) ==> ((real_lt ((dot a) x)) b))) /\\ (!x. ((x IN t) ==> ((real_gt ((dot a) x)) b))))))))))"}
{"_id": "50370", "text": "ABSOLUTELY_SETCONTINUOUS_ON_LIFT_ABS |- (!f. (!s. (((absolutely_setcontinuous_on (\\x. (lift (real_abs (f x))))) s) = ((absolutely_setcontinuous_on (\\x. (lift (f x)))) s))))"}
{"_id": "50371", "text": "INT_SGN_POW_2 |- (!x. ((int_sgn ((int_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = (int_sgn (int_abs x))))"}
{"_id": "50372", "text": "BOUNDED_EQ_TOTALLY_BOUNDED |- (!s. ((bounded 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": "50373", "text": "BOUNDED_SUMS_IMAGES |- (!f. (!t. (!s. (((FINITE s) /\\ (!a. ((a IN s) ==> (bounded (GSPEC (\\GEN%PVAR%591. (?x. (((SETSPEC GEN%PVAR%591) (x IN t)) ((f x) a))))))))) ==> (bounded (GSPEC (\\GEN%PVAR%592. (?x. (((SETSPEC GEN%PVAR%592) (x IN t)) ((vsum s) (f x)))))))))))"}
{"_id": "50374", "text": "ANR_SIMPLICIAL_COMPLEX |- (!c. ((simplicial_complex c) ==> (ANR (UNIONS c))))"}
{"_id": "50375", "text": "DIM_OPEN_IN |- (!s. (!t. (((~ (s = EMPTY)) /\\ (((open_in ((subtopology euclidean) t)) s) /\\ (subspace t))) ==> ((dim s) = (dim t)))))"}
{"_id": "50376", "text": "ABSOLUTELY_INTEGRABLE_CONVOLUTION_LINF_L1 |- (!bop. (!f. (!g. (!x. (((bilinear bop) /\\ (((measurable_on f) UNIV) /\\ ((bounded ((IMAGE f) UNIV)) /\\ ((absolutely_integrable_on g) UNIV)))) ==> ((absolutely_integrable_on (\\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))"}
{"_id": "50377", "text": "REAL_MEASURABLE_ON_MEASURABLE_SUBSET |- (!f. (!s. (!t. (((s SUBSET t) /\\ (((real_measurable_on f) t) /\\ (real_measurable s))) ==> ((real_measurable_on f) s)))))"}
{"_id": "50378", "text": "OPEN_IN_CLOSED_IN |- (!s. ((s SUBSET (topspace top)) ==> (((open_in top) s) = ((closed_in top) ((topspace top) DIFF s)))))"}
{"_id": "50379", "text": "REAL_BEPPO_LEVI_INCREASING |- (!f. (!s. (((!k. ((real_integrable_on (f k)) s)) /\\ ((!k. (!x. ((x IN s) ==> ((real_le ((f k) x)) ((f (SUC k)) x))))) /\\ (real_bounded (GSPEC (\\GEN%PVAR%4926. (?k. (((SETSPEC GEN%PVAR%4926) (k IN UNIV)) ((real_integral s) (f k))))))))) ==> (?g. (?k. ((real_negligible k) /\\ (!x. ((x IN (s DIFF k)) ==> (((---> (\\k. ((f k) x))) (g x)) sequentially)))))))))"}
{"_id": "50380", "text": "INTEGRAL_INTERVALS_DIFF_INCLUSION_EXCLUSION |- (!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))))) ==> (((vector_sub ((integral (closed_interval ((CONS (a , d)) NIL))) f)) ((integral (closed_interval ((CONS (a , c)) NIL))) f)) = ((vsum (GSPEC (\\GEN%PVAR%2930. (?s. (((SETSPEC GEN%PVAR%2930) ((~ (s = EMPTY)) /\\ (s SUBSET ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))))) s))))) (\\s. ((% ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (CARD (GSPEC (\\GEN%PVAR%2931. (?i. (((SETSPEC GEN%PVAR%2931) ((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": "50381", "text": "Cuxvzoz.vv_split_azim_alt |- (!vv. (!k. (!i. (!j. ((LET (\\V. (LET_END ((LET (\\E. (LET_END ((LET (\\f. (LET_END ((((periodic vv) k) /\\ (((NUMERAL (BIT1 (BIT1 _0))) <= k) /\\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv j) INSERT EMPTY))))) /\\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv (i + (k - (NUMERAL (BIT1 _0))))) INSERT EMPTY))))) /\\ ((~ (collinear ((vec (NUMERAL _0)) INSERT ((vv i) INSERT ((vv (SUC i)) INSERT EMPTY))))) /\\ ((convex_local_fan (V , (E , f))) /\\ (!i. (!j. (((i < k) /\\ ((j < k) /\\ ((vv i) = (vv j)))) ==> (i = j)))))))))) ==> (((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + (k - (NUMERAL (BIT1 _0)))))) = ((real_add ((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv j))) ((((azim (vec (NUMERAL _0))) (vv i)) (vv j)) (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": "50382", "text": "COMPLEX_IN_SPHERE_0 |- (!v. (!r. ((v IN (sphere ((Cx (real_of_num (NUMERAL _0))) , r))) = ((vector_norm v) = r))))"}
{"_id": "50383", "text": "Tame_list.minimal_min_num |- (minimal = min_num)"}
{"_id": "50384", "text": "Ssrnat.minnAC |- (!m1. (!m2. (!m3. (((minn ((minn m1) m2)) m3) = ((minn ((minn m1) m3)) m2)))))"}
{"_id": "50385", "text": "Lp_ineqs_proofs.dart3_azim3_fan_eq_dih3_y |- (!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)))) ==> (((azim3_fan (V , E)) d) = ((((((dih3_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": "50386", "text": "LOCALLY_CONNECTED_SPHERE |- (!a. (!r. ((locally connected) (sphere (a , r)))))"}
{"_id": "50387", "text": "Merge_ineq.nonf_mud_234_x_v1 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_234_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul a) a)) x2) x3) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) (sqrt (((((((((delta_234_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))"}
{"_id": "50388", "text": "ANTIDERIVATIVE_INTEGRAL_CONTINUOUS |- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?g. (!u. (!v. (((u IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((v IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le (drop u)) (drop v)))) ==> (((has_integral f) ((vector_sub (g v)) (g u))) (closed_interval ((CONS (u , v)) NIL)))))))))))"}
{"_id": "50389", "text": "INT_LT_NEGTOTAL |- (!x. ((x = (int_of_num (NUMERAL _0))) \\/ (((int_lt (int_of_num (NUMERAL _0))) x) \\/ ((int_lt (int_of_num (NUMERAL _0))) (int_neg x)))))"}
{"_id": "50390", "text": "TAN_PERIODIC_NPI |- (!x. (!n. ((tan ((real_add x) ((real_mul (real_of_num n)) pi))) = (tan x))))"}
{"_id": "50391", "text": "CONVEX_ON_IS_INDEFINITE_INTEGRAL |- (!f. (!s. ((((convex_on (drop o f)) s) /\\ ((is_interval s) /\\ (open s))) ==> (?f'. ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f' x))) (drop (f' y)))))) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((real_le (drop x)) (drop y)))) ==> (((has_integral f') ((vector_sub (f y)) (f x))) (closed_interval ((CONS (x , y)) NIL)))))))))))"}
{"_id": "50392", "text": "SERIES_DIRICHLET_COMPLEX_EXPLICIT |- (!f. (!g. (!p. (!q. ((((NUMERAL (BIT1 _0)) <= p) /\\ ((bounded (GSPEC (\\GEN%PVAR%4783. (?n. (((SETSPEC GEN%PVAR%4783) (n IN UNIV)) ((vsum (q .. n)) f)))))) /\\ ((!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)))))))) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\\ (!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) (\\k. ((complex_mul (f k)) (g k)))))) ((real_mul B) (vector_norm (g m))))))))))))))"}
{"_id": "50393", "text": "REAL_UNIFORMLY_CONTINUOUS_ON_ID |- (!s. ((real_uniformly_continuous_on (\\x. x)) s))"}
{"_id": "50394", "text": "AFF_LE_INJECTIVE_LINEAR_IMAGE |- (!f. (!s. (!t. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((aff_le ((IMAGE f) s)) ((IMAGE f) t)) = ((IMAGE f) ((aff_le s) t)))))))"}
{"_id": "50395", "text": "Planarity.remark012_fan |- (!x. (!V. (!E. (!u. (!w. (!a. (((FAN (x , (V , E))) /\\ (((u INSERT (w INSERT EMPTY)) IN E) /\\ (((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 V)))))))))"}
{"_id": "50396", "text": "Ssrfun.inj_id |- (injective I)"}
{"_id": "50397", "text": "Tame_list.indexf_mem1 |- (!x. (!y. (!s. ((~ ((MEM x) s)) ==> ((((indexf x) y) s) = (sizel s))))))"}
{"_id": "50398", "text": "FIXED_POINT_INESSENTIAL_SPHERE_MAP |- (!f. (!a. (!r. (!c. ((((real_lt (real_of_num (NUMERAL _0))) r) /\\ ((((homotopic_with (\\x. T)) ((sphere (a , r)) , (sphere (a , r)))) f) (\\x. c))) ==> (?x. ((x IN (sphere (a , r))) /\\ ((f x) = x))))))))"}
{"_id": "50399", "text": "Lp_ineqs_proofs.y1_def |- (!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))) ==> ((y1_fan d) = (yn_fan (FST d)))))))))))"}
{"_id": "50400", "text": "POLYTOPE_LINEAR_IMAGE_EQ |- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\\ (!y. (?x. ((f x) = y))))) ==> ((polytope ((IMAGE f) s)) = (polytope s)))))"}
{"_id": "50401", "text": "REAL_OPEN_DIFF |- (!s. (!t. (((real_open s) /\\ (real_closed t)) ==> (real_open (s DIFF t)))))"}
{"_id": "50402", "text": "Ssrnat.geqE |- (!m. (!n. ((m >= n) = (n <= m))))"}
{"_id": "50403", "text": "TC_RTC_CASES_L |- ((((TC R) x) z) = (?y. ((((RTC R) x) y) /\\ ((R y) z))))"}
{"_id": "50404", "text": "REAL_MEASURABLE_EMPTY |- (real_measurable EMPTY)"}
{"_id": "50405", "text": "HAS_BOUNDED_REAL_VARIATION_ON_COMBINE |- (!f. (!a. (!b. (!c. ((((real_le a) c) /\\ ((real_le c) b)) ==> (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL))) = (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , c)) NIL))) /\\ ((has_bounded_real_variation_on f) (closed_real_interval ((CONS (c , b)) NIL))))))))))"}
{"_id": "50406", "text": "POSITIVE_SEMIDEFINITE_ZERO_COLUMN |- (!A. (!i. (((positive_semidefinite A) /\\ (((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (((A $ i) $ i) = (real_of_num (NUMERAL _0)))))) ==> (((column i) A) = (vec (NUMERAL _0))))))"}
{"_id": "50407", "text": "UNCOUNTABLE_HAS_CONDENSATION_POINT |- (!s. ((~ (COUNTABLE s)) ==> (?x. ((condensation_point_of x) s))))"}
{"_id": "50408", "text": "RANK_NEG |- (!A. ((rank (matrix_neg A)) = (rank A)))"}
{"_id": "50409", "text": "LOCALLY_COMPACT_CLOSED_INTER_OPEN |- (!s. (((locally compact) s) = (?t. (?u. ((closed t) /\\ ((open u) /\\ (s = (t INTER u))))))))"}
{"_id": "50410", "text": "CLOSED_MAP_FROM_COMPOSITION_SURJECTIVE |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\\ ((((IMAGE f) s) = t) /\\ ((((IMAGE g) t) SUBSET u) /\\ (!k. (((closed_in ((subtopology euclidean) s)) k) ==> ((closed_in ((subtopology euclidean) u)) ((IMAGE (g o f)) k))))))) ==> (!k. (((closed_in ((subtopology euclidean) t)) k) ==> ((closed_in ((subtopology euclidean) u)) ((IMAGE g) k))))))))))"}
{"_id": "50411", "text": "HAS_REAL_DERIVATIVE_SERIES |- (!s. (!f. (!f'. (!g'. (!k. (((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 ((sum (k INTER ((NUMERAL _0) .. n))) (\\i. ((f' i) x)))) (g' x)))) e))))))) /\\ (?x. (?l. ((x IN s) /\\ (((real_sums (\\n. ((f n) x))) l) k))))))) ==> (?g. (!x. ((x IN s) ==> ((((real_sums (\\n. ((f n) x))) (g x)) k) /\\ (((has_real_derivative g) (g' x)) ((within (atreal x)) s))))))))))))"}
{"_id": "50412", "text": "CONIC_HULL_POINTLESS_AS_IMAGE |- (!s. ((((hull conic) s) DELETE (vec (NUMERAL _0))) = ((IMAGE (\\z. ((% (drop (fstcart z))) (sndcart z)))) ((GSPEC (\\GEN%PVAR%1459. (?t. (((SETSPEC GEN%PVAR%1459) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))) PCROSS (s DELETE (vec (NUMERAL _0)))))))"}
{"_id": "50413", "text": "HAS_INTEGRAL_COMPLEX_RMUL |- (!f. (!y. (!i. (!c. ((((has_integral f) y) i) ==> (((has_integral (\\x. ((complex_mul (f x)) c))) ((complex_mul y) c)) i))))))"}
{"_id": "50414", "text": "SUMMABLE_NEG |- (!x. (!s. (((summable s) x) ==> ((summable s) (\\n. (vector_neg (x n)))))))"}
{"_id": "50415", "text": "CENTRE_IN_MCBALL |- (!m. (!x. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\\ (x IN (mspace m))) ==> (x IN ((mcball m) (x , r)))))))"}
{"_id": "50416", "text": "Nonlinear_lemma.ineq_expand10 |- (!a1. (!a2. (!a3. (!a4. (!a5. (!a6. (!a7. (!a8. (!a9. (!a10. (!b1. (!b2. (!b3. (!b4. (!b5. (!b6. (!b7. (!b8. (!b9. (!b10. (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. (!y10. (!P. (((ineq ((CONS (a1 , (y1 , b1))) ((CONS (a2 , (y2 , b2))) ((CONS (a3 , (y3 , b3))) ((CONS (a4 , (y4 , b4))) ((CONS (a5 , (y5 , b5))) ((CONS (a6 , (y6 , b6))) ((CONS (a7 , (y7 , b7))) ((CONS (a8 , (y8 , b8))) ((CONS (a9 , (y9 , b9))) ((CONS (a10 , (y10 , b10))) NIL))))))))))) P) = ((((real_le a1) y1) /\\ ((real_le y1) b1)) ==> ((((real_le a2) y2) /\\ ((real_le y2) b2)) ==> ((((real_le a3) y3) /\\ ((real_le y3) b3)) ==> ((((real_le a4) y4) /\\ ((real_le y4) b4)) ==> ((((real_le a5) y5) /\\ ((real_le y5) b5)) ==> ((((real_le a6) y6) /\\ ((real_le y6) b6)) ==> ((((real_le a7) y7) /\\ ((real_le y7) b7)) ==> ((((real_le a8) y8) /\\ ((real_le y8) b8)) ==> ((((real_le a9) y9) /\\ ((real_le y9) b9)) ==> ((((real_le a10) y10) /\\ ((real_le y10) b10)) ==> P))))))))))))))))))))))))))))))))))))))))))"}
{"_id": "50417", "text": "REALLIM_AT_INFINITY_COMPLEX_0 |- (!f. (!l. ((((---> f) l) at_infinity) = (((---> (f o complex_inv)) l) (at (Cx (real_of_num (NUMERAL _0))))))))"}
{"_id": "50418", "text": "LIM_INFINITY_POSINFINITY_LIFT |- (!f. (!l. ((((--> f) l) at_infinity) ==> (((--> (f o lift)) l) at_posinfinity))))"}
{"_id": "50419", "text": "STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION_AFFINE |- (!f. (!s. (!e. (!t. (((compact s) /\\ (((continuous_on f) s) /\\ (((real_lt (real_of_num (NUMERAL _0))) e) /\\ ((affine 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": "50420", "text": "LEFT_INVERSE_LINEAR |- (!f. (!g. (((linear f) /\\ ((g o f) = I)) ==> (linear g))))"}
{"_id": "50421", "text": "Add_triangle.contracted_in_dart |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d_fan ((vec (NUMERAL _0)) , (V , E)))) ==> ((contracted_dart d) IN (dart_of_fan (V , E))))))))"}
{"_id": "50422", "text": "Lp_ineqs_proofs.tau_hi |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> ((contravening V) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le ((tau_fan (V , E)) f)) tgt)))))))))))"}
{"_id": "50423", "text": "Lp_ineqs_proofs.face_last_dart |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) - (NUMERAL (BIT1 _0)))) (v , w)) = ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v))))))))"}
{"_id": "50424", "text": "BINOMIAL_THEOREM |- (!n. (!x. (!y. (((x + y) EXP n) = ((nsum ((NUMERAL _0) .. n)) (\\k. ((binom (n , k)) * ((x EXP k) * (y EXP (n - k))))))))))"}
{"_id": "50425", "text": "TRANSLATION_EQ_IMP |- (!P. ((!a. (!s. ((P ((IMAGE (\\x. ((vector_add a) x))) s)) = (P s)))) = (!a. (!s. ((P s) ==> (P ((IMAGE (\\x. ((vector_add a) x))) s)))))))"}
{"_id": "50426", "text": "Fan.azim_is_zero_fan |- (!x. (!v. (!u. (((~ (v = x)) /\\ ((~ (u = x)) /\\ (~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))))) ==> (((((azim x) v) u) u) = (real_of_num (NUMERAL _0)))))))"}
{"_id": "50427", "text": "RPOW_MINUS1_QUOTIENT_LE |- (!a. (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) a) /\\ (((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_le x) y))) ==> ((real_le ((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": "50428", "text": "RC_IDEMP |- (!R. ((RC (RC R)) = (RC R)))"}
{"_id": "50429", "text": "Miqmcsn.J_SCS_3T4_prime2 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3T4_prime2) i)) i1) j) = F)"}
{"_id": "50430", "text": "LIM_WITHINREAL_LE |- (!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_le (real_abs ((real_sub x) a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))"}
{"_id": "50431", "text": "HAS_VECTOR_DERIVATIVE_NEG_EQ |- (!f. (!f'. (!net. ((((has_vector_derivative (\\x. (vector_neg (f x)))) (vector_neg f')) net) = (((has_vector_derivative f) f') net)))))"}
{"_id": "50432", "text": "RPOW_1_LE |- (!a. (!x. ((((real_le (real_of_num (NUMERAL _0))) x) /\\ (((real_le x) (real_of_num (NUMERAL (BIT1 _0)))) /\\ ((real_le (real_of_num (NUMERAL _0))) a))) ==> ((real_le ((rpow x) a)) (real_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "50433", "text": "CONNECTED_SUMS |- (!s. (!t. (((connected s) /\\ (connected t)) ==> (connected (GSPEC (\\GEN%PVAR%917. (?x. (?y. (((SETSPEC GEN%PVAR%917) ((x IN s) /\\ (y IN t))) ((vector_add x) y))))))))))"}
{"_id": "50434", "text": "REAL_MUL_COS_SIN |- (!x. (!y. (((real_mul (cos x)) (sin y)) = ((real_div ((real_sub (sin ((real_add x) y))) (sin ((real_sub x) y)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "50435", "text": "Ssrbool.implyNb |- (!a. (!b. (((~ a) ==> b) = (a \\/ b))))"}
{"_id": "50436", "text": "Merge_ineq.nonf_mud_135_x_v1 |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (((real_le (real_of_num (NUMERAL _0))) a) ==> ((((((((((mud_135_x_v1 a) b) c) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul a) a)) x1) x3) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) (sqrt (((((((((delta_135_x ((real_mul a) a)) ((real_mul b) b)) ((real_mul c) c)) x1) x2) x3) x4) x5) x6))))))))))))))"}
{"_id": "50437", "text": "LEBESGUE_MEASURABLE_PREIMAGE_OPEN |- (!f. (!t. ((((measurable_on f) UNIV) /\\ (open t)) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3550. (?x. (((SETSPEC GEN%PVAR%3550) ((f x) IN t)) x))))))))"}
{"_id": "50438", "text": "Nonlinear_lemma.edge_quadratic |- (!x1. (!x2. (!x3. (!x5. (!x6. ((quadratic_root_plus (x1 , (((real_add ((real_mul x1) x1)) ((real_sub ((real_mul ((real_sub x3) x5)) ((real_sub x2) x6))) ((real_mul x1) ((real_add x2) ((real_add x3) ((real_add x5) x6)))))) , ((real_add ((real_mul x1) ((real_mul x3) x5))) ((real_sub ((real_sub ((real_mul x1) ((real_mul x2) x6))) ((real_mul x3) ((real_mul ((real_add x1) ((real_add ((real_sub x2) x3)) ((real_sub x5) x6)))) x6)))) ((real_mul x2) ((real_mul x5) ((real_add ((real_sub x1) x2)) ((real_add ((real_sub x3) x5)) x6))))))))) = ((real_div ((real_add (real_neg ((real_mul x1) x1))) ((real_add ((real_mul x1) x2)) ((real_add ((real_sub ((real_mul x1) x3)) ((real_mul x2) x3))) ((real_add ((real_mul x1) x5)) ((real_add ((real_mul x2) x5)) ((real_add ((real_mul x1) x6)) ((real_add ((real_sub ((real_mul x3) x6)) ((real_mul x5) x6))) (sqrt ((real_mul (((ups_x x1) x3) x5)) (((ups_x x1) x2) x6))))))))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x1))))))))"}
{"_id": "50439", "text": "Seq.leq_size_perm |- (!s1. (!s2. ((uniq s1) ==> ((!x. (((MEM x) s1) ==> ((MEM x) s2))) ==> (((sizel s2) <= (sizel s1)) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) /\\ ((sizel s1) = (sizel s2))))))))"}
{"_id": "50440", "text": "LIM_CONG_ATREAL |- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\\x. (f x))) l) (atreal a)) = (((--> g) l) (atreal a))))"}
{"_id": "50441", "text": "HOMEOMORPHIC_SCALING_RIGHT |- (!c. (((real_lt (real_of_num (NUMERAL _0))) c) ==> (!s. (!t. (((homeomorphic s) ((IMAGE (\\x. ((% c) x))) t)) = ((homeomorphic s) t))))))"}
{"_id": "50442", "text": "REAL_INTEGRAL_REFLECT |- (!f. (!a. (!b. (((real_integral (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) (\\x. (f (real_neg x)))) = ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) f)))))"}
{"_id": "50443", "text": "Ssrnat.ltn_pexp2l |- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> (((m EXP n1) < (m EXP n2)) ==> (n1 < n2))))))"}
{"_id": "50444", "text": "STC_INC_TC |- (!R. (!x. (!y. ((((TC R) x) y) ==> (((STC R) x) y)))))"}
{"_id": "50445", "text": "OPEN_MAP_IFF_CLOSED_MAP_BIJECTIVE |- (!f. (!s. (!t. (((((IMAGE f) s) = t) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y))))) ==> ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) = (!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c)))))))))"}
{"_id": "50446", "text": "Nonlinear_lemma.vol2r_y |- (!y. (!r. (((vol2r y) r) = ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul pi) ((real_mul r) ((real_mul r) (real_inv (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) ((real_mul pi) ((real_mul (real_inv (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0)))))))))))"}
{"_id": "50447", "text": "LOCALLY_CONNECTED_SPHERE_GEN |- (!s. (((bounded s) /\\ (convex s)) ==> ((locally connected) (relative_frontier s))))"}
{"_id": "50448", "text": "Wrgcvdr_cizmrrh.hypermap |- (!H. (((tuple_hypermap H) = (D , (e , (n , f)))) ==> (((permutes e) D) /\\ (((permutes n) D) /\\ (((permutes f) D) /\\ ((e o (n o f)) = I))))))"}
{"_id": "50449", "text": "SUPERADMISSIBLE_MATCH_SEQPATTERN |- (!<<. (!p. (!s. (!c1. (!c2. (!e. ((((((admissible <<) p) s) (\\f. (\\x. (?y. ((((c1 f) x) ((e f) x)) y))))) /\\ (((((superadmissible <<) (\\f. (\\x. (((p f) x) /\\ (?y. ((((c1 f) x) ((e f) x)) y)))))) s) (\\f. (\\x. ((_MATCH ((e f) x)) ((c1 f) x))))) /\\ ((((superadmissible <<) (\\f. (\\x. (((p f) x) /\\ (~ (?y. ((((c1 f) x) ((e f) x)) y))))))) s) (\\f. (\\x. ((_MATCH ((e f) x)) ((c2 f) x))))))) ==> ((((superadmissible <<) p) s) (\\f. (\\x. ((_MATCH ((e f) x)) ((_SEQPATTERN ((c1 f) x)) ((c2 f) x)))))))))))))"}
{"_id": "50450", "text": "SEPARATION_NORMAL_COMPACT |- (!s. (!t. (((compact s) /\\ ((closed t) /\\ ((s INTER t) = EMPTY))) ==> (?u. (?v. ((open u) /\\ ((compact (closure u)) /\\ ((open v) /\\ ((s SUBSET u) /\\ ((t SUBSET v) /\\ ((u INTER v) = EMPTY)))))))))))"}
{"_id": "50451", "text": "Upfzbzm_support_lemmas.john_harrison_lemma1 |- ((GABS (\\f. (!x. (!y. ((GEQ (f (x , y))) ((P x) y)))))) = (\\p. ((P (FST p)) (SND p))))"}
{"_id": "50452", "text": "INT_LT_ADDL |- (!x. (!y. (((int_lt y) ((int_add x) y)) = ((int_lt (int_of_num (NUMERAL _0))) x))))"}
{"_id": "50453", "text": "ORTHOGONAL_BASIS_EXISTS |- (!v. (?b. ((independent b) /\\ ((b SUBSET (span v)) /\\ ((v SUBSET (span b)) /\\ ((b HAS_SIZE (dim v)) /\\ ((pairwise orthogonal) b)))))))"}
{"_id": "50454", "text": "FSIGMA_IMP_LEBESGUE_MEASURABLE |- (!s. ((fsigma s) ==> (lebesgue_measurable s)))"}
{"_id": "50455", "text": "NPRODUCT_DELTA |- (!s. (!a. (((nproduct s) (\\x. (((COND (x = a)) b) (NUMERAL (BIT1 _0))))) = (((COND (a IN s)) b) (NUMERAL (BIT1 _0))))))"}
{"_id": "50456", "text": "CLOSED_MAP_CLOSURES |- (!f. ((!s. ((closed s) ==> (closed ((IMAGE f) s)))) = (!s. ((closure ((IMAGE f) s)) SUBSET ((IMAGE f) (closure s))))))"}
{"_id": "50457", "text": "Ssrnat.sqrn_add_sub |- (!m. (!n. ((n <= m) ==> ((((m + n) EXP (NUMERAL (BIT0 (BIT1 _0)))) - ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) * (m * n))) = ((m - n) EXP (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "50458", "text": "COMPLEX_IN_BALL_0 |- (!v. (!r. ((v IN (ball ((Cx (real_of_num (NUMERAL _0))) , r))) = ((real_lt (vector_norm v)) r))))"}
{"_id": "50459", "text": "NORM_CCOS_POW_2 |- (!z. (((real_pow (vector_norm (ccos z))) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_add ((real_pow (cos (Re z))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_div ((real_pow ((real_sub (exp (Im z))) (real_inv (exp (Im z))))) (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))"}
{"_id": "50460", "text": "Seq.unzip2_zip |- (!s. (!t. (((sizel t) <= (sizel s)) ==> ((unzip2 ((zip s) t)) = t))))"}
{"_id": "50461", "text": "EXPOSED_POINT_OF_FURTHEST_POINT |- (!s. (!a. (!b. (((b IN s) /\\ (!x. ((x IN s) ==> ((real_le (distance (a , x))) (distance (a , b)))))) ==> ((exposed_face_of (b INSERT EMPTY)) s)))))"}
{"_id": "50462", "text": "CONVEX_NORM |- (!s. ((convex_on vector_norm) s))"}
{"_id": "50463", "text": "Tame_defs2.good_list_nodes |- (!L. ((good_list_nodes L) = ((node_set (hypermap_of_list L)) = (set_of_list (nodes_of_list L)))))"}
{"_id": "50464", "text": "DEPENDENT_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\\ ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) /\\ (dependent s))) ==> (dependent ((IMAGE f) s)))))"}
{"_id": "50465", "text": "SUBSPACE_REAL |- (subspace real)"}
{"_id": "50466", "text": "ANR_INSERT |- (!s. (!a. (((closed s) /\\ (ANR s)) ==> (ANR (a INSERT s)))))"}
{"_id": "50467", "text": "HAS_VECTOR_DERIVATIVE_ID |- (!net. (((has_vector_derivative (\\x. x)) (vec (NUMERAL (BIT1 _0)))) net))"}
{"_id": "50468", "text": "REAL_INTEGRABLE_REAL_BOUNDED_VARIATION_PRODUCT |- (!f. (!g. (!a. (!b. ((((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((has_bounded_real_variation_on g) (closed_real_interval ((CONS (a , b)) NIL)))) ==> ((real_integrable_on (\\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "50469", "text": "HAS_REAL_INTEGRAL_NEGLIGIBLE_EQ |- (!f. (!s. ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f x)))) ==> ((((has_real_integral f) (real_of_num (NUMERAL _0))) s) = (real_negligible (GSPEC (\\GEN%PVAR%4964. (?x. (((SETSPEC GEN%PVAR%4964) ((x IN s) /\\ (~ ((f x) = (real_of_num (NUMERAL _0)))))) x)))))))))"}
{"_id": "50470", "text": "CARD_EXP_C |- (!s. (!t. (((FINITE s) /\\ (FINITE t)) ==> ((CARD ((^_c s) t)) = ((CARD s) EXP (CARD t))))))"}
{"_id": "50471", "text": "Nonlinear_lemma.gcy_high_hplus |- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) hplus)) y) ==> ((gcy y) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_div mm1) pi)))))"}
{"_id": "50472", "text": "NPRODUCT_FACT |- (!n. (((nproduct ((NUMERAL (BIT1 _0)) .. n)) (\\m. m)) = (FACT n)))"}
{"_id": "50473", "text": "INT_NEG_LT0 |- (!x. (((int_lt (int_neg x)) (int_of_num (NUMERAL _0))) = ((int_lt (int_of_num (NUMERAL _0))) x)))"}
{"_id": "50474", "text": "CONVEX_CONE_PCROSS_EQ |- (!s. (!t. ((convex_cone (s PCROSS t)) = ((convex_cone s) /\\ (convex_cone t)))))"}
{"_id": "50475", "text": "REAL_UNIFORMLY_CONTINUOUS_ON_COMPOSE |- (!f. (!g. (!s. ((((real_uniformly_continuous_on f) s) /\\ ((real_uniformly_continuous_on g) ((IMAGE f) s))) ==> ((real_uniformly_continuous_on (g o f)) s)))))"}
{"_id": "50476", "text": "TAGGED_DIVISION_OF_NONTRIVIAL |- (!s. (!a. (!b. ((((tagged_division_of s) (closed_interval ((CONS (a , b)) NIL))) /\\ (~ ((content (closed_interval ((CONS (a , b)) NIL))) = (real_of_num (NUMERAL _0))))) ==> ((tagged_division_of (GSPEC (\\GEN%PVAR%2765. (?x. (?k. (((SETSPEC GEN%PVAR%2765) (((x , k) IN s) /\\ (~ ((content k) = (real_of_num (NUMERAL _0)))))) (x , k))))))) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "50477", "text": "Lp_gen_theory.iso_face_trans |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!P. ((!f. ((f IN (face_set H)) ==> (P f))) ==> (!f. ((f IN (face_set G)) ==> (P ((IMAGE g) f))))))))))"}
{"_id": "50478", "text": "Seq.mem_seq4 |- (!x. (!y1. (!y2. (!y3. (!y4. (((MEM x) ((CONS y1) ((CONS y2) ((CONS y3) ((CONS y4) NIL))))) = ((x = y1) \\/ ((x = y2) \\/ ((x = y3) \\/ (x = y4))))))))))"}
{"_id": "50479", "text": "REAL_OPEN_REAL_CLOSED |- (!s. ((real_open s) = (real_closed (UNIV DIFF s))))"}
{"_id": "50480", "text": "Ssrbool.implybE |- (!a. (!b. ((a ==> b) = ((~ a) \\/ b))))"}
{"_id": "50481", "text": "IMAGE_MATRIX_INV |- (!A. (((IMAGE (\\x. ((matrix_vector_mul (matrix_inv A)) x))) UNIV) = ((IMAGE (\\x. ((matrix_vector_mul (transp A)) x))) UNIV)))"}
{"_id": "50482", "text": "ANR_DELETE |- (!s. (!a. ((ANR s) ==> (ANR (s DELETE a)))))"}
{"_id": "50483", "text": "RECTIFIABLE_PATH_DIFFERENTIABLE |- (!g. (!s. (((COUNTABLE s) /\\ ((path g) /\\ (!t. ((t IN ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) DIFF s)) ==> ((differentiable g) (at t)))))) ==> ((rectifiable_path g) = ((absolutely_integrable_on (\\t. ((vector_derivative g) (at t)))) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))))))"}
{"_id": "50484", "text": "COVERING_SPACE_COUNTABLE_SHEETS |- (!p. (!c. (!s. (!b. (((covering_space (c , p)) s) ==> (COUNTABLE (GSPEC (\\GEN%PVAR%2073. (?x. (((SETSPEC GEN%PVAR%2073) ((x IN c) /\\ ((p x) = b))) x))))))))))"}
{"_id": "50485", "text": "COUNTABLE_DIFF_FINITE |- (!s. (!t. ((FINITE s) ==> ((COUNTABLE (t DIFF s)) = (COUNTABLE t)))))"}
{"_id": "50486", "text": "Hypermap_and_fan.F_HAS_NO_FIXED_POINTS_IN_D1 |- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\\ (x IN (dart1_of_fan (V , E)))) ==> (~ (((f_fan_pair (V , E)) x) = x))))))"}
{"_id": "50487", "text": "NOT_EMPTY_INSERT |- (!x. (!s. (~ (EMPTY = (x INSERT s)))))"}
{"_id": "50488", "text": "Lp_ineqs_proofs.azim3_hi |- (!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 ((azim3_fan (V , E)) d)) pi)))))))))"}
{"_id": "50489", "text": "Functional_equation.functional_ups234_x |- (ups234_x = (((((((compose6 ups_126) proj_x2) proj_x3) dummy6) dummy6) dummy6) proj_x4))"}
{"_id": "50490", "text": "Add_triangle.fan_hypermap_map2 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d1_fan ((vec (NUMERAL _0)) , (V , E)))) ==> ((((face_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((f1_fan (vec (NUMERAL _0))) V) E) d)) /\\ ((((node_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((n_fan (vec (NUMERAL _0))) V) E) d)) /\\ (((edge_map (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))) d) = ((((e_fan (vec (NUMERAL _0))) V) E) d)))))))))"}
{"_id": "50491", "text": "Float.INT_POS_NEG |- (!n. (((int_lt (int_of_num (NUMERAL _0))) (int_neg (int_of_num n))) = F))"}
{"_id": "50492", "text": "Ssrnat.distn0 |- (!n. (((distn n) (NUMERAL _0)) = n))"}
{"_id": "50493", "text": "VOLUME_FRUSTT |- (!v0. (!v1. (!h. (!a. (((real_lt (real_of_num (NUMERAL _0))) 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": "50494", "text": "LIMPT_OF_OPEN |- (!s. (!x. (((open s) /\\ (x IN s)) ==> ((limit_point_of x) s))))"}
{"_id": "50495", "text": "COMPLEX_DIFFERENTIABLE_CARATHEODORY_WITHIN |- (!f. (!z. (!s. (((complex_differentiable 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))))))))"}
{"_id": "50496", "text": "Pack2.POLYTOPE_VORONOI_CLOSED |- (!s. (!v. (((packing s) /\\ ((saturated s) /\\ (v IN s))) ==> (polytope ((voronoi_closed s) v)))))"}
{"_id": "50497", "text": "PROPER_MAP_FROM_COMPOSITION_LEFT |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\\ ((((IMAGE f) s) = t) /\\ (((continuous_on g) t) /\\ ((((IMAGE g) t) SUBSET u) /\\ (!k. (((k SUBSET u) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%738. (?x. (((SETSPEC GEN%PVAR%738) ((x IN s) /\\ (((g o f) x) IN k))) x))))))))))) ==> (!k. (((k SUBSET u) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%739. (?x. (((SETSPEC GEN%PVAR%739) ((x IN t) /\\ ((g x) IN k))) x)))))))))))))"}
{"_id": "50498", "text": "UNIFORMLY_CONTINUOUS_ON_COMPLEX_MUL |- (!f. (!g. (!s. ((((uniformly_continuous_on f) s) /\\ (((uniformly_continuous_on g) s) /\\ ((bounded ((IMAGE f) s)) /\\ (bounded ((IMAGE g) s))))) ==> ((uniformly_continuous_on (\\x. ((complex_mul (f x)) (g x)))) s)))))"}
{"_id": "50499", "text": "NSUM_EQ_GENERAL_INVERSES |- (!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))))))) ==> (((nsum s) f) = ((nsum t) g)))))))))"}
{"_id": "50500", "text": "HAS_BOUNDED_REAL_VARIATION_ON_SUB |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\\ ((has_bounded_real_variation_on g) s)) ==> ((has_bounded_real_variation_on (\\x. ((real_sub (f x)) (g x)))) s)))))"}
{"_id": "50501", "text": "RECTIFIABLE_PATH_REPARAMETRIZATION |- (!g. (!h. (!h'. (((rectifiable_path g) /\\ ((homeomorphism ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) , (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)))) (h , h'))) ==> (rectifiable_path (g o h))))))"}
{"_id": "50502", "text": "Tame_list.hideDups_eq_nil |- (!s. (((hideDups s) = NIL) = (s = NIL)))"}
{"_id": "50503", "text": "Nonlinear_lemma.sqrt_x2 |- (((((((sqrt_x2 x1) x2) x3) x4) x5) x6) = (sqrt x2))"}
{"_id": "50504", "text": "HAS_REAL_DERIVATIVE_DERIVATIVE |- (!f. (!f'. (!x. ((((has_real_derivative f) f') (atreal x)) ==> (((real_derivative f) x) = f')))))"}
{"_id": "50505", "text": "CONVEX_CONE_HULL_CONVEX_HULL |- (!s. (((hull convex_cone) s) = ((vec (NUMERAL _0)) INSERT (GSPEC (\\GEN%PVAR%1701. (?c. (?x. (((SETSPEC GEN%PVAR%1701) (((real_le (real_of_num (NUMERAL _0))) c) /\\ (x IN ((hull convex) s)))) ((% c) x)))))))))"}
{"_id": "50506", "text": "Pqcsxwg.REAL_CONTINUOUS_WITHIN_SQRT_COMPOSE |- (!f. (!s. (!a. ((((real_continuous f) ((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)))))) ==> ((real_continuous (\\x. (sqrt (f x)))) ((within (at a)) s))))))"}
{"_id": "50507", "text": "Lp_ineqs_proofs.rhazim_lo |- (!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 _0))) ((rhazim_fan (V , E)) d))))))))))"}
{"_id": "50508", "text": "REAL_INFSUM_RESTRICT |- (!k. (!a. (((real_infsum UNIV) (\\n. (((COND (n IN k)) (a n)) (real_of_num (NUMERAL _0))))) = ((real_infsum k) a))))"}
{"_id": "50509", "text": "Tame_lemmas.reflection_eq2 |- (!n. (!x. ((((reflection n) x) = x) = (((dot x) n) = (real_of_num (NUMERAL _0))))))"}
{"_id": "50510", "text": "Tame_list.nextElem_ALT |- (((((nextElem NIL) b) x) = b) /\\ ((((nextElem ((CONS a) as)) b) x) = (((COND (x = a)) (((COND (as = NIL)) b) ((LET (\\a_. (LET_END ((LET (\\as_. (LET_END a_))) (TL as))))) (HD as)))) (((nextElem as) b) x))))"}
{"_id": "50511", "text": "Seq.has_mask_cons |- (!a. (!b. (!m. (!x. (!s. (((has a) ((mask ((CONS b) m)) ((CONS x) s))) = ((b /\\ (a x)) \\/ ((has a) ((mask m) s)))))))))"}
{"_id": "50512", "text": "ORTHOGONAL_MATRIX_NORM_EQ |- (!A. ((orthogonal_matrix A) = (!x. ((vector_norm ((matrix_vector_mul A) x)) = (vector_norm x)))))"}
{"_id": "50513", "text": "Tame_general.RUNOQPQ |- (!H. ((tame_planar_hypermap H) ==> (restricted_hypermap H)))"}
{"_id": "50514", "text": "Bump.MCELL2_PARAM_UL |- (!V. (!ul. (!vl. (!X. (((packing V) /\\ ((saturated V) /\\ ((X = ((mcell2 V) ul)) /\\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\\ ((~ (negligible X)) /\\ (vl = (SND ((cell_params V) X)))))))) ==> ((X = ((mcell2 V) vl)) /\\ (vl IN ((barV V) (NUMERAL (BIT1 (BIT1 _0)))))))))))"}
{"_id": "50515", "text": "DECREASING_LEFT_LIMIT |- (!f. (!a. (!b. (!c. (((!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))))) /\\ (c IN (closed_real_interval ((CONS (a , b)) NIL)))) ==> (?l. (((---> f) l) ((within (atreal c)) (closed_real_interval ((CONS (a , c)) NIL))))))))))"}
{"_id": "50516", "text": "MEASURE_LIPSCHITZ_IMAGE |- (!f. (!s. (!B. ((((dimindex UNIV) <= (dimindex UNIV)) /\\ ((measurable 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))))))))) ==> ((real_le (measure ((IMAGE f) s))) ((real_mul ((real_pow B) (dimindex UNIV))) (measure s)))))))"}
{"_id": "50517", "text": "IM_COMPLEX_INV_EQ_0 |- (!z. (((Im (complex_inv z)) = (real_of_num (NUMERAL _0))) = ((Im z) = (real_of_num (NUMERAL _0)))))"}
{"_id": "50518", "text": "CARD_EQ_COUNTABLE_SUBSETS_REAL |- ((GSPEC (\\GEN%PVAR%73. (?s. (((SETSPEC GEN%PVAR%73) (COUNTABLE s)) s)))) =_c UNIV)"}
{"_id": "50519", "text": "QUOTIENT_MAP_FROM_SUBSET |- (!f. (!s. (!t. (!u. ((((continuous_on f) t) /\\ ((((IMAGE f) t) SUBSET u) /\\ ((s SUBSET t) /\\ ((((IMAGE f) s) = u) /\\ (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%669. (?x. (((SETSPEC GEN%PVAR%669) ((x IN s) /\\ ((f x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))) ==> (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) t)) (GSPEC (\\GEN%PVAR%670. (?x. (((SETSPEC GEN%PVAR%670) ((x IN t) /\\ ((f x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))))"}
{"_id": "50520", "text": "REAL_DIFFERENTIABLE_BOUND |- (!f. (!f'. (!s. (!B. (((is_realinterval s) /\\ (!x. ((x IN s) ==> ((((has_real_derivative f) (f' x)) ((within (atreal x)) s)) /\\ ((real_le (real_abs (f' x))) B))))) ==> (!x. (!y. (((x IN s) /\\ (y IN s)) ==> ((real_le (real_abs ((real_sub (f x)) (f y)))) ((real_mul B) (real_abs ((real_sub x) y))))))))))))"}
{"_id": "50521", "text": "RELATIVE_ORIENTATION_COMPOSE |- (!f. (!g. (!c. ((((simplex ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))) c) /\\ (~ (((relative_orientation f) c) = (real_of_num (NUMERAL _0))))) ==> (((relative_orientation (g o f)) c) = ((real_mul ((relative_orientation g) ((vertex_image f) c))) ((relative_orientation f) c)))))))"}
{"_id": "50522", "text": "DIFFERENTIABLE_ON_MUL |- (!f. (!g. (!s. ((((differentiable_on (lift o f)) s) /\\ ((differentiable_on g) s)) ==> ((differentiable_on (\\x. ((% (f x)) (g x)))) s)))))"}
{"_id": "50523", "text": "CONTINUOUS_ON_CX_IM |- (!s. ((continuous_on (Cx o Im)) s))"}
{"_id": "50524", "text": "ORTHOGONAL_TRANSFORMATION_BALL |- (!f. (!r. ((orthogonal_transformation f) ==> (((IMAGE f) (ball ((vec (NUMERAL _0)) , r))) = (ball ((vec (NUMERAL _0)) , r))))))"}
{"_id": "50525", "text": "REAL_SUMMABLE_IMP_REAL_SUMS_BOUNDED |- (!f. (!k. (((real_summable (from k)) f) ==> (real_bounded (GSPEC (\\GEN%PVAR%4863. (?n. (((SETSPEC GEN%PVAR%4863) (n IN UNIV)) ((sum (k .. n)) f)))))))))"}
{"_id": "50526", "text": "PATH_CONNECTED_SCALING |- (!s. (!c. ((path_connected s) ==> (path_connected ((IMAGE (\\x. ((% c) x))) s)))))"}
{"_id": "50527", "text": "INT_LE_SQUARE_ABS |- (!x. (!y. (((int_le (int_abs x)) (int_abs y)) = ((int_le ((int_pow x) (NUMERAL (BIT0 (BIT1 _0))))) ((int_pow y) (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "50528", "text": "INT_LTE_TOTAL |- (!x. (!y. (((int_lt x) y) \\/ ((int_le y) x))))"}
{"_id": "50529", "text": "CLOG_NEG |- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((clog (vector_neg z)) = (((COND (((real_le (Im z)) (real_of_num (NUMERAL _0))) /\\ (~ (((real_lt (Re z)) (real_of_num (NUMERAL _0))) /\\ ((Im z) = (real_of_num (NUMERAL _0))))))) ((vector_add (clog z)) ((complex_mul ii) (Cx pi)))) ((vector_sub (clog z)) ((complex_mul ii) (Cx pi)))))))"}
{"_id": "50530", "text": "Seq2.next_el_hd_belast |- (!h. (!t. (((NUMERAL (BIT1 _0)) < (sizel t)) ==> (((next_el ((belast h) t)) h) = (HD t)))))"}
{"_id": "50531", "text": "HOMOTOPIC_NEARBY_PATHS |- (!g. (!s. (((path g) /\\ ((open s) /\\ ((path_image g) SUBSET s))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (!h. (((path h) /\\ (((pathstart h) = (pathstart g)) /\\ (((pathfinish h) = (pathfinish g)) /\\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> ((real_lt (vector_norm ((vector_sub (h t)) (g t)))) e)))))) ==> (((homotopic_paths s) g) h))))))))"}
{"_id": "50532", "text": "REAL_INTEGRABLE_RESTRICT_INTER |- (!f. (!s. (!t. (((real_integrable_on (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) t) = ((real_integrable_on f) (s INTER t))))))"}
{"_id": "50533", "text": "PSUBSET_INSERT_SUBSET |- (!s. (!t. ((s PSUBSET t) = (?x. ((~ (x IN s)) /\\ ((x INSERT s) SUBSET t))))))"}
{"_id": "50534", "text": "LIM_UNION_UNIV |- (!f. (!x. (!l. (!s. (!t. (((((--> f) l) ((within (at x)) s)) /\\ ((((--> f) l) ((within (at x)) t)) /\\ ((s UNION t) = UNIV))) ==> (((--> f) l) (at x))))))))"}
{"_id": "50535", "text": "DIST_FSTCART |- (!x. (!y. ((real_le (distance ((fstcart x) , (fstcart y)))) (distance (x , y)))))"}
{"_id": "50536", "text": "INT_DIFFSQ |- (!x. (!y. (((int_mul ((int_add x) y)) ((int_sub x) y)) = ((int_sub ((int_mul x) x)) ((int_mul y) y)))))"}
{"_id": "50537", "text": "INT_NEG_SUB |- (!x. (!y. ((int_neg ((int_sub x) y)) = ((int_sub y) x))))"}
{"_id": "50538", "text": "COMPLEX_EQ_ADD_RCANCEL |- (!x. (!y. (!z. ((((vector_add x) z) = ((vector_add y) z)) = (x = y)))))"}
{"_id": "50539", "text": "PRODUCT_NEG_NUMSEG_1 |- (!f. (!n. (((product ((NUMERAL (BIT1 _0)) .. n)) (\\i. (real_neg (f i)))) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) n)) ((product ((NUMERAL (BIT1 _0)) .. n)) f)))))"}
{"_id": "50540", "text": "REAL_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG |- (!f. (!s. (((FINITE (GSPEC (\\GEN%PVAR%4934. (?x. (((SETSPEC GEN%PVAR%4934) ((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))))))) ==> ((real_measure (UNIONS ((IMAGE f) s))) = ((sum s) (\\x. (real_measure (f x))))))))"}
{"_id": "50541", "text": "HELLY_COMPACT |- (!f. ((((FINITE f) ==> ((CARD f) >= ((dimindex UNIV) + (NUMERAL (BIT1 _0))))) /\\ ((!s. ((s IN f) ==> ((convex s) /\\ (compact s)))) /\\ (!t. (((t SUBSET f) /\\ ((FINITE t) /\\ ((CARD t) = ((dimindex UNIV) + (NUMERAL (BIT1 _0)))))) ==> (~ ((INTERS t) = EMPTY)))))) ==> (~ ((INTERS f) = EMPTY))))"}
{"_id": "50542", "text": "Tame_list.BUTLAST_NIL |- (!xs. (((NUMERAL (BIT1 _0)) < (sizel xs)) ==> (~ ((BUTLAST xs) = NIL))))"}
{"_id": "50543", "text": "REAL_SUMMABLE_ZETA_INTEGER |- (!n. (!m. (((NUMERAL (BIT0 (BIT1 _0))) <= m) ==> ((real_summable (from n)) (\\k. (real_inv ((real_pow (real_of_num k)) m)))))))"}
{"_id": "50544", "text": "REAL_CLOSED_OPEN_INTERVAL |- (!a. (!b. (((real_le a) b) ==> ((closed_real_interval ((CONS (a , b)) NIL)) = ((open_real_interval (a , b)) UNION (a INSERT (b INSERT EMPTY)))))))"}
{"_id": "50545", "text": "Packing3.HULL_INTER_EQ_INTER |- (!P. (!s. (!t. (((P s) /\\ (P t)) ==> (((hull P) (s INTER t)) = (s INTER t))))))"}
{"_id": "50546", "text": "Constants_approx.tgt_interval |- ((interval_arith 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)))))))))))) , ((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": "50547", "text": "Lp_main_estimate.ineq_tau4_diag3_std |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\\ ((packing V) /\\ ((V SUBSET ball_annulus) /\\ (((CARD V) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\\ ((!v. (!w. (((~ (v = w)) /\\ ((v IN V) /\\ ((w IN V) /\\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) (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 ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((tau_fun V) E) FF)))))) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((y4'_fan (V , E)) d)) /\\ ((real_le (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d)))) ==> ((real_le ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))))"}
{"_id": "50548", "text": "ADMISSIBLE_MATCH_SEQPATTERN |- (!<<. (!p. (!s. (!c1. (!c2. (!e. ((((((admissible <<) p) s) (\\f. (\\x. (?y. ((((c1 f) x) ((e f) x)) y))))) /\\ (((((admissible <<) (\\f. (\\x. (((p f) x) /\\ (?y. ((((c1 f) x) ((e f) x)) y)))))) s) (\\f. (\\x. ((_MATCH ((e f) x)) ((c1 f) x))))) /\\ ((((admissible <<) (\\f. (\\x. (((p f) x) /\\ (~ (?y. ((((c1 f) x) ((e f) x)) y))))))) s) (\\f. (\\x. ((_MATCH ((e f) x)) ((c2 f) x))))))) ==> ((((admissible <<) p) s) (\\f. (\\x. ((_MATCH ((e f) x)) ((_SEQPATTERN ((c1 f) x)) ((c2 f) x)))))))))))))"}
{"_id": "50549", "text": "REAL_DIFFERENTIABLE_WITHIN |- (!f. (!a. (!s. (((real_differentiable f) ((within (atreal x)) s)) = ((differentiable (lift o (f o drop))) ((within (at (lift x))) ((IMAGE lift) s)))))))"}
{"_id": "50550", "text": "REAL_LOG_CONVEX_PRODUCT |- (!f. (!s. (!k. (((FINITE k) /\\ (!i. ((i IN k) ==> ((real_log_convex_on (\\x. ((f x) i))) s)))) ==> ((real_log_convex_on (\\x. ((product k) (f x)))) s)))))"}
{"_id": "50551", "text": "HOLOMORPHIC_ON_CLOG |- (!s. ((!z. (((z IN s) /\\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z)))) ==> ((holomorphic_on clog) s)))"}
{"_id": "50552", "text": "REAL_INTEGRABLE_SPIKE_EQ |- (!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": "50553", "text": "AFFINE_PCROSS_EQ |- (!s. (!t. ((affine (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((affine s) /\\ (affine t)))))))"}
{"_id": "50554", "text": "Rdwkarc.SATURATED_TRANS |- (!V. (!x. ((saturated V) ==> (saturated (GSPEC (\\GEN%PVAR%6925. (?u. (((SETSPEC GEN%PVAR%6925) (((vector_add u) x) IN V)) u))))))))"}
{"_id": "50555", "text": "EMPTY_INTERIOR_AFF_DIM |- (!s. (((int_lt (aff_dim s)) (int_of_num (dimindex UNIV))) ==> ((interior s) = EMPTY)))"}
{"_id": "50556", "text": "CONVEX_AFF_LT |- (!s. (!t. (convex ((aff_lt s) t))))"}
{"_id": "50557", "text": "Ssrbool.iffRLn |- (!Q. (!P. ((P = Q) ==> ((~ Q) ==> (~ P)))))"}
{"_id": "50558", "text": "REAL_SUMMABLE_ADD |- (!x. (!y. (!s. ((((real_summable s) x) /\\ ((real_summable s) y)) ==> ((real_summable s) (\\n. ((real_add (x n)) (y n))))))))"}
{"_id": "50559", "text": "Seq.all_predI |- (!a1. (!a2. (!s. (((all ((predI a1) a2)) s) = (((all a1) s) /\\ ((all a2) s))))))"}
{"_id": "50560", "text": "REAL_CONTINUOUS_INV_ATREAL |- (!f. (!a. ((((real_continuous f) (atreal a)) /\\ (~ ((f a) = (real_of_num (NUMERAL _0))))) ==> ((real_continuous (\\x. (real_inv (f x)))) (atreal a)))))"}
{"_id": "50561", "text": "COMPLEX_ADD_SUB |- (!x. (!y. (((vector_sub ((vector_add x) y)) x) = y)))"}
{"_id": "50562", "text": "SUM_PERMUTE_NUMSEG |- (!f. (!p. (!m. (!n. (((permutes p) (m .. n)) ==> (((sum (m .. n)) f) = ((sum (m .. n)) (f o p))))))))"}
{"_id": "50563", "text": "FARKAS_LEMMA_ALT |- (!A. (!b. ((?x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (((matrix_vector_mul A) x) $ i)) (b $ i))))) = (~ (?y. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (y $ i)))) /\\ ((((vector_matrix_mul y) A) = (vec (NUMERAL _0))) /\\ ((real_lt ((dot b) y)) (real_of_num (NUMERAL _0))))))))))"}
{"_id": "50564", "text": "LOCALLY_CONNECTED_PATH_IMAGE |- (!p. ((path p) ==> ((locally connected) (path_image p))))"}
{"_id": "50565", "text": "HAS_BOUNDED_VARIATION_ON_LINEAR_IMAGE |- (!f. (!g. (!a. (!b. (((linear f) /\\ ((has_bounded_variation_on g) ((IMAGE f) (closed_interval ((CONS (a , b)) NIL))))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "50566", "text": "COMPACT_CONTINUOUS_IMAGE_EQ |- (!f. (!s. ((!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((f x) = (f y)))) ==> (x = y)))) ==> (((continuous_on f) s) = (!t. (((compact t) /\\ (t SUBSET s)) ==> (compact ((IMAGE f) t))))))))"}
{"_id": "50567", "text": "EMPTY_INSIDE_PSUBSET_CONVEX_FRONTIER |- (!s. (!t. (((convex s) /\\ (t PSUBSET (frontier s))) ==> ((inside t) = EMPTY))))"}
{"_id": "50568", "text": "Geomdetail.IN_SET3 |- ((x IN (a INSERT (b INSERT (c INSERT EMPTY)))) = ((x = a) \\/ ((x = b) \\/ (x = c))))"}
{"_id": "50569", "text": "MBOUNDED_INSERT |- (!m. (!a. (!s. (((mbounded m) (a INSERT s)) = ((a IN (mspace m)) /\\ ((mbounded m) s))))))"}
{"_id": "50570", "text": "INSIDE_SUBSET_CONVEX_HULL |- (!s. ((inside s) SUBSET ((hull convex) s)))"}
{"_id": "50571", "text": "Hdplygy.INDEX_J1_SY |- (!s. ((x IN (J1_SY s)) ==> (((NUMERAL (BIT1 _0)) <= (FST x)) /\\ ((FST x) <= (k_sy s)))))"}
{"_id": "50572", "text": "Fan.remark_fan1 |- (!v. (!w. (!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": "50573", "text": "Float.REAL_EQ_RCANCEL_IMP' |- (!x. (!y. (!z. ((((real_mul x) z) = ((real_mul y) z)) ==> ((~ (z = (real_of_num (NUMERAL _0)))) ==> (x = y))))))"}
{"_id": "50574", "text": "COMPLEX_POW_DIV |- (!x. (!y. (!n. (((complex_pow ((complex_div x) y)) n) = ((complex_div ((complex_pow x) n)) ((complex_pow y) n))))))"}
{"_id": "50575", "text": "Fan.FAN5_LINEAR_IMAGE_EQ |- (!f. (!x. (!V. (!E. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((fan5 ((f x) , (((IMAGE f) V) , ((IMAGE (IMAGE f)) E)))) = (fan5 (x , (V , E)))))))))"}
{"_id": "50576", "text": "REAL_CONVEX_DISTANCE |- (!s. (!a. ((real_convex_on (\\x. (real_abs ((real_sub a) x)))) s)))"}
{"_id": "50577", "text": "CONIC_HALFSPACE_LE |- (!a. (conic (GSPEC (\\GEN%PVAR%1464. (?x. (((SETSPEC GEN%PVAR%1464) ((real_le ((dot a) x)) (real_of_num (NUMERAL _0)))) x))))))"}
{"_id": "50578", "text": "HAS_REAL_DERIVATIVE_DIFFERENTIABLE |- (!f. (!x. ((((has_real_derivative f) ((real_derivative f) x)) (atreal x)) = ((real_differentiable f) (atreal x)))))"}
{"_id": "50579", "text": "Lp_ineqs_quads.ineq119_std |- (!V. (!E. ((lp_fan (V , E)) ==> (quad_nonlinear_v10 ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y5 , ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) ((CONS ((real_of_num (NUMERAL (BIT0 (BIT1 _0)))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_gt ((real_add ((real_sub ((((((taum y1) y2) y3) y4) y5) y6)) ((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))))))))))))) ((((((dih_y y1) y2) y3) y4) y5) y6)))) ((real_div ((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))))))))))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))))))))) ==> ((!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)))))))))) ==> ((!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)))))) ==> ((lp_tau (V , E)) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((real_ge ((real_add ((real_sub ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))) ((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))))))))))))) ((azim_dart (V , E)) d)))) ((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)))))))))))))) (real_of_num (NUMERAL _0))))))))))))))"}
{"_id": "50580", "text": "NEGATIONS_BALL |- (!r. (((IMAGE vector_neg) (ball ((vec (NUMERAL _0)) , r))) = (ball ((vec (NUMERAL _0)) , r))))"}
{"_id": "50581", "text": "SUMMABLE_COMPONENT |- (!f. (!s. (!k. ((((summable s) f) /\\ (((NUMERAL (BIT1 _0)) <= k) /\\ (k <= (dimindex UNIV)))) ==> ((summable s) (\\i. (lift ((f i) $ k))))))))"}
{"_id": "50582", "text": "REAL_VARIATION_AFFINITY2 |- (!m. (!c. (!f. (!s. (((real_variation ((IMAGE (\\x. ((real_add ((real_mul (real_inv m)) x)) (real_neg ((real_mul (real_inv m)) c))))) s)) (\\x. (f ((real_add ((real_mul m) x)) c)))) = (((COND (m = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) ((real_variation s) f)))))))"}
{"_id": "50583", "text": "NSUM_RESTRICT |- (!f. (!s. ((FINITE s) ==> (((nsum s) (\\x. (((COND (x IN s)) (f x)) (NUMERAL _0)))) = ((nsum s) f)))))"}
{"_id": "50584", "text": "FORALL_SUBSET_IMAGE_INJ |- (!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": "50585", "text": "CONTINUOUS_ON_COMPONENTS_OPEN_EQ |- (!f. (!s. ((open s) ==> (((continuous_on f) s) = (!c. ((c IN (components s)) ==> ((continuous_on f) c)))))))"}
{"_id": "50586", "text": "Leaf_cell.AFF_GE_MONO_TRANS |- (!X. (!Y. (!S. ((S SUBSET X) ==> (((aff_ge (X DIFF S)) (Y UNION S)) SUBSET ((aff_ge X) Y))))))"}
{"_id": "50587", "text": "REALLIM_1_OVER_LOG |- (((---> (\\n. (real_inv (log (real_of_num n))))) (real_of_num (NUMERAL _0))) sequentially)"}
{"_id": "50588", "text": "CONTINUOUS_CONTINUOUS_ATREAL |- (!f. (!x. (((continuous f) (atreal x)) = ((continuous (f o drop)) (at (lift x))))))"}
{"_id": "50589", "text": "DIAMETER_ATTAINED_FRONTIER |- (!s. (((bounded s) /\\ (~ ((diameter s) = (real_of_num (NUMERAL _0))))) ==> (?x. (?y. ((x IN (frontier s)) /\\ ((y IN (frontier s)) /\\ ((vector_norm ((vector_sub x) y)) = (diameter s))))))))"}
{"_id": "50590", "text": "CARD_EQ_ARC_IMAGE |- (!g. ((arc g) ==> ((path_image g) =_c UNIV)))"}
{"_id": "50591", "text": "COMPLEX_DERIVATIVE_LMUL_AT |- (!f. (!c. (!z. (((analytic_on f) (z INSERT EMPTY)) ==> (((complex_derivative (\\w. ((complex_mul c) (f w)))) z) = ((complex_mul c) ((complex_derivative f) z)))))))"}
{"_id": "50592", "text": "INT_EQ_ADD_LCANCEL |- (!x. (!y. (!z. ((((int_add x) y) = ((int_add x) z)) = (y = z)))))"}
{"_id": "50593", "text": "Collect_geom2.INSERT_SUBSET |- ((EMPTY SUBSET s) /\\ (((a INSERT s) SUBSET ss) = ((a IN ss) /\\ (s SUBSET ss))))"}
{"_id": "50594", "text": "EXISTS_PATH_SUBPATH_TO_FRONTIER_CLOSED |- (!g. (!s. (((closed 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) INTER s)) /\\ ((pathfinish h) IN (frontier s)))))))))"}
{"_id": "50595", "text": "OPEN_CONTAINS_CBALL_EQ |- (!s. ((open s) ==> (!x. ((x IN s) = (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ ((cball (x , e)) SUBSET s)))))))"}
{"_id": "50596", "text": "TAN_BOUND_PI2 |- (!x. (((real_lt (real_abs x)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) ==> ((real_lt (real_abs (tan x))) (real_of_num (NUMERAL (BIT1 _0))))))"}
{"_id": "50597", "text": "REAL_MEASURABLE_REAL_NEGLIGIBLE_SYMDIFF |- (!s. (!t. (((real_measurable s) /\\ (real_negligible ((s DIFF t) UNION (t DIFF s)))) ==> (real_measurable t))))"}
{"_id": "50598", "text": "OPEN_MAP_INTERIORS |- (!f. ((!s. ((open s) ==> (open ((IMAGE f) s)))) = (!s. (((IMAGE f) (interior s)) SUBSET (interior ((IMAGE f) s))))))"}
{"_id": "50599", "text": "INTERIOR_BALL |- (!a. (!r. ((interior (ball (a , r))) = (ball (a , r)))))"}
{"_id": "50600", "text": "INT_LE_TRANS_LT |- (!x. (!y. (((int_le x) y) = (!z. (((int_lt y) z) ==> ((int_lt x) z))))))"}
{"_id": "50601", "text": "LOCALLY_NEGLIGIBLE |- (!s. (((locally negligible) s) = (negligible s)))"}
{"_id": "50602", "text": "SUM_INTEGRAL_BOUNDS_INCREASING |- (!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_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_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) 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 (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))))))) /\\ ((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": "50603", "text": "COND_ABS |- (!b. (!f. (!g. ((\\x. (((COND b) (f x)) (g x))) = (((COND b) f) g)))))"}
{"_id": "50604", "text": "ABSOLUTELY_CONTINUOUS_ON_NEG |- (!f. (!s. (((absolutely_continuous_on f) s) ==> ((absolutely_continuous_on (\\x. (vector_neg (f x)))) s))))"}
{"_id": "50605", "text": "Ssrnat.distn_eq0 |- (!m. (!n. ((((distn m) n) = (NUMERAL _0)) = (m = n))))"}
{"_id": "50606", "text": "HOLOMORPHIC_ON_COMPOSE_GEN |- (!f. (!g. (!s. (!t. ((((holomorphic_on f) s) /\\ (((holomorphic_on g) t) /\\ (!z. ((z IN s) ==> ((f z) IN t))))) ==> ((holomorphic_on (g o f)) s))))))"}
{"_id": "50607", "text": "REAL_DIFFERENTIABLE_WITHIN_SIN |- (!s. (!x. ((real_differentiable sin) ((within (atreal x)) s))))"}
{"_id": "50608", "text": "REAL_MEASURABLE_ON_CASES |- (!P. (!f. (!g. (!s. (((real_lebesgue_measurable (GSPEC (\\GEN%PVAR%5045. (?x. (((SETSPEC GEN%PVAR%5045) (P x)) x))))) /\\ (((real_measurable_on f) s) /\\ ((real_measurable_on g) s))) ==> ((real_measurable_on (\\x. (((COND (P x)) (f x)) (g x)))) s))))))"}
{"_id": "50609", "text": "REAL_INTEGRABLE_DECREASING_PRODUCT |- (!f. (!g. (!a. (!b. ((((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 y)) (g x)))))) ==> ((real_integrable_on (\\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "50610", "text": "Ckqowsa_3_points.in_aff_ge_0_2_imp_dot_pos |- (!v1. (!v2. (!w. ((((real_lt (real_of_num (NUMERAL _0))) ((dot v1) v2)) /\\ ((w IN ((aff_ge ((vec (NUMERAL _0)) INSERT EMPTY)) (v1 INSERT (v2 INSERT EMPTY)))) /\\ (~ (w = (vec (NUMERAL _0)))))) ==> (((real_lt (real_of_num (NUMERAL _0))) ((dot v1) w)) /\\ ((real_lt (real_of_num (NUMERAL _0))) ((dot v2) w)))))))"}
{"_id": "50611", "text": "REAL_LE_ROOT |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\\ ((real_le (real_of_num (NUMERAL _0))) x)) ==> (((real_le x) ((root n) y)) = ((real_le ((real_pow x) n)) y))))))"}
{"_id": "50612", "text": "CROSS_MONO |- (!s. (!t. (!s'. (!t'. (((s SUBSET s') /\\ (t SUBSET t')) ==> ((s CROSS t) SUBSET (s' CROSS t')))))))"}
{"_id": "50613", "text": "REAL_INTEGRAL_SUBSET_LE |- (!f. (!s. (!t. (((s SUBSET t) /\\ (((real_integrable_on f) s) /\\ (((real_integrable_on f) t) /\\ (!x. ((x IN t) ==> ((real_le (real_of_num (NUMERAL _0))) (f x))))))) ==> ((real_le ((real_integral s) f)) ((real_integral t) f))))))"}
{"_id": "50614", "text": "Audit_formal_proof.chk_basic_definitions |- ((((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))) /\\ ((((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ ((((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n))) /\\ ((((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))) /\\ ((((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)))) /\\ (((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))))))) /\\ (((A HAS_SIZE n) = ((FINITE A) /\\ ((CARD A) = n))) /\\ ((EMPTY HAS_SIZE (NUMERAL _0)) /\\ (((a INSERT EMPTY) HAS_SIZE (NUMERAL (BIT1 _0))) /\\ ((((A HAS_SIZE m) /\\ ((B HAS_SIZE n) /\\ ((A INTER B) HAS_SIZE p))) ==> ((A UNION B) HAS_SIZE (m + (n - p)))) /\\ (((triple_of_real3 o real3_of_triple) = (\\w. w)) /\\ (((real3_of_triple o triple_of_real3) = (\\v. v)) /\\ (((vec (NUMERAL _0)) = (real3_of_triple ((real_of_num (NUMERAL _0)) , ((real_of_num (NUMERAL _0)) , (real_of_num (NUMERAL _0)))))) /\\ (((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)))))))) /\\ ((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": "50615", "text": "REAL_POLYNOMIAL_FUNCTION_EXPLICIT_NZ |- (!f. ((real_polynomial_function f) = (?s. (?a. ((FINITE s) /\\ ((!k. ((k IN s) ==> (~ ((a k) = (real_of_num (NUMERAL _0)))))) /\\ (f = (\\x. ((sum s) (\\k. ((real_mul (a k)) ((product ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. ((real_pow (x $ i)) (k $ i)))))))))))))))"}
{"_id": "50616", "text": "HAS_BOUNDED_REAL_VARIATION_ON_RMUL |- (!f. (!c. (!s. (((has_bounded_real_variation_on f) s) ==> ((has_bounded_real_variation_on (\\x. ((real_mul (f x)) c))) s)))))"}
{"_id": "50617", "text": "MATRIX_INV_EQ_0 |- (!A. (((matrix_inv A) = (mat (NUMERAL _0))) = (A = (mat (NUMERAL _0)))))"}
{"_id": "50618", "text": "SIMPLICIAL_COMPLEX_DISJOINT_RELATIVE_INTERIORS |- (!t. (!c. (!d. (((simplicial_complex t) /\\ ((c IN t) /\\ ((d IN t) /\\ (~ (c = d))))) ==> (((relative_interior c) INTER (relative_interior d)) = EMPTY)))))"}
{"_id": "50619", "text": "HOMEOMORPHIC_CBALLS |- (!a. (!b. (!d. (!e. ((((real_lt (real_of_num (NUMERAL _0))) d) /\\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> ((homeomorphic (cball (a , d))) (cball (b , e))))))))"}
{"_id": "50620", "text": "CTAN_DOUBLE |- (!z. (((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\\ (~ ((ccos ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = (Cx (real_of_num (NUMERAL _0)))))) ==> ((ctan ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = ((complex_div ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (ctan z))) ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow (ctan z)) (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "50621", "text": "EVENTUALLY_AT_INFINITY_POS |- (!p. (((eventually p) at_infinity) = (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\\ (!x. (((real_ge (vector_norm x)) b) ==> (p x)))))))"}
{"_id": "50622", "text": "TAGGED_DIVISION_UNIONS_EXISTS |- (!d. (!iset. (!i. (((FINITE iset) /\\ ((!i. ((i IN iset) ==> (?p. (((tagged_division_of p) i) /\\ ((fine d) p))))) /\\ ((!i1. (!i2. (((i1 IN iset) /\\ ((i2 IN iset) /\\ (~ (i1 = i2)))) ==> (((interior i1) INTER (interior i2)) = EMPTY)))) /\\ ((UNIONS iset) = i)))) ==> (?p. (((tagged_division_of p) i) /\\ ((fine d) p)))))))"}
{"_id": "50623", "text": "Betwn_corek_z_x.core_rot |- (((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))))) ==> (((core L) ((rot n) r)) = ((rot n) ((core L) r))))"}
{"_id": "50624", "text": "COMPONENTS_OPEN_UNIQUE |- (!f. (!s. (((!c. ((c IN f) ==> ((open c) /\\ ((connected c) /\\ (~ (c = EMPTY)))))) /\\ (((pairwise DISJOINT) f) /\\ ((UNIONS f) = s))) ==> ((components s) = f))))"}
{"_id": "50625", "text": "Ssrbool.andbK |- (!a. (!b. (((a /\\ b) \\/ a) = a)))"}
{"_id": "50626", "text": "INT_MIN |- (!x. (!y. (((int_min x) y) = (((COND ((int_le x) y)) x) y))))"}
{"_id": "50627", "text": "NORM_CPOW |- (!w. (!z. (((real w) /\\ ((real_lt (real_of_num (NUMERAL _0))) (Re w))) ==> ((vector_norm ((cpow w) z)) = ((rpow (vector_norm w)) (Re z))))))"}
{"_id": "50628", "text": "COMPLEX_DIFFSQ |- (!x. (!y. (((complex_mul ((vector_add x) y)) ((vector_sub x) y)) = ((vector_sub ((complex_mul x) x)) ((complex_mul y) y)))))"}
{"_id": "50629", "text": "LIM_NULL_COMPLEX_MUL |- (!net. (!f. (!g. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> (\\x. ((complex_mul (f x)) (g x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))"}
{"_id": "50630", "text": "REAL_NEGLIGIBLE_TRANSLATION_EQ |- (!s. (!a. ((real_negligible ((IMAGE (\\x. ((real_add a) x))) s)) = (real_negligible s))))"}
{"_id": "50631", "text": "NSUM_OFFSET_0 |- (!f. (!m. (!n. ((m <= n) ==> (((nsum (m .. n)) f) = ((nsum ((NUMERAL _0) .. (n - m))) (\\i. (f (i + m)))))))))"}
{"_id": "50632", "text": "BOUNDED_HYPERPLANE_EQ_TRIVIAL |- (!a. (!b. ((bounded (GSPEC (\\GEN%PVAR%1578. (?x. (((SETSPEC GEN%PVAR%1578) (((dot a) x) = b)) x))))) = (((COND (a = (vec (NUMERAL _0)))) (~ (b = (real_of_num (NUMERAL _0))))) ((dimindex UNIV) = (NUMERAL (BIT1 _0)))))))"}
{"_id": "50633", "text": "FRONTIER_INTER_SUBSET_INTER |- (!s. (!t. ((frontier (s INTER t)) SUBSET (((closure s) INTER (frontier t)) UNION ((frontier s) INTER (closure t))))))"}
{"_id": "50634", "text": "INTEGRABLE_SUBINTERVALS_IMP_REAL_MEASURABLE |- (!f. ((!a. (!b. ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))))) ==> ((real_measurable_on f) UNIV)))"}
{"_id": "50635", "text": "Functional_equation.functional_proj_y3 |- (proj_y3 = (rotate3 proj_y1))"}
{"_id": "50636", "text": "REAL_MEASURABLE_ON_PREIMAGE_HALFSPACE_GE |- (!f. (((real_measurable_on f) UNIV) = (!a. (real_lebesgue_measurable (GSPEC (\\GEN%PVAR%5049. (?x. (((SETSPEC GEN%PVAR%5049) ((real_ge (f x)) a)) x))))))))"}
{"_id": "50637", "text": "HAS_REAL_INTEGRAL_EQ_EQ |- (!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": "50638", "text": "PATH_IMAGE_SYM |- (!p. (!q. ((((pathfinish p) = (pathstart q)) /\\ ((pathfinish q) = (pathstart p))) ==> ((path_image ((++ p) q)) = (path_image ((++ q) p))))))"}
{"_id": "50639", "text": "INTEGRAL_PASTECART_CONTINUOUS |- (!f. (!a. (!b. (!c. (!d. (((continuous_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": "50640", "text": "PROPER_MAP_FROM_COMPOSITION_RIGHT |- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((continuous_on g) t) /\\ ((((IMAGE g) t) SUBSET u) /\\ (!k. (((k SUBSET u) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%740. (?x. (((SETSPEC GEN%PVAR%740) ((x IN s) /\\ (((g o f) x) IN k))) x))))))))))) ==> (!k. (((k SUBSET t) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%741. (?x. (((SETSPEC GEN%PVAR%741) ((x IN s) /\\ ((f x) IN k))) x)))))))))))))"}
{"_id": "50641", "text": "Localization.wedge_in_fan_ge |- (!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%7533. (?x. (((SETSPEC GEN%PVAR%7533) T) x)))))))))"}
{"_id": "50642", "text": "LEBESGUE_MEASURABLE_ALMOST_GDELTA |- (!s. ((lebesgue_measurable s) ==> (?c. (?t. ((gdelta c) /\\ ((negligible t) /\\ (((s UNION t) = c) /\\ ((DISJOINT t) s))))))))"}
{"_id": "50643", "text": "AR_INTERVAL |- ((!a. (!b. ((AR (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\\ (!a. (!b. ((AR (open_interval (a , b))) = (~ ((open_interval (a , b)) = EMPTY))))))"}
{"_id": "50644", "text": "RELATIVE_INTERIOR_RELATIVE_INTERIOR |- (!s. ((relative_interior (relative_interior s)) = (relative_interior s)))"}
{"_id": "50645", "text": "MEASURABLE_SMALL_IMP_NEGLIGIBLE |- (!s. (((measurable s) /\\ (s <_c UNIV)) ==> (negligible s)))"}
{"_id": "50646", "text": "LEBESGUE_DENSITY_THEOREM_INTEGRAL_BALL |- (!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 (ball (x , (drop e)))))) ((integral (s INTER (ball (x , (drop e))))) f)))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\\GEN%PVAR%3725. (?t. (((SETSPEC GEN%PVAR%3725) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))))))))))))"}
{"_id": "50647", "text": "CCOS_DOUBLE_CCOS |- (!z. ((ccos ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) z)) = ((vector_sub ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0)))))) (Cx (real_of_num (NUMERAL (BIT1 _0)))))))"}
{"_id": "50648", "text": "LINEAR_FRECHET_DERIVATIVE |- (!f. (!net. (((differentiable f) net) ==> (linear ((frechet_derivative f) net)))))"}
{"_id": "50649", "text": "HOLOMORPHIC_ON_CACS |- (!s. ((!z. (((z IN s) /\\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((holomorphic_on cacs) s)))"}
{"_id": "50650", "text": "DIFFERENTIABLE_REAL_COMPLEX |- (!f. (!a. (((complex_differentiable f) (at (Cx (drop a)))) ==> ((differentiable (\\x. (f (Cx (drop x))))) (at a)))))"}
{"_id": "50651", "text": "COMPLEX_L1_LE_NORM |- (!z. ((real_le ((real_mul ((real_div (sqrt (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add (real_abs (Re z))) (real_abs (Im z))))) (vector_norm z)))"}
{"_id": "50652", "text": "Merge_ineq.nonf_sol_euler156_x_div_sqrtdelta |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((sol_euler156_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 x6) ((real_mul x1) x5)))) ((real_add ((real_mul (sqrt x6)) ((real_div ((real_add x1) ((real_sub x5) x3))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_add ((real_mul (sqrt x1)) ((real_div ((real_add x6) ((real_sub x5) x4))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul (sqrt x5)) ((real_div ((real_add x6) ((real_sub x1) x2))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))"}
{"_id": "50653", "text": "INDEFINITE_INTEGRAL_UNIFORMLY_CONTINUOUS_EXPLICIT |- (!f. (!a. (!b. (!e. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?k. (((real_lt (real_of_num (NUMERAL _0))) k) /\\ (!c. (!d. (!c'. (!d'. (((c IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((d IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((c' IN (closed_interval ((CONS (a , b)) NIL))) /\\ ((d' IN (closed_interval ((CONS (a , b)) NIL))) /\\ (((real_le (vector_norm ((vector_sub c') c))) k) /\\ ((real_le (vector_norm ((vector_sub d') d))) k)))))) ==> ((real_lt (vector_norm ((vector_sub ((integral (closed_interval ((CONS (c' , d')) NIL))) f)) ((integral (closed_interval ((CONS (c , d)) NIL))) f)))) e)))))))))))))"}
{"_id": "50654", "text": "MULT_DIV_LE |- (!m. (!n. (!p. ((~ (p = (NUMERAL _0))) ==> ((m * (n DIV p)) <= ((m * n) DIV p))))))"}
{"_id": "50655", "text": "ABSOLUTELY_REAL_INTEGRABLE_INF |- (!fs. (!s. (!k. (((FINITE k) /\\ ((~ (k = EMPTY)) /\\ (!i. ((i IN k) ==> ((absolutely_real_integrable_on (\\x. ((fs x) i))) s))))) ==> ((absolutely_real_integrable_on (\\x. (inf ((IMAGE (fs x)) k)))) s)))))"}
{"_id": "50656", "text": "REAL_CONVEX_ON_EQ |- (!f. (!g. (!s. (((is_realinterval s) /\\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((real_convex_on f) s))) ==> ((real_convex_on g) s)))))"}
{"_id": "50657", "text": "HAS_BOUNDED_REAL_VARIATION_DARBOUX_STRICT |- (!f. (!a. (!b. (((has_bounded_real_variation_on f) (closed_real_interval ((CONS (a , b)) NIL))) = (?g. (?h. ((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_lt x) y))) ==> ((real_lt (g x)) (g y))))) /\\ ((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_lt x) y))) ==> ((real_lt (h x)) (h y))))) /\\ (!x. ((f x) = ((real_sub (g x)) (h x))))))))))))"}
{"_id": "50658", "text": "ORTHOGONAL_TO_ORTHOGONAL_2D |- (!x. (!y. (!z. (((~ (x = (vec (NUMERAL _0)))) /\\ (((orthogonal x) y) /\\ ((orthogonal x) z))) ==> (collinear ((vec (NUMERAL _0)) INSERT (y INSERT (z INSERT EMPTY))))))))"}
{"_id": "50659", "text": "PERMUTES_INDUCT |- (!P. (!s. (((FINITE s) /\\ ((P I) /\\ (!a. (!b. (!p. (((a IN s) /\\ ((b IN s) /\\ ((P p) /\\ (permutation p)))) ==> (P ((swap (a , b)) o p)))))))) ==> (!p. (((permutes p) s) ==> (P p))))))"}
{"_id": "50660", "text": "Nonlinear_lemma.halfbump_x_expand |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((halfbump_x x) = ((real_add (real_neg ((real_div (real_of_num (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))))))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))) ((real_sub ((real_mul ((real_div (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))) (real_of_num (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))) (sqrt x))) ((real_mul ((real_div (real_of_num (NUMERAL (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))) (real_of_num (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) x))))))"}
{"_id": "50661", "text": "Functional_equation.functional_proj_y1 |- (proj_y1 = (promote1_to_6 sqrt))"}
{"_id": "50662", "text": "MEASURABLE_CLOSED_IN |- (!s. (!t. ((((closed_in ((subtopology euclidean) s)) t) /\\ (measurable s)) ==> (measurable t))))"}
{"_id": "50663", "text": "REAL_MEASURABLE_ON_MIN |- (!f. (!g. (!s. ((((real_measurable_on f) s) /\\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\\x. ((real_min (f x)) (g x)))) s)))))"}
{"_id": "50664", "text": "DIFFERENTIABLE_ON_EQ |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((differentiable_on f) s)) ==> ((differentiable_on g) s)))))"}
{"_id": "50665", "text": "NORMAL_MATRIX_KERNEL_TRANSP |- (!A. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> ((GSPEC (\\GEN%PVAR%234. (?x. (((SETSPEC GEN%PVAR%234) (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))) x)))) = (GSPEC (\\GEN%PVAR%235. (?x. (((SETSPEC GEN%PVAR%235) (((matrix_vector_mul A) x) = (vec (NUMERAL _0)))) x)))))))"}
{"_id": "50666", "text": "MATRIX_CMUL_ADD_RDISTRIB |- (!A. (!b. (!c. (((%% ((real_add b) c)) A) = ((matrix_add ((%% b) A)) ((%% c) A))))))"}
{"_id": "50667", "text": "Lp_ineqs_proofs.tau_sum5 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_add pi) sol0))))))))))))))"}
{"_id": "50668", "text": "REALLIM_NULL_LMUL |- (!net. (!f. (!c. ((((---> f) (real_of_num (NUMERAL _0))) net) ==> (((---> (\\x. ((real_mul c) (f x)))) (real_of_num (NUMERAL _0))) net)))))"}
{"_id": "50669", "text": "Lp_gen_theory.ALL_list_dart_pairs_split |- (!L. (!P. (((ALL P) (list_dart_pairs L)) = ((ALL (\\d1. ((ALL (\\d2. (P (d1 , d2)))) (list_of_darts L)))) (list_of_darts L)))))"}
{"_id": "50670", "text": "REAL_INTEGRABLE_RMUL_EQ |- (!f. (!s. (!c. (((real_integrable_on (\\x. ((real_mul (f x)) c))) s) = ((c = (real_of_num (NUMERAL _0))) \\/ ((real_integrable_on f) s))))))"}
{"_id": "50671", "text": "AR_BALL |- (!a. (!r. ((AR (ball (a , r))) = ((real_lt (real_of_num (NUMERAL _0))) r))))"}
{"_id": "50672", "text": "Functional_equation.functional_proj_y2 |- (proj_y2 = (rotate2 proj_y1))"}
{"_id": "50673", "text": "Lp_ineqs_proofs.y5_hi_std |- (!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)) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))))))))"}
{"_id": "50674", "text": "ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES |- (!f. (!g. (!g'. (!s. (((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 f) ((IMAGE g) s)) = ((absolutely_integrable_on (\\x. ((% (real_abs (det (matrix (g' x))))) (f (g x))))) s)))))))"}
{"_id": "50675", "text": "UNION_INTERIOR_SUBSET |- (!s. (!t. (((interior s) UNION (interior t)) SUBSET (interior (s UNION t)))))"}
{"_id": "50676", "text": "NEGATIONS_SPHERE |- (!r. (((IMAGE vector_neg) (sphere ((vec (NUMERAL _0)) , r))) = (sphere ((vec (NUMERAL _0)) , r))))"}
{"_id": "50677", "text": "CARD_UNION_OVERLAP |- (!s. (!t. (((FINITE s) /\\ ((FINITE t) /\\ ((CARD (s UNION t)) < ((CARD s) + (CARD t))))) ==> (~ ((s INTER t) = EMPTY)))))"}
{"_id": "50678", "text": "NEGLIGIBLE_IFF_MEASURABLE_SUBSETS |- (!s. ((negligible s) = (!t. ((t SUBSET s) ==> (measurable t)))))"}
{"_id": "50679", "text": "ARC_JOIN_EQ_ALT |- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((arc ((++ g1) g2)) = ((arc g1) /\\ ((arc g2) /\\ (((path_image g1) INTER (path_image g2)) = ((pathstart g2) INSERT EMPTY))))))))"}
{"_id": "50680", "text": "Appendix.LENGTH_s_init_list |- ((LENGTH s_init_list_v39) = (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))"}
{"_id": "50681", "text": "Lp_ineqs_proofs.sol_sum3 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT1 (BIT1 _0)))) ==> (((sum f) (azim_dart (V , E))) = ((real_add ((sol_fan (V , E)) f)) pi)))))))))))"}
{"_id": "50682", "text": "VSUM_INJECTION |- (!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)))))) ==> (((vsum s) (f o p)) = ((vsum s) f))))))"}
{"_id": "50683", "text": "NEIGHBOURHOOD_EXTENSION_INTO_ANR |- (!f. (!s. (!t. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ ((ANR t) /\\ (closed s)))) ==> (?v. (?g. ((s SUBSET v) /\\ ((open v) /\\ (((continuous_on g) v) /\\ ((((IMAGE g) v) SUBSET t) /\\ (!x. ((x IN s) ==> ((g x) = (f x))))))))))))))"}
{"_id": "50684", "text": "BERNOULLI_NUMBER_ALT |- (!n. (((sum ((NUMERAL _0) .. n)) (\\k. ((real_mul (real_of_num (binom ((n + (NUMERAL (BIT1 _0))) , k)))) ((bernoulli k) (real_of_num (NUMERAL _0)))))) = (((COND (n = (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))"}
{"_id": "50685", "text": "LIM_WITHIN_EMPTY |- (!f. (!l. (!x. (((--> f) l) ((within (at x)) EMPTY)))))"}
{"_id": "50686", "text": "Miqmcsn.J_SCS_4M8_13 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_4M8_13) i)) i1) j) = F)"}
{"_id": "50687", "text": "STC_IDEMP |- (!R. ((STC (STC R)) = (STC R)))"}
{"_id": "50688", "text": "ROTATE2D_SUB_ARG |- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> ((rotate2d ((real_sub (Arg w)) (Arg z))) = (rotate2d (Arg ((complex_div w) z)))))))"}
{"_id": "50689", "text": "HOMOTOPIC_COMPOSE |- (!f. (!f'. (!g. (!g'. (!s. (!t. (!u. ((((((homotopic_with (\\x. T)) (s , t)) f) f') /\\ ((((homotopic_with (\\x. T)) (t , u)) g) g')) ==> ((((homotopic_with (\\x. T)) (s , u)) (g o f)) (g' o f'))))))))))"}
{"_id": "50690", "text": "CLOSEST_POINT_TRANSLATION |- (!s. (!a. (((convex s) /\\ ((closed s) /\\ (~ (s = EMPTY)))) ==> (((closest_point ((IMAGE (\\x. ((vector_add a) x))) s)) ((vector_add a) x)) = ((vector_add a) ((closest_point s) x))))))"}
{"_id": "50691", "text": "LT_MULT2 |- (!m. (!n. (!p. (!q. (((m < n) /\\ (p < q)) ==> ((m * p) < (n * q)))))))"}
{"_id": "50692", "text": "INT_MIN_LE |- (!x. (!y. (!z. (((int_le ((int_min x) y)) z) = (((int_le x) z) \\/ ((int_le y) z))))))"}
{"_id": "50693", "text": "COLLINEAR_TRANSLATION |- (!s. (!a. ((collinear s) ==> (collinear ((IMAGE (\\x. ((vector_add a) x))) s)))))"}
{"_id": "50694", "text": "Functional_equation.functional_proj_x2 |- (proj_x2 = (rotate2 proj_x1))"}
{"_id": "50695", "text": "TIETZE_CLOSED_INTERVAL_1 |- (!f. (!u. (!s. (!a. (!b. ((((real_le (drop a)) (drop b)) /\\ (((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": "50696", "text": "Trigonometry.QAFHJNM |- (!v. (!w. (!x. ((LET (\\e3. (LET_END ((LET (\\r. (LET_END ((LET (\\phi. (LET_END (((~ (v = x)) /\\ (~ (v = w))) ==> (?u'. ((((dot u') e3) = (real_of_num (NUMERAL _0))) /\\ (x = ((vector_add v) ((vector_add u') ((% ((real_mul r) (cos phi))) e3)))))))))) (((arcV v) w) x))))) (vector_norm ((vector_sub x) v)))))) ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (vector_norm ((vector_sub w) v)))) ((vector_sub w) v))))))"}
{"_id": "50697", "text": "HAS_BOUNDED_REAL_VARIATION_AFFINITY_EQ |- (!m. (!c. (!f. (!s. (((has_bounded_real_variation_on (\\x. (f ((real_add ((real_mul m) x)) c)))) s) = ((m = (real_of_num (NUMERAL _0))) \\/ ((has_bounded_real_variation_on f) ((IMAGE (\\x. ((real_add ((real_mul m) x)) c))) s))))))))"}
{"_id": "50698", "text": "Lp_ineqs_proofs.SURJ_pair |- (!h1. (!h2. (!g. ((!x. ((g x) = ((h1 (FST x)) , (h2 (SND x))))) ==> (!s1. (!t1. (!s2. (!t2. (((((SURJ h1) s1) t1) /\\ (((SURJ h2) s2) t2)) ==> (((SURJ g) (s1 CROSS s2)) (t1 CROSS t2)))))))))))"}
{"_id": "50699", "text": "CARD_LE_COMPONENTS |- (!f. (!s. (((continuous_on f) s) ==> ((components ((IMAGE f) s)) <=_c (components s)))))"}
{"_id": "50700", "text": "Tame_inequalities.DELTA_X4_MONO_LE_4 |- (!x1. (!x2. (!x3. (!a. (!x5. (!x6. (!b. ((((real_lt a) b) /\\ ((real_le (real_of_num (NUMERAL _0))) x1)) ==> ((real_le ((((((delta_x4 x1) x2) x3) b) x5) x6)) ((((((delta_x4 x1) x2) x3) a) x5) x6))))))))))"}
{"_id": "50701", "text": "REAL_DIFFERENTIABLE_WITHIN_EXP |- (!s. (!x. ((real_differentiable exp) ((within (atreal x)) s))))"}
{"_id": "50702", "text": "Seq.eq_filter |- (!a1. (!a2. ((!x. ((a1 x) = (a2 x))) ==> (!s. (((filter a1) s) = ((filter a2) s))))))"}
{"_id": "50703", "text": "SUMMABLE_SUB |- (!x. (!y. (!s. ((((summable s) x) /\\ ((summable s) y)) ==> ((summable s) (\\n. ((vector_sub (x n)) (y n))))))))"}
{"_id": "50704", "text": "HOMOTOPIC_INVERTIBLE_LINEAR_MAPS |- (!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 (det (matrix f))) (det (matrix g)))))))))"}
{"_id": "50705", "text": "INT_LT_POW2 |- (!n. ((int_lt (int_of_num (NUMERAL _0))) ((int_pow (int_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)))"}
{"_id": "50706", "text": "INT_SGN_POW |- (!x. (!n. ((int_sgn ((int_pow x) n)) = ((int_pow (int_sgn x)) n))))"}
{"_id": "50707", "text": "CONTINUOUS_TRANSFORM_WITHINREAL_SET_IMP |- (!f. (!a. (!s. (!t. ((((eventually (\\x. ((x IN t) ==> (x IN s)))) (atreal a)) /\\ ((continuous f) ((within (atreal a)) s))) ==> ((continuous f) ((within (atreal a)) t)))))))"}
{"_id": "50708", "text": "CONIC_IMP_SIMPLY_CONNECTED |- (!s. ((conic s) ==> (simply_connected s)))"}
{"_id": "50709", "text": "More_list_hypermap.contravening_dist_dart_le |- (!V. (!g. (!L. ((contravening V) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , (ESTD V))))) ==> ((ALL (\\d. ((real_le (distance (g d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (list_of_darts L))))))))"}
{"_id": "50710", "text": "AFFINE_EXPLICIT |- (!s. ((affine s) = (!t. (!u. (((FINITE t) /\\ ((t SUBSET s) /\\ (((sum t) u) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> (((vsum t) (\\x. ((% (u x)) x))) IN s))))))"}
{"_id": "50711", "text": "CONVEX |- (!V. ((convex V) = (!s. (!u. (((FINITE s) /\\ ((~ (s = EMPTY)) /\\ ((s SUBSET V) /\\ ((!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))) IN V))))))"}
{"_id": "50712", "text": "COMPLEX_NEG_RMUL |- (!x. (!y. ((vector_neg ((complex_mul x) y)) = ((complex_mul x) (vector_neg y)))))"}
{"_id": "50713", "text": "INT_EQ_MUL_LCANCEL |- (!x. (!y. (!z. ((((int_mul x) y) = ((int_mul x) z)) = ((x = (int_of_num (NUMERAL _0))) \\/ (y = z))))))"}
{"_id": "50714", "text": "Seq2.butlast_cons |- (!h. (!t. ((butlast ((CONS h) t)) = (((COND (t = NIL)) NIL) ((CONS h) (butlast t))))))"}
{"_id": "50715", "text": "FRONTIER_SPHERE |- (!a. (!r. ((frontier (sphere (a , r))) = (sphere (a , r)))))"}
{"_id": "50716", "text": "SUMMABLE_COMPLEX_RMUL |- (!f. (!c. (!s. (((summable s) f) ==> ((summable s) (\\x. ((complex_mul (f x)) c)))))))"}
{"_id": "50717", "text": "HOMOTOPIC_PATHS_RCANCEL_EQ |- (!p. (!q. (!r. (!s. ((((pathfinish p) = (pathstart r)) /\\ ((pathfinish q) = (pathstart r))) ==> ((((homotopic_paths s) ((++ p) r)) ((++ q) r)) = ((((homotopic_paths s) p) q) /\\ ((path r) /\\ ((path_image r) SUBSET s)))))))))"}
{"_id": "50718", "text": "DEFORMATION_RETRACT_OF_CONTRACTIBLE_SING |- (!s. (!a. (((contractible s) /\\ (a IN s)) ==> (?r. (((((homotopic_with (\\x. T)) (s , s)) (\\x. x)) r) /\\ ((retraction (s , (a INSERT EMPTY))) r))))))"}
{"_id": "50719", "text": "IS_INTERVAL_SIMPLY_CONNECTED_1 |- (!s. ((is_interval s) = (simply_connected s)))"}
{"_id": "50720", "text": "REGULAR_OPEN_INTER |- (!s. (!t. ((((interior (closure s)) = s) /\\ ((interior (closure t)) = t)) ==> ((interior (closure (s INTER t))) = (s INTER t)))))"}
{"_id": "50721", "text": "HAS_INTEGRAL_LIM_SEQUENTIALLY |- (!f. (!l. (((((--> (f o lift)) (vec (NUMERAL _0))) at_posinfinity) /\\ ((!n. ((integrable_on f) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec n))) NIL)))) /\\ (((--> (\\n. ((integral (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec n))) NIL))) f))) l) sequentially))) ==> (((has_integral f) l) (GSPEC (\\GEN%PVAR%2954. (?t. (((SETSPEC GEN%PVAR%2954) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t))))))))"}
{"_id": "50722", "text": "INT_LE_NEGR |- (!x. (((int_le x) (int_neg x)) = ((int_le x) (int_of_num (NUMERAL _0)))))"}
{"_id": "50723", "text": "Lp_ineqs_proofs.rho_lo |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((real_le (real_of_num (NUMERAL (BIT1 _0)))) (rho_fan v))))))))))"}
{"_id": "50724", "text": "VITALI_COVERING_LEMMA_CBALLS |- (!s. (!a. (!r. (!k. (!B. (((s SUBSET (UNIONS ((IMAGE (\\i. (cball ((a i) , (r i))))) k))) /\\ (!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) /\\ (s SUBSET (UNIONS ((IMAGE (\\i. (cball ((a i) , ((real_mul (real_of_num (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) (r i)))))) c)))))))))))))"}
{"_id": "50725", "text": "VECTOR_EXPAND_1 |- (!x. (x = (vector ((CONS (x $ (NUMERAL (BIT1 _0)))) NIL))))"}
{"_id": "50726", "text": "STEINHAUS_DIFFS |- (!s. (!t. (((lebesgue_measurable s) /\\ ((~ (negligible s)) /\\ ((lebesgue_measurable t) /\\ (~ (negligible t))))) ==> (~ ((interior (GSPEC (\\GEN%PVAR%4608. (?x. (?y. (((SETSPEC GEN%PVAR%4608) ((x IN s) /\\ (y IN t))) ((vector_sub x) y))))))) = EMPTY)))))"}
{"_id": "50727", "text": "INTEGRABLE_UNION_EQ |- (!f. (!s. (!t. ((((integrable_on f) s) /\\ ((integrable_on f) t)) ==> (((integrable_on f) (s UNION t)) = ((integrable_on f) (s INTER t)))))))"}
{"_id": "50728", "text": "SING_STRONG_DEFORMATION_RETRACT_OF_AR |- (!s. (!a. (((AR s) /\\ (a IN s)) ==> (?r. (((((homotopic_with (\\h. ((h a) = a))) (s , s)) (\\x. x)) r) /\\ ((retraction (s , (a INSERT EMPTY))) r))))))"}
{"_id": "50729", "text": "INT_SUB_ADD2 |- (!x. (!y. (((int_add y) ((int_sub x) y)) = x)))"}
{"_id": "50730", "text": "DIFFERENTIABLE_ON_EMPTY |- (!f. ((differentiable_on f) EMPTY))"}
{"_id": "50731", "text": "REAL_MEASURE_DISJOINT_UNIONS |- (!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)))))) ==> ((real_measure (UNIONS f)) = ((sum f) m)))))"}
{"_id": "50732", "text": "IN_BALL_RE |- (!x. (!z. (!e. ((x IN (ball (z , e))) ==> ((real_lt (real_abs ((real_sub (Re x)) (Re z)))) e)))))"}
{"_id": "50733", "text": "Tame_list.rev2 |- (!x. (!y. ((rev ((CONS x) ((CONS y) NIL))) = ((CONS y) ((CONS x) NIL)))))"}
{"_id": "50734", "text": "INT_SGN_INEQS |- ((!x. (((int_le (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_le (int_of_num (NUMERAL _0))) x))) /\\ ((!x. (((int_lt (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_lt (int_of_num (NUMERAL _0))) x))) /\\ ((!x. (((int_ge (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_ge (int_of_num (NUMERAL _0))) x))) /\\ ((!x. (((int_gt (int_of_num (NUMERAL _0))) (int_sgn x)) = ((int_gt (int_of_num (NUMERAL _0))) x))) /\\ ((!x. (((int_of_num (NUMERAL _0)) = (int_sgn x)) = ((int_of_num (NUMERAL _0)) = x))) /\\ ((!x. (((int_le (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_le x) (int_of_num (NUMERAL _0))))) /\\ ((!x. (((int_lt (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_lt x) (int_of_num (NUMERAL _0))))) /\\ ((!x. (((int_ge (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_ge x) (int_of_num (NUMERAL _0))))) /\\ ((!x. (((int_gt (int_sgn x)) (int_of_num (NUMERAL _0))) = ((int_gt x) (int_of_num (NUMERAL _0))))) /\\ (!x. (((int_sgn x) = (int_of_num (NUMERAL _0))) = (x = (int_of_num (NUMERAL _0))))))))))))))"}
{"_id": "50735", "text": "CONVEX_CONE_HULL_NONEMPTY |- (!s. (~ (((hull convex_cone) s) = EMPTY)))"}
{"_id": "50736", "text": "INT_LT_ADD1 |- (!x. (!y. (((int_le x) y) ==> ((int_lt x) ((int_add y) (int_of_num (NUMERAL (BIT1 _0))))))))"}
{"_id": "50737", "text": "Seq.subseq_uniqP |- (!s1. (!s2. ((uniq s2) ==> (((subseq s1) s2) = (s1 = ((filter (\\x. ((MEM x) s1))) s2))))))"}
{"_id": "50738", "text": "Lp_ineqs_proofs.azim3c |- (!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))) ==> (((azim3_fan (V , E)) d) = ((azim_dart (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))"}
{"_id": "50739", "text": "Tactics_jordan.switch_lemma_le |- (!A. (!B. (!C. ((((real_le (real_of_num (NUMERAL _0))) A) ==> (((real_le (real_of_num (NUMERAL _0))) B) ==> C)) = (((real_le (real_of_num (NUMERAL _0))) B) ==> (((real_le (real_of_num (NUMERAL _0))) A) ==> C))))))"}
{"_id": "50740", "text": "HOLOMORPHIC_ON_DIV |- (!f. (!g. (!s. ((((holomorphic_on f) s) /\\ (((holomorphic_on g) s) /\\ (!z. ((z IN s) ==> (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((holomorphic_on (\\z. ((complex_div (f z)) (g z)))) s)))))"}
{"_id": "50741", "text": "ABSOLUTE_RETRACT_FROM_UNION_AND_INTER |- (!s. (!t. ((((retract_of (s UNION t)) UNIV) /\\ (((retract_of (s INTER t)) UNIV) /\\ ((closed s) /\\ (closed t)))) ==> ((retract_of s) UNIV))))"}
{"_id": "50742", "text": "INTEGRABLE_SPIKE_EQ |- (!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": "50743", "text": "CSQRT_EQ_0 |- (!z. (((csqrt z) = (Cx (real_of_num (NUMERAL _0)))) = (z = (Cx (real_of_num (NUMERAL _0))))))"}
{"_id": "50744", "text": "LIM_COMPLEX_REAL_0 |- (!f. (!g. ((((eventually (\\n. ((Re (g n)) = (f n)))) sequentially) /\\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) sequentially)) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs (f n))) e)))))))))"}
{"_id": "50745", "text": "CONTINUOUS_ON_CX_NORM |- (!s. ((continuous_on (\\z. (Cx (vector_norm z)))) s))"}
{"_id": "50746", "text": "Add_triangle.ext_in_dart1 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((ext_dart (V , E)) d) IN (d1_fan ((vec (NUMERAL _0)) , (V , E)))))))))"}
{"_id": "50747", "text": "INFSUM_0 |- (((infsum s) (\\i. (vec (NUMERAL _0)))) = (vec (NUMERAL _0)))"}
{"_id": "50748", "text": "Local_lemmas1.CON_ATREAL_REAL_CON2 |- (!f. ((((continuous f) (atreal t)) /\\ ((continuous g) (atreal t))) ==> ((real_continuous (\\t. (distance ((f t) , (g t))))) (atreal t))))"}
{"_id": "50749", "text": "Seq2.foldr |- (!x. (!s. (!f. (!z0. (((((foldr f) z0) ((CONS x) s)) = ((f x) (((foldr f) z0) s))) /\\ ((((foldr f) z0) NIL) = z0))))))"}
{"_id": "50750", "text": "REAL_INTEGRABLE_SPIKE_FINITE |- (!f. (!g. (!s. (((FINITE s) /\\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((real_integrable_on f) t) ==> ((real_integrable_on g) t))))))"}
{"_id": "50751", "text": "NSUM_UNIONS_NONZERO |- (!f. (!s. (((FINITE s) /\\ ((!t. ((t IN s) ==> (FINITE t))) /\\ (!t1. (!t2. (!x. (((t1 IN s) /\\ ((t2 IN s) /\\ ((~ (t1 = t2)) /\\ ((x IN t1) /\\ (x IN t2))))) ==> ((f x) = (NUMERAL _0)))))))) ==> (((nsum (UNIONS s)) f) = ((nsum s) (\\t. ((nsum t) f)))))))"}
{"_id": "50752", "text": "Localization.order |- (!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": "50753", "text": "COMPLEX_DIFFERENTIABLE_WITHIN_CLOG |- (!s. (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> ((complex_differentiable clog) ((within (at z)) s)))))"}
{"_id": "50754", "text": "INTEGRAL_PASTECART_CONST |- (!a. (!b. (!c. (!d. (!k. (((integral (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) (\\x. k)) = ((integral (closed_interval ((CONS (a , b)) NIL))) (\\x. ((integral (closed_interval ((CONS (c , d)) NIL))) (\\y. k))))))))))"}
{"_id": "50755", "text": "FSIGMA_PCROSS_EQ |- (!s. (!t. ((fsigma (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((fsigma s) /\\ (fsigma t)))))))"}
{"_id": "50756", "text": "FSIGMA_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((fsigma ((IMAGE f) s)) = (fsigma s)))))"}
{"_id": "50757", "text": "INTEGER_SUB_EQ |- ((!x. (!y. ((integer x) ==> ((integer ((real_sub x) y)) = (integer y))))) /\\ (!x. (!y. ((integer y) ==> ((integer ((real_sub x) y)) = (integer x))))))"}
{"_id": "50758", "text": "Topology.union_aff |- (!x. (!v. (UNIV = ((aff (x INSERT (v INSERT EMPTY))) UNION (complement_set (x INSERT (v INSERT EMPTY)))))))"}
{"_id": "50759", "text": "ANR_IMP_CLOSED_NEIGHBOURHOOD_RETRACT |- (!s. (!u. (((ANR 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": "50760", "text": "DIVMOD_ELIM_THM' |- (((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": "50761", "text": "SUMS_COMPLEX_0 |- (!f. (!s. ((!n. ((n IN s) ==> ((f n) = (Cx (real_of_num (NUMERAL _0)))))) ==> (((sums f) (Cx (real_of_num (NUMERAL _0)))) s))))"}
{"_id": "50762", "text": "MBOUNDED_SUBSET_MSPACE |- (!m. (!s. (((mbounded m) s) ==> (s SUBSET (mspace m)))))"}
{"_id": "50763", "text": "INT_ADD_LDISTRIB |- (!x. (!y. (!z. (((int_mul x) ((int_add y) z)) = ((int_add ((int_mul x) y)) ((int_mul x) z))))))"}
{"_id": "50764", "text": "TC_TC_RTC_CASES |- (!R. (!x. (!z. ((((TC R) x) z) = (?y. ((((TC R) x) y) /\\ (((RTC R) y) z)))))))"}
{"_id": "50765", "text": "LOCALLY_COMPACT_IMP_BOREL |- (!s. (((locally compact) s) ==> (borel s)))"}
{"_id": "50766", "text": "LIM_CONG_WITHINREAL |- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\\x. (f x))) l) ((within (atreal a)) s)) = (((--> g) l) ((within (atreal a)) s))))"}
{"_id": "50767", "text": "NOT_PSUBSET_EMPTY |- (!s. (~ (s PSUBSET EMPTY)))"}
{"_id": "50768", "text": "LOG_LE |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (log ((real_add (real_of_num (NUMERAL (BIT1 _0)))) x))) x)))"}
{"_id": "50769", "text": "CONVERGENT_REAL_BOUNDED_MONOTONE |- (!s. (((real_bounded ((IMAGE s) UNIV)) /\\ ((!n. ((real_le (s n)) (s (SUC n)))) \\/ (!n. ((real_le (s (SUC n))) (s n))))) ==> (?l. (((---> s) l) sequentially))))"}
{"_id": "50770", "text": "CENTRE_IN_MCBALL_EQ |- (!m. (!x. (!r. ((x IN ((mcball m) (x , r))) = ((x IN (mspace m)) /\\ ((real_le (real_of_num (NUMERAL _0))) r))))))"}
{"_id": "50771", "text": "SETDIST_FRONTIERS |- (!s. (!t. ((setdist (s , t)) = (((COND ((DISJOINT s) t)) (setdist ((frontier s) , (frontier t)))) (real_of_num (NUMERAL _0))))))"}
{"_id": "50772", "text": "Nonlinear_lemma.sqrt_x1 |- (((((((sqrt_x1 x1) x2) x3) x4) x5) x6) = (sqrt x1))"}
{"_id": "50773", "text": "Ssrnat.odd_double |- (!n. ((odd (double n)) = F))"}
{"_id": "50774", "text": "Seq.nth_incr_nth |- (!v. (!i. (!j. ((((nth (NUMERAL _0)) ((incr_nth v) i)) j) = ((((COND (i = j)) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + (((nth (NUMERAL _0)) v) j))))))"}
{"_id": "50775", "text": "Lp_main_estimate.ineq_tau6 |- (!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 (BIT1 (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 (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ ((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) 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 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d))))))))))"}
{"_id": "50776", "text": "INT_POW_LT |- (!x. (!n. (((int_lt (int_of_num (NUMERAL _0))) x) ==> ((int_lt (int_of_num (NUMERAL _0))) ((int_pow x) n)))))"}
{"_id": "50777", "text": "NPRODUCT_CONST_NUMSEG_1 |- (!c. (!n. (((nproduct ((NUMERAL (BIT1 _0)) .. n)) (\\x. c)) = (c EXP n))))"}
{"_id": "50778", "text": "COMPLEX_NEG_LMUL |- (!x. (!y. ((vector_neg ((complex_mul x) y)) = ((complex_mul (vector_neg x)) y))))"}
{"_id": "50779", "text": "COMPLEX_DIFFERENTIABLE_INV_AT |- (!f. (!z. ((((complex_differentiable f) (at z)) /\\ (~ ((f z) = (Cx (real_of_num (NUMERAL _0)))))) ==> ((complex_differentiable (\\z. (complex_inv (f z)))) (at z)))))"}
{"_id": "50780", "text": "REAL_VARIATION_TRIANGLE |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\\ ((has_bounded_real_variation_on g) s)) ==> ((real_le ((real_variation s) (\\x. ((real_add (f x)) (g x))))) ((real_add ((real_variation s) f)) ((real_variation s) g)))))))"}
{"_id": "50781", "text": "Packing3.VORONOI_SET_2_BIS_LE |- (!V. (!u. (!v. (((u IN V) /\\ (v IN V)) ==> (((voronoi_set V) (u INSERT (v INSERT EMPTY))) = (((voronoi_closed V) v) INTER ((bis_le u) v)))))))"}
{"_id": "50782", "text": "INSIDE_INSIDE_COMPACT_CONNECTED |- (!s. (!t. (((closed s) /\\ ((compact t) /\\ ((s SUBSET (inside t)) /\\ (connected t)))) ==> ((inside s) SUBSET (inside t)))))"}
{"_id": "50783", "text": "Lp_ineqs_proofs.rhazim_fan_eq_rhazim |- (!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))) ==> (((rhazim_fan (V , E)) d) = ((((((rhazim (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": "50784", "text": "Ssrnat.neq_ltn |- (!m. (!n. ((~ (m = n)) = ((m < n) \\/ (n < m)))))"}
{"_id": "50785", "text": "INT_MUL_ASSOC |- (!x. (!y. (!z. (((int_mul x) ((int_mul y) z)) = ((int_mul ((int_mul x) y)) z)))))"}
{"_id": "50786", "text": "Ssrbool.orKb |- (!a. (!b. ((a /\\ (b \\/ a)) = a)))"}
{"_id": "50787", "text": "ATN_LE_X |- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (atn x)) x)))"}
{"_id": "50788", "text": "FORALL_ALL |- (!P. (!l. ((!x. ((ALL (P x)) l)) = ((ALL (\\s. (!x. ((P x) s)))) l))))"}
{"_id": "50789", "text": "REAL_SUMMABLE_ZETA |- (!n. (!x. (((real_lt (real_of_num (NUMERAL (BIT1 _0)))) x) ==> ((real_summable (from n)) (\\k. (real_inv ((rpow (real_of_num k)) x)))))))"}
{"_id": "50790", "text": "Seq2.list_of_empty_set |- ((list_of_set EMPTY) = NIL)"}
{"_id": "50791", "text": "CONVEX_STARCENTRES |- (!s. (convex (GSPEC (\\GEN%PVAR%1721. (?a. (((SETSPEC GEN%PVAR%1721) ((a IN s) /\\ (!x. ((x IN s) ==> ((closed_segment ((CONS (a , x)) NIL)) SUBSET s))))) a))))))"}
{"_id": "50792", "text": "RE_POS_SEGMENT |- (!a. (!b. (!x. (((x IN (closed_segment ((CONS (a , b)) NIL))) /\\ (((real_lt (real_of_num (NUMERAL _0))) (Re a)) /\\ ((real_lt (real_of_num (NUMERAL _0))) (Re b)))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re x))))))"}
{"_id": "50793", "text": "Merge_ineq.nonf_flat_term2_234_x |- (!d. (!a. (!b. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((((flat_term2_234_x d) a) b) x1) x2) x3) x4) x5) x6) = (flat_term_x (((((((((edge2_234_x d) a) b) x1) x2) x3) x4) x5) x6))))))))))))"}
{"_id": "50794", "text": "SUM_MULTICOUNT |- (!R. (!s. (!t. (!k. (((FINITE s) /\\ ((FINITE t) /\\ (!j. ((j IN t) ==> ((CARD (GSPEC (\\GEN%PVAR%205. (?i. (((SETSPEC GEN%PVAR%205) ((i IN s) /\\ ((R i) j))) i))))) = k))))) ==> (((sum s) (\\i. (real_of_num (CARD (GSPEC (\\GEN%PVAR%206. (?j. (((SETSPEC GEN%PVAR%206) ((j IN t) /\\ ((R i) j))) j)))))))) = (real_of_num (k * (CARD t)))))))))"}
{"_id": "50795", "text": "ANALYTIC_ON_COMPOSE_GEN |- (!f. (!g. (!s. (!t. ((((analytic_on f) s) /\\ (((analytic_on g) t) /\\ (!z. ((z IN s) ==> ((f z) IN t))))) ==> ((analytic_on (g o f)) s))))))"}
{"_id": "50796", "text": "Merge_ineq.ineq_mp_simple |- (!a. (!b. (!c. (((ineq a) (b ==> c)) = (b ==> ((ineq a) c))))))"}
{"_id": "50797", "text": "MEASURABLE_ON_PREIMAGE_HALFSPACE_COMPONENT_GT_DENSE |- (!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%3601. (?x. (((SETSPEC GEN%PVAR%3601) ((real_gt ((f x) $ k)) a)) x))))))))))))"}
{"_id": "50798", "text": "CROSS_INTER |- ((!s. (!t. (!u. ((s CROSS (t INTER u)) = ((s CROSS t) INTER (s CROSS u)))))) /\\ (!s. (!t. (!u. (((s INTER t) CROSS u) = ((s CROSS u) INTER (t CROSS u)))))))"}
{"_id": "50799", "text": "REAL_POW |- (!z. (!n. ((real z) ==> (real ((complex_pow z) n)))))"}
{"_id": "50800", "text": "BOUNDED_SPHERE |- (!a. (!r. (bounded (sphere (a , r)))))"}
{"_id": "50801", "text": "NSUM_IMAGE_NONZERO |- (!d. (!i. (!s. (((FINITE s) /\\ (!x. (!y. (((x IN s) /\\ ((y IN s) /\\ ((~ (x = y)) /\\ ((i x) = (i y))))) ==> ((d (i x)) = (NUMERAL _0)))))) ==> (((nsum ((IMAGE i) s)) d) = ((nsum s) (d o i)))))))"}
{"_id": "50802", "text": "STC_REFL |- (!R. ((!x. ((R x) x)) ==> (!x. (((STC R) x) x))))"}
{"_id": "50803", "text": "HAS_DERIVATIVE_VSUM_NUMSEG |- (!f. (!net. (!m. (!n. ((!i. (((m <= i) /\\ (i <= n)) ==> (((has_derivative (f i)) (f' i)) net))) ==> (((has_derivative (\\x. ((vsum (m .. n)) (\\i. ((f i) x))))) (\\h. ((vsum (m .. n)) (\\i. ((f' i) h))))) net))))))"}
{"_id": "50804", "text": "EXTENSION_INTO_AR |- (!f. (!s. (!t. ((((continuous_on f) s) /\\ ((((IMAGE f) s) SUBSET t) /\\ (((retract_of t) UNIV) /\\ (closed s)))) ==> (?g. (((continuous_on g) UNIV) /\\ ((((IMAGE g) UNIV) SUBSET t) /\\ (!x. ((x IN s) ==> ((g x) = (f x)))))))))))"}
{"_id": "50805", "text": "HULL_UNIONS_SUBSET |- (!P. (!f. ((UNIONS (GSPEC (\\GEN%PVAR%96. (?s. (((SETSPEC GEN%PVAR%96) (s IN f)) ((hull P) s)))))) SUBSET ((hull P) (UNIONS f)))))"}
{"_id": "50806", "text": "CONTINUOUS_WITHIN_CTAN |- (!s. (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> ((continuous ctan) ((within (at z)) s)))))"}
{"_id": "50807", "text": "Upfzbzm_support_lemmas.FINITE_PERMUTE_3 |- (FINITE (GSPEC (\\GEN%PVAR%6211. (?p. (((SETSPEC GEN%PVAR%6211) ((permutes p) ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT ((NUMERAL (BIT0 (BIT1 _0))) INSERT EMPTY))))) p)))))"}
{"_id": "50808", "text": "INTEGRAL_COMBINE_TAGGED_DIVISION_BOTTOMUP |- (!f. (!p. (!a. (!b. ((((tagged_division_of p) (closed_interval ((CONS (a , b)) NIL))) /\\ (!x. (!k. (((x , k) IN p) ==> ((integrable_on f) k))))) ==> (((integral (closed_interval ((CONS (a , b)) NIL))) f) = ((vsum p) (GABS (\\f. (!x. (!k. ((GEQ (f (x , k))) ((integral k) f)))))))))))))"}
{"_id": "50809", "text": "INT_LE_LT |- (!x. (!y. (((int_le x) y) = (((int_lt x) y) \\/ (x = y)))))"}
{"_id": "50810", "text": "Nonlinear_lemma.sqrt_x6 |- (((((((sqrt_x6 x1) x2) x3) x4) x5) x6) = (sqrt x6))"}
{"_id": "50811", "text": "AFFINE_DEPENDENT_BIGGERSET_GENERAL |- (!s. (((FINITE s) ==> ((CARD s) >= ((dim s) + (NUMERAL (BIT0 (BIT1 _0)))))) ==> (affine_dependent s)))"}
{"_id": "50812", "text": "Lp_gen_theory.components_iso_image_face_map |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((((node H) (g ((face_map G) d))) = ((IMAGE g) ((node G) ((face_map G) d)))) /\\ ((((face H) (g ((face_map G) d))) = ((IMAGE g) ((face G) ((face_map G) d)))) /\\ ((((node H) (g ((inverse (face_map G)) d))) = ((IMAGE g) ((node G) ((inverse (face_map G)) d)))) /\\ (((face H) (g ((inverse (face_map G)) d))) = ((IMAGE g) ((face G) ((inverse (face_map G)) d)))))))))))))"}
{"_id": "50813", "text": "IM_COMPLEX_INV_GT_0 |- (!z. (((real_lt (real_of_num (NUMERAL _0))) (Im (complex_inv z))) = ((real_lt (Im z)) (real_of_num (NUMERAL _0)))))"}
{"_id": "50814", "text": "List_hypermap.flatten_filter_empty |- (!L. ((flatten ((filter (\\l. (~ (l = NIL)))) L)) = (flatten L)))"}
{"_id": "50815", "text": "Collect_geom.IN_SET4 |- ((x IN (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))) = ((x = a) \\/ ((x = b) \\/ ((x = c) \\/ (x = d)))))"}
{"_id": "50816", "text": "CFUNSPACE_MDIST_LT |- (!top. (!m. (!f. (!g. (!a. (!x. ((((compact_in (topspace top)) top) /\\ ((f IN (mspace ((cfunspace top) m))) /\\ ((g IN (mspace ((cfunspace top) m))) /\\ (((real_lt ((mdist ((cfunspace top) m)) (f , g))) a) /\\ (x IN (topspace top)))))) ==> ((real_lt ((mdist m) ((f x) , (g x)))) a))))))))"}
{"_id": "50817", "text": "COMPLEX_SUB_TRIANGLE |- (!a. (!b. (!c. (((vector_add ((vector_sub a) b)) ((vector_sub b) c)) = ((vector_sub a) c)))))"}
{"_id": "50818", "text": "REAL_SECOND_MEAN_VALUE_THEOREM_GEN |- (!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))) /\\ (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\\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)))))))))))))"}
{"_id": "50819", "text": "REALLIM_AT_NEGINFINITY |- (!f. (!l. ((((---> f) l) at_neginfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_le x) b) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))"}
{"_id": "50820", "text": "REAL_NEGLIGIBLE_ON_INTERVALS |- (!s. ((real_negligible s) = (!a. (!b. (real_negligible (s INTER (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "50821", "text": "REALLIM_WITHIN_OPEN |- (!f. (!l. (!a. (!s. (((a IN s) /\\ (open s)) ==> ((((---> f) l) ((within (at a)) s)) = (((---> f) l) (at a))))))))"}
{"_id": "50822", "text": "NO_RETRACTION_FRONTIER_BOUNDED |- (!s. (((bounded s) /\\ (~ ((interior s) = EMPTY))) ==> (~ ((retract_of (frontier s)) s))))"}
{"_id": "50823", "text": "Upfzbzm_support_lemmas.FINITE_PERMUTE_4 |- (FINITE (GSPEC (\\GEN%PVAR%6212. (?p. (((SETSPEC GEN%PVAR%6212) ((permutes p) ((NUMERAL _0) INSERT ((NUMERAL (BIT1 _0)) INSERT ((NUMERAL (BIT0 (BIT1 _0))) INSERT ((NUMERAL (BIT1 (BIT1 _0))) INSERT EMPTY)))))) p)))))"}
{"_id": "50824", "text": "SUMMABLE_SUBSET_ABSCONV |- (!x. (!s. (!t. ((((summable s) (\\n. (lift (vector_norm (x n))))) /\\ (t SUBSET s)) ==> ((summable t) (\\n. (lift (vector_norm (x n)))))))))"}
{"_id": "50825", "text": "BOHL_ALT |- (!f. (!s. (!a. ((((continuous_on f) s) /\\ ((convex s) /\\ ((compact s) /\\ ((a IN (interior s)) /\\ (((IMAGE f) s) SUBSET (UNIV DELETE a)))))) ==> (?x. ((x IN (frontier s)) /\\ (a IN (open_segment (x , (f x))))))))))"}
{"_id": "50826", "text": "Ssrbool.negbFE |- (!b. (((~ b) = F) ==> b))"}
{"_id": "50827", "text": "Nonlinear_lemma.rho_sqrtx |- (!x. ((rho (sqrt x)) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_div ((real_sub (sqrt x)) (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": "50828", "text": "OPEN_OPEN_LEFT_PROJECTION |- (!s. (!t. (((open s) /\\ (open t)) ==> (open (GSPEC (\\GEN%PVAR%1073. (?x. (((SETSPEC GEN%PVAR%1073) ((x IN s) /\\ (?y. (((pastecart x) y) IN t)))) x))))))))"}
{"_id": "50829", "text": "Lp_ineqs_proofs.y8_def |- (!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))) ==> (((y8_fan (V , E)) d) = ((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))"}
{"_id": "50830", "text": "HOLOMORPHIC_ON_CTAN |- (!s. ((!z. ((z IN s) ==> (~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((holomorphic_on ctan) s)))"}
{"_id": "50831", "text": "COMPLEX_DIFFERENTIABLE_DIV_AT |- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\\ (((complex_differentiable g) (at z)) /\\ (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((complex_differentiable (\\z. ((complex_div (f z)) (g z)))) (at z))))))"}
{"_id": "50832", "text": "NSUM_PERMUTE_NUMSEG |- (!f. (!p. (!m. (!n. (((permutes p) (m .. n)) ==> (((nsum (m .. n)) f) = ((nsum (m .. n)) (f o p))))))))"}
{"_id": "50833", "text": "REAL_ROOT_LE |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> (((real_le ((root n) x)) y) = ((real_le x) ((real_pow y) n)))))))"}
{"_id": "50834", "text": "ANALYTIC_ON_DIV |- (!f. (!g. (!s. ((((analytic_on f) s) /\\ (((analytic_on g) s) /\\ (!z. ((z IN s) ==> (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((analytic_on (\\z. ((complex_div (f z)) (g z)))) s)))))"}
{"_id": "50835", "text": "Ssrnat.ltn_trans |- (!n. (!m. (!p. ((m < n) ==> ((n < p) ==> (m < p))))))"}
{"_id": "50836", "text": "Ckqowsa_3_points.pos_vector_angle_bounds |- (!v. (!u. (((real_gt ((dot v) u)) (real_of_num (NUMERAL _0))) ==> (((real_le (real_of_num (NUMERAL _0))) ((vector_angle v) u)) /\\ ((real_le ((vector_angle v) u)) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))"}
{"_id": "50837", "text": "LIPSCHITZ_EXTENSION_EXISTS |- (!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)))))))) ==> (?g. ((?B. (!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": "50838", "text": "Pack2.sum_measure_voronoi_closed_le_ball |- (!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_closed S) v))))) (measure (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))"}
{"_id": "50839", "text": "Lp_ineqs_proofs.dart4_azim2_eq_sum |- (!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)))) ==> (((azim2_fan (V , E)) d) = ((real_add ((((((dih_y (y2_fan d)) ((y3_fan (V , E)) d)) (y1_fan d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y4'_fan (V , E)) d))) ((((((dih_y (y2_fan d)) ((y7_fan (V , E)) d)) ((y3_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y9_fan (V , E)) d))))))))))))"}
{"_id": "50840", "text": "Float.INT_POW_ODD_NEG1 |- (!x. (!n. (((int_pow (int_neg (int_of_num x))) (NUMERAL (BIT1 n))) = (int_neg ((int_pow (int_of_num x)) (NUMERAL (BIT1 n)))))))"}
{"_id": "50841", "text": "PATH_CONNECTED_SUMS |- (!s. (!t. (((path_connected s) /\\ (path_connected t)) ==> (path_connected (GSPEC (\\GEN%PVAR%1816. (?x. (?y. (((SETSPEC GEN%PVAR%1816) ((x IN s) /\\ (y IN t))) ((vector_add x) y))))))))))"}
{"_id": "50842", "text": "FUNDAMENTAL_GROUP_SIMPLY_CONNECTED |- (!s. (!a. (((simply_connected s) /\\ (a IN s)) ==> ((fundamental_group (s , a)) = (((homotopic_paths s) (linepath (a , a))) INSERT EMPTY)))))"}
{"_id": "50843", "text": "HAS_REAL_MEASURE_SCALING_EQ |- (!s. (!m. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> (((has_real_measure ((IMAGE (\\x. ((real_mul c) x))) s)) ((real_mul (real_abs c)) m)) = ((has_real_measure s) m))))))"}
{"_id": "50844", "text": "PATH_CONNECTED_COMPLEMENT_CARD_LT |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ (s <_c UNIV)) ==> (path_connected (UNIV DIFF s))))"}
{"_id": "50845", "text": "LIM_NULL_COMPLEX_LMUL |- (!net. (!f. (!c. ((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) ==> (((--> (\\x. ((complex_mul c) (f x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))"}
{"_id": "50846", "text": "REAL_NEGLIGIBLE_DIFF |- (!s. (!t. ((real_negligible s) ==> (real_negligible (s DIFF t)))))"}
{"_id": "50847", "text": "CLOSED_CONNECTED_PREIMAGES_IMP_CONTINUOUS_ON |- (!f. (!s. (!t. (((((IMAGE f) s) SUBSET t) /\\ ((compact t) /\\ ((!y. (connected (GSPEC (\\GEN%PVAR%1942. (?x. (((SETSPEC GEN%PVAR%1942) ((x IN s) /\\ ((f x) = y))) x)))))) /\\ ((!c. (((closed_in ((subtopology euclidean) s)) c) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) c)))) /\\ (!c. (((c SUBSET s) /\\ (connected c)) ==> (connected ((IMAGE f) c)))))))) ==> ((continuous_on f) s)))))"}
{"_id": "50848", "text": "NOT_ON_PATH_CBALL |- (!g. (!z. (((path g) /\\ (~ (z IN (path_image g)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (((cball (z , e)) INTER (path_image g)) = EMPTY))))))"}
{"_id": "50849", "text": "CONTINUOUS_COUNTABLE_RANGE_CONSTANT |- (!f. (!s. (((connected s) /\\ (((continuous_on f) s) /\\ (COUNTABLE ((IMAGE f) s)))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a)))))))"}
{"_id": "50850", "text": "ARG_ROTATE2D_UNIQUE_2PI |- (!s. (!t. (!z. (((~ (z = (Cx (real_of_num (NUMERAL _0))))) /\\ (((real_le (real_of_num (NUMERAL _0))) s) /\\ (((real_lt s) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\\ (((real_le (real_of_num (NUMERAL _0))) t) /\\ (((real_lt t) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\\ ((Arg ((rotate2d s) z)) = (Arg ((rotate2d t) z)))))))) ==> (s = t)))))"}
{"_id": "50851", "text": "INSIDE_CONNECTED_COMPONENT_LE |- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ (bounded s)) ==> ((inside s) = (GSPEC (\\GEN%PVAR%2016. (?x. (((SETSPEC GEN%PVAR%2016) ((~ (x IN s)) /\\ (!B. (?y. (((real_le B) (vector_norm y)) /\\ (~ (((connected_component (UNIV DIFF s)) x) y))))))) x)))))))"}
{"_id": "50852", "text": "CLOG_EQ |- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> (((clog w) = (clog z)) = (w = z)))))"}
{"_id": "50853", "text": "Ssrnat.distnn |- (!m. (((distn m) m) = (NUMERAL _0)))"}
{"_id": "50854", "text": "REAL_SGN_SIGN |- (!p. ((real_sgn (sign p)) = (sign p)))"}
{"_id": "50855", "text": "REAL_SERIES_RATIO |- (!c. (!a. (!s. (!N. ((((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\\ (!n. ((n >= N) ==> ((real_le (real_abs (a (SUC n)))) ((real_mul c) (real_abs (a n))))))) ==> (?l. (((real_sums a) l) s)))))))"}
{"_id": "50856", "text": "NORM_EQ_0_DOT |- (!x. (((vector_norm x) = (real_of_num (NUMERAL _0))) = (((dot x) x) = (real_of_num (NUMERAL _0)))))"}
{"_id": "50857", "text": "COMPLEX_SUB_NEG2 |- (!x. (!y. (((vector_sub (vector_neg x)) (vector_neg y)) = ((vector_sub y) x))))"}
{"_id": "50858", "text": "BILINEAR_CROSS |- (bilinear cross)"}
{"_id": "50859", "text": "RATIONAL_ALT |- (!x. ((rational x) = (?p. (?q. ((~ (q = (NUMERAL _0))) /\\ ((real_abs x) = ((real_div (real_of_num p)) (real_of_num q))))))))"}
{"_id": "50860", "text": "Merge_ineq.nonf_mudLs_126_x |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a. (!b. (!c. (!d. (!e. ((((((((((((mudLs_126_x a) b) c) d) e) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((mu6_x ((real_mul c) c)) x1) x2) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL _0)))) ((real_add ((real_mul ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_add a) b))) ((real_sub (((((((((delta_126_x ((real_mul c) c)) ((real_mul d) d)) ((real_mul e) e)) x1) x2) x3) x4) x5) x6)) ((real_mul a) a)))) a))))))))))))))"}
{"_id": "50861", "text": "REALLIM_TRANSFORM_WITHIN_SET_IMP |- (!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": "50862", "text": "REAL_CONVEX_ON_DERIVATIVE_INCREASING_IMP |- (!f. (!f'x. (!f'y. (!s. (!x. (!y. ((((real_convex_on f) s) /\\ (((closed_real_interval ((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_lt x) y))))) ==> ((real_le f'x) f'y))))))))"}
{"_id": "50863", "text": "CONTINUOUS_ON_DIST_CLOSEST_POINT |- (!s. (!t. (((closed s) /\\ (~ (s = EMPTY))) ==> ((continuous_on (\\x. (lift (distance (x , ((closest_point s) x)))))) t))))"}
{"_id": "50864", "text": "Lp_ineqs_proofs.dart3_azim_fan_eq_dih_y |- (!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)))) ==> (((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": "50865", "text": "POSITIVE_SEMIDEFINITE_MUL |- (!A. (!B. (((positive_semidefinite A) /\\ ((positive_semidefinite B) /\\ (((matrix_mul A) B) = ((matrix_mul B) A)))) ==> (positive_semidefinite ((matrix_mul A) B)))))"}
{"_id": "50866", "text": "DIFFERENTIABLE_ON_ADD |- (!f. (!g. (!s. ((((differentiable_on f) s) /\\ ((differentiable_on g) s)) ==> ((differentiable_on (\\z. ((vector_add (f z)) (g z)))) s)))))"}
{"_id": "50867", "text": "CONTINUOUS_ON_CACS |- (!s. ((!z. (((z IN s) /\\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((continuous_on cacs) s)))"}
{"_id": "50868", "text": "SETDIST_RELATIVE_INTERIOR |- ((!s. (!t. ((convex s) ==> ((setdist ((relative_interior s) , t)) = (setdist (s , t)))))) /\\ (!s. (!t. ((convex t) ==> ((setdist (s , (relative_interior t))) = (setdist (s , t)))))))"}
{"_id": "50869", "text": "ABSOLUTELY_INTEGRABLE_CONVOLUTION_L2 |- (!bop. (!f. (!g. (!x. (((bilinear bop) /\\ (((measurable_on f) UNIV) /\\ (((measurable_on g) UNIV) /\\ (((absolutely_integrable_on (\\x. (lift ((real_pow (vector_norm (f x))) (NUMERAL (BIT0 (BIT1 _0))))))) UNIV) /\\ ((absolutely_integrable_on (\\x. (lift ((real_pow (vector_norm (g x))) (NUMERAL (BIT0 (BIT1 _0))))))) UNIV))))) ==> ((absolutely_integrable_on (\\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))"}
{"_id": "50870", "text": "CCOS_CSIN_CSQRT |- (!z. ((((real_lt (real_of_num (NUMERAL _0))) (cos (Re z))) \\/ (((cos (Re z)) = (real_of_num (NUMERAL _0))) /\\ ((real_le ((real_mul (Im z)) (sin (Re z)))) (real_of_num (NUMERAL _0))))) ==> ((ccos z) = (csqrt ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow (csin z)) (NUMERAL (BIT0 (BIT1 _0)))))))))"}
{"_id": "50871", "text": "one_Axiom |- (!e. (?!fn. ((fn one) = e)))"}
{"_id": "50872", "text": "Ssrbool.contraFN |- (!c. (!b. ((c ==> b) ==> ((b = F) ==> (~ c)))))"}
{"_id": "50873", "text": "Hijqaha.J_SCS_4M5 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_4M5') i)) i1) j) = F)"}
{"_id": "50874", "text": "ANALYTIC_ON_SUB |- (!f. (!g. (!s. ((((analytic_on f) s) /\\ ((analytic_on g) s)) ==> ((analytic_on (\\z. ((vector_sub (f z)) (g z)))) s)))))"}
{"_id": "50875", "text": "LOG_ROOT |- (!n. (!x. (((~ (n = (NUMERAL _0))) /\\ ((real_lt (real_of_num (NUMERAL _0))) x)) ==> ((log ((root n) x)) = ((real_div (log x)) (real_of_num n))))))"}
{"_id": "50876", "text": "CONTINUOUS_ON_ABS |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\\x. (lambdai. (real_abs ((f x) $ i))))) s))))"}
{"_id": "50877", "text": "COMPLEX_DIFFERENTIABLE_WITHIN_CSIN |- (!s. (!z. ((complex_differentiable csin) ((within (at z)) s))))"}
{"_id": "50878", "text": "INSEG_ORDINAL |- (!l. (!m. ((((inseg m) l) /\\ (ordinal l)) ==> (ordinal m))))"}
{"_id": "50879", "text": "DIV_MULT2 |- (!m. (!n. (!p. ((~ ((m * p) = (NUMERAL _0))) ==> (((m * n) DIV (m * p)) = (n DIV p))))))"}
{"_id": "50880", "text": "Seq.mask_cons |- (!b. (!m. (!x. (!s. (((mask ((CONS b) m)) ((CONS x) s)) = ((cat ((nseq (((COND b) (NUMERAL (BIT1 _0))) (NUMERAL _0))) x)) ((mask m) s)))))))"}
{"_id": "50881", "text": "REAL_INTEGRAL_RESTRICT |- (!f. (!s. (!t. ((s SUBSET t) ==> (((real_integral t) (\\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) = ((real_integral s) f))))))"}
{"_id": "50882", "text": "AFF_LE_SPECIAL_SCALE |- (!s. (!t. (!a. (!v. (((FINITE s) /\\ ((FINITE t) /\\ ((~ ((vec (NUMERAL _0)) IN t)) /\\ ((~ (v IN t)) /\\ ((~ (((% a) v) IN t)) /\\ ((real_lt (real_of_num (NUMERAL _0))) a)))))) ==> (((aff_le ((vec (NUMERAL _0)) INSERT (((% a) v) INSERT s))) t) = ((aff_le ((vec (NUMERAL _0)) INSERT (v INSERT s))) t)))))))"}
{"_id": "50883", "text": "Ssrnat.subn_sqr |- (!m. (!n. (((m EXP (NUMERAL (BIT0 (BIT1 _0)))) - (n EXP (NUMERAL (BIT0 (BIT1 _0))))) = ((m - n) * (m + n)))))"}
{"_id": "50884", "text": "AFF_DIM_PSUBSET |- (!s. (!t. ((((hull affine) s) PSUBSET ((hull affine) t)) ==> ((int_lt (aff_dim s)) (aff_dim t)))))"}
{"_id": "50885", "text": "INSIDE_EQ_OUTSIDE |- (!s. (((inside s) = (outside s)) = (s = UNIV)))"}
{"_id": "50886", "text": "VSUM_SUPPORT |- (!f. (!s. (((vsum (GSPEC (\\GEN%PVAR%110. (?x. (((SETSPEC GEN%PVAR%110) ((x IN s) /\\ (~ ((f x) = (vec (NUMERAL _0)))))) x))))) f) = ((vsum s) f))))"}
{"_id": "50887", "text": "DIFFERENTIABLE_ON_NORM |- (!s. ((~ ((vec (NUMERAL _0)) IN s)) ==> ((differentiable_on (\\x. (lift (vector_norm x)))) s)))"}
{"_id": "50888", "text": "DIM_OPEN |- (!s. (((open s) /\\ (~ (s = EMPTY))) ==> ((dim s) = (dimindex UNIV))))"}
{"_id": "50889", "text": "CONVEX_FINITE |- (!s. ((FINITE s) ==> ((convex s) = (!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))) IN s))))))"}
{"_id": "50890", "text": "Misc_defs_and_lemmas.SURJ_SECTION |- (!a. (!b. (!f. ((((SURJ f) a) b) ==> (?g. (((INJ g) b) a))))))"}
{"_id": "50891", "text": "Nonlinear_lemma.sqrt_x4 |- (((((((sqrt_x4 x1) x2) x3) x4) x5) x6) = (sqrt x4))"}
{"_id": "50892", "text": "INT_MIN_MIN |- (!x. (!y. (((int_le ((int_min x) y)) x) /\\ ((int_le ((int_min x) y)) y))))"}
{"_id": "50893", "text": "ANALYTIC_AT_MUL |- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\\ ((analytic_on g) (z INSERT EMPTY))) ==> ((analytic_on (\\w. ((complex_mul (f w)) (g w)))) (z INSERT EMPTY))))))"}
{"_id": "50894", "text": "CASN_UNIQUE |- (!w. (!z. ((((csin z) = w) /\\ (((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 w) = z))))"}
{"_id": "50895", "text": "COUNTABLE_CASES |- (!s. ((COUNTABLE s) = ((FINITE s) \\/ (s =_c UNIV))))"}
{"_id": "50896", "text": "AFF_LOWDIM_SUBSET_HYPERPLANE |- (!s. (((int_lt (aff_dim s)) (int_of_num (dimindex UNIV))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\\ (s SUBSET (GSPEC (\\GEN%PVAR%1597. (?x. (((SETSPEC GEN%PVAR%1597) (((dot a) x) = b)) x))))))))))"}
{"_id": "50897", "text": "HAS_INTEGRAL_SPIKE_FINITE_EQ |- (!f. (!g. (!s. (!y. (((FINITE s) /\\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> ((((has_integral f) y) t) = (((has_integral g) y) t)))))))"}
{"_id": "50898", "text": "AFFINE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((affine ((IMAGE (\\x. ((% c) x))) s)) = (affine s)))))"}
{"_id": "50899", "text": "ABSOLUTELY_CONTINUOUS_ON_VMUL |- (!f. (!s. (!v. (((absolutely_continuous_on (\\x. (lift (f x)))) s) ==> ((absolutely_continuous_on (\\x. ((% (f x)) v))) s)))))"}
{"_id": "50900", "text": "MANHATTAN |- (((mspace manhattan) = UNIV) /\\ (!x. (!y. (((mdist manhattan) (x , y)) = ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. (real_abs ((real_sub (x $ i)) (y $ i)))))))))"}
{"_id": "50901", "text": "VECTOR_POLYNOMIAL_FUNCTION_o |- (!f. (!g. (((vector_polynomial_function f) /\\ (vector_polynomial_function g)) ==> (vector_polynomial_function (g o f)))))"}
{"_id": "50902", "text": "Tactics_jordan.switch_lemma_let |- (!A. (!B. (!C. ((((real_lt (real_of_num (NUMERAL _0))) A) ==> (((real_le (real_of_num (NUMERAL _0))) B) ==> C)) = (((real_le (real_of_num (NUMERAL _0))) B) ==> (((real_lt (real_of_num (NUMERAL _0))) A) ==> C))))))"}
{"_id": "50903", "text": "CONTINUOUS_ON_LIFT_ABS_COMPONENT |- (!f. (!s. (((continuous_on f) s) ==> ((continuous_on (\\x. (lift (real_abs ((f x) $ k))))) s))))"}
{"_id": "50904", "text": "REAL_SERIES_BOUND |- (!f. (!g. (!s. (!a. (!b. (((((real_sums f) a) s) /\\ ((((real_sums g) b) s) /\\ (!i. ((i IN s) ==> ((real_le (real_abs (f i))) (g i)))))) ==> ((real_le (real_abs a)) b)))))))"}
{"_id": "50905", "text": "MEASURE_CBALL_SCALING |- (!a. (!c. (!r. (((real_le (real_of_num (NUMERAL _0))) c) ==> ((measure (cball (a , ((real_mul c) r)))) = ((real_mul ((real_pow c) (dimindex UNIV))) (measure (cball (a , r)))))))))"}
{"_id": "50906", "text": "REAL_MEASURABLE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((real_measurable ((IMAGE (\\x. ((real_mul c) x))) s)) = (real_measurable s)))))"}
{"_id": "50907", "text": "LIM_INFINITY_SEQUENTIALLY_COMPLEX |- (!f. (!l. ((((--> f) l) at_infinity) ==> (((--> (\\n. (f (Cx (real_of_num n))))) l) sequentially))))"}
{"_id": "50908", "text": "VECTOR_AFFINITY_EQ |- (!m. (!c. (!x. (!y. ((~ (m = (real_of_num (NUMERAL _0)))) ==> ((((vector_add ((% m) x)) c) = y) = (x = ((vector_add ((% (real_inv m)) y)) (vector_neg ((% (real_inv m)) c))))))))))"}
{"_id": "50909", "text": "IS_INTERVAL_IMP_FSIGMA |- (!s. ((is_interval s) ==> (fsigma s)))"}
{"_id": "50910", "text": "FORALL_TRIPLED_THM |- (!P. ((! (GABS (\\f. (!x. (!y. (!z. ((GEQ (f (x , (y , z)))) (((P x) y) z)))))))) = (!x. (!y. (!z. (((P x) y) z))))))"}
{"_id": "50911", "text": "INT_SUB_SUB2 |- (!x. (!y. (((int_sub x) ((int_sub x) y)) = y)))"}
{"_id": "50912", "text": "CONTINUOUS_ON_MDIST |- (!m. (!a. ((a IN (mspace m)) ==> (((topcontinuous (mtopology m)) euclideanreal) (\\x. ((mdist m) (a , x)))))))"}
{"_id": "50913", "text": "SUBSET_CONVEX_HULL_RELATIVE_FRONTIER |- (!s. (((bounded s) /\\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (s SUBSET ((hull convex) (relative_frontier s)))))"}
{"_id": "50914", "text": "Tactics_jordan.expand_prod_le |- (!B1. (!B2. (!C. ((((real_le (real_of_num (NUMERAL _0))) ((real_mul B1) B2)) ==> C) ==> (((real_le (real_of_num (NUMERAL _0))) B1) ==> (((real_le (real_of_num (NUMERAL _0))) B2) ==> C))))))"}
{"_id": "50915", "text": "Ssrnat.half_gt0 |- (!n. (((NUMERAL _0) < (half n)) = ((NUMERAL (BIT1 _0)) < n)))"}
{"_id": "50916", "text": "Hypermap_iso.finite_inj_eq_bij |- (!s. (!t. ((FINITE s) ==> ((FINITE t) ==> (((CARD s) = (CARD t)) ==> (!f. ((((INJ f) s) t) = (((BIJ f) s) t))))))))"}
{"_id": "50917", "text": "FLOOR_DIV_DIV |- (!m. (!n. ((~ (m = (NUMERAL _0))) ==> ((floor ((real_div (real_of_num n)) (real_of_num m))) = (real_of_num (n DIV m))))))"}
{"_id": "50918", "text": "CARD_EQ_FULLSIZE_POWERSET |- (!s. ((INFINITE s) ==> ((GSPEC (\\GEN%PVAR%51. (?t. (((SETSPEC GEN%PVAR%51) ((t SUBSET s) /\\ (t =_c s))) t)))) =_c (GSPEC (\\GEN%PVAR%52. (?t. (((SETSPEC GEN%PVAR%52) (t SUBSET s)) t)))))))"}
{"_id": "50919", "text": "ABSOLUTELY_REAL_INTEGRABLE_INCREASING |- (!f. (!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 f) (closed_real_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "50920", "text": "Nonlinear_lemma.c2001 |- (!y. (((real_le y) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))))"}
{"_id": "50921", "text": "REAL_ABS_SIN_BOUND_LE |- (!x. ((real_le (real_abs (sin x))) (real_abs x)))"}
{"_id": "50922", "text": "FACES_OF_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((GSPEC (\\GEN%PVAR%2138. (?t. (((SETSPEC GEN%PVAR%2138) ((face_of t) ((IMAGE f) s))) t)))) = ((IMAGE (IMAGE f)) (GSPEC (\\GEN%PVAR%2139. (?t. (((SETSPEC GEN%PVAR%2139) ((face_of t) s)) t)))))))))"}
{"_id": "50923", "text": "VECTOR_DIFF_CHAIN_AT |- (!f. (!g. (!f'. (!g'. (!x. (((((has_vector_derivative f) f') (at x)) /\\ (((has_vector_derivative g) g') (at (f x)))) ==> (((has_vector_derivative (g o f)) ((% (drop f')) g')) (at x))))))))"}
{"_id": "50924", "text": "GRASSMANN_PLUCKER_3 |- (!x1. (!x2. (!x3. (!y1. (!y2. (!y3. (((real_mul (det (vector ((CONS x1) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS y2) ((CONS y3) NIL)))))) = ((real_add ((real_mul (det (vector ((CONS y1) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS x1) ((CONS y2) ((CONS y3) NIL))))))) ((real_add ((real_mul (det (vector ((CONS y2) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS x1) ((CONS y3) NIL))))))) ((real_mul (det (vector ((CONS y3) ((CONS x2) ((CONS x3) NIL)))))) (det (vector ((CONS y1) ((CONS y2) ((CONS x1) NIL)))))))))))))))"}
{"_id": "50925", "text": "SET_VARIATION_LBOUND_ON_INTERVAL |- (!f. (!a. (!b. (!B. ((((has_bounded_setvariation_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ (?d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) /\\ ((real_le B) ((sum d) (\\k. (vector_norm (f k)))))))) ==> ((real_le B) ((set_variation (closed_interval ((CONS (a , b)) NIL))) f)))))))"}
{"_id": "50926", "text": "REAL_ADD_COS |- (!x. (!y. (((real_add (cos x)) (cos y)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (cos ((real_div ((real_add x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) (cos ((real_div ((real_sub x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))"}
{"_id": "50927", "text": "Float.ABS_NUM |- (!m. (!n. ((real_abs ((real_sub (real_of_num n)) (real_of_num m))) = (real_of_num ((m - n) + (n - m))))))"}
{"_id": "50928", "text": "COMPLEX_DIFFERENTIABLE_WITHIN_SUBSET |- (!f. (!s. (!t. ((((complex_differentiable f) ((within (at x)) s)) /\\ (t SUBSET s)) ==> ((complex_differentiable f) ((within (at x)) t))))))"}
{"_id": "50929", "text": "GDELTA_PCROSS_EQ |- (!s. (!t. ((gdelta (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ ((gdelta s) /\\ (gdelta t)))))))"}
{"_id": "50930", "text": "COMPLEX_NEG_EQ |- (!x. (!y. (((vector_neg x) = y) = (x = (vector_neg y)))))"}
{"_id": "50931", "text": "NPRODUCT_CLAUSES_LEFT |- (!f. (!m. (!n. ((m <= n) ==> (((nproduct (m .. n)) f) = ((f m) * ((nproduct ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))"}
{"_id": "50932", "text": "Add_triangle.fully_surrounded_card_set_of_edge |- (!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 (BIT1 _0)))))))))"}
{"_id": "50933", "text": "DIM_SPECIAL_SUBSPACE |- (!k. ((dim (GSPEC (\\GEN%PVAR%216. (?x. (((SETSPEC GEN%PVAR%216) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ ((i <= (dimindex UNIV)) /\\ (i IN k))) ==> ((x $ i) = (real_of_num (NUMERAL _0)))))) x))))) = (CARD (((NUMERAL (BIT1 _0)) .. (dimindex UNIV)) DIFF k))))"}
{"_id": "50934", "text": "HOMEOMORPHIC_COMPACT_ARNESS |- (!s. (!s'. (((homeomorphic s) s') ==> (((compact s) /\\ ((retract_of s) UNIV)) = ((compact s') /\\ ((retract_of s') UNIV))))))"}
{"_id": "50935", "text": "REAL_SUMS_SUMMABLE |- (!f. (!l. (!s. ((((real_sums f) l) s) ==> ((real_summable s) f)))))"}
{"_id": "50936", "text": "INTERIOR_SURJECTIVE_LINEAR_IMAGE |- (!f. (!s. (((linear f) /\\ (!y. (?x. ((f x) = y)))) ==> ((interior ((IMAGE f) s)) = ((IMAGE f) (interior s))))))"}
{"_id": "50937", "text": "INT_POW_LT2 |- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\\ (((int_le (int_of_num (NUMERAL _0))) x) /\\ ((int_lt x) y))) ==> ((int_lt ((int_pow x) n)) ((int_pow y) n))))))"}
{"_id": "50938", "text": "HOLOMORPHIC_ON_CASN |- (!s. ((!z. (((z IN s) /\\ ((Im z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((holomorphic_on casn) s)))"}
{"_id": "50939", "text": "CART_EQ_FULL |- (!x. (!y. ((x = y) = (!i. ((x $ i) = (y $ i))))))"}
{"_id": "50940", "text": "MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE_AE |- (!f. (!g. (!s. (!t. ((((measurable_on f) s) /\\ (((integrable_on g) s) /\\ ((negligible t) /\\ (!x. ((x IN (s DIFF t)) ==> ((real_le (vector_norm (f x))) (drop (g x)))))))) ==> ((absolutely_integrable_on f) s))))))"}
{"_id": "50941", "text": "DOT_4 |- (((dot x) y) = ((real_add ((real_mul (x $ (NUMERAL (BIT1 _0)))) (y $ (NUMERAL (BIT1 _0))))) ((real_add ((real_mul (x $ (NUMERAL (BIT0 (BIT1 _0))))) (y $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul (x $ (NUMERAL (BIT1 (BIT1 _0))))) (y $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul (x $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (y $ (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))"}
{"_id": "50942", "text": "AFFINE_HULL_AFFINE_INTER_OPEN_IN |- (!s. (!t. (((affine s) /\\ (((open_in ((subtopology euclidean) s)) t) /\\ (~ ((s INTER t) = EMPTY)))) ==> (((hull affine) (s INTER t)) = s))))"}
{"_id": "50943", "text": "Nonlinear_lemma.rad2_y_e |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((rad2_y y1) y2) y3) y4) y5) y6) = (((((((y_of_x rad2_x) y1) y2) y3) y4) y5) y6))))))))"}
{"_id": "50944", "text": "MIDPOINTS_IN_CONVEX_HULL |- (!x. (!s. (((x IN ((hull convex) s)) /\\ (y IN ((hull convex) s))) ==> ((midpoint (x , y)) IN ((hull convex) s)))))"}
{"_id": "50945", "text": "Asfutbf.INTER_NEG |- (!A. (!B. ((((IMAGE vector_neg) A) INTER ((IMAGE vector_neg) B)) = ((IMAGE vector_neg) (A INTER B)))))"}
{"_id": "50946", "text": "ABSOLUTELY_INTEGRABLE_ON_MUL_BERNOULLI_FRAC |- (!f. (!s. (!n. (((absolutely_integrable_on f) s) ==> ((absolutely_integrable_on (\\x. ((% ((bernoulli n) (frac (drop x)))) (f x)))) s)))))"}
{"_id": "50947", "text": "REAL_SUMMABLE_LMUL |- (!s. (!x. (!c. (((real_summable s) x) ==> ((real_summable s) (\\n. ((real_mul c) (x n))))))))"}
{"_id": "50948", "text": "HAS_BOUNDED_REAL_VARIATION_REFLECT_EQ_INTERVAL |- (!f. (!u. (!v. (((has_bounded_real_variation_on (\\x. (f (real_neg x)))) (closed_real_interval ((CONS (u , v)) NIL))) = ((has_bounded_real_variation_on f) (closed_real_interval ((CONS ((real_neg v) , (real_neg u))) NIL)))))))"}
{"_id": "50949", "text": "REAL_INTEGRABLE_SPIKE_INTERIOR |- (!f. (!g. (!a. (!b. ((!x. ((x IN (open_real_interval (a , b))) ==> ((g x) = (f x)))) ==> (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) ==> ((real_integrable_on g) (closed_real_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "50950", "text": "Float.FLOAT_ADD_PPv2 |- (!a. (!b. (!m. (!n. ((m < n) ==> (((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 (n - m)))) b)) a)) (int_of_num m))))))))"}
{"_id": "50951", "text": "More_list_hypermap.nodes_hypermap_of_list_all |- (!L. ((good_list_nodes L) ==> ((ALL (\\n. ((ALL (\\d. (((node (hypermap_of_list L)) d) = (set_of_list n)))) n))) (list_of_nodes L))))"}
{"_id": "50952", "text": "SUP_INSERT_INSERT |- (!a. (!b. (!s. ((sup (b INSERT (a INSERT s))) = (sup (((real_max a) b) INSERT s))))))"}
{"_id": "50953", "text": "AFF_LT_EQ_AFFINE_HULL |- (!s. (!t. (((aff_lt s) EMPTY) = ((hull affine) s))))"}
{"_id": "50954", "text": "Nonlinear_lemma.lmdih5_ldih5' |- (!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))) ==> (((((((lmdih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6)) = ((((((ldih5_x_div_sqrtdelta_posbranch ((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": "50955", "text": "TC_RTC_CASES_R |- (!R. (!x. (!z. ((((TC R) x) z) = (?y. (((R x) y) /\\ (((RTC R) y) z)))))))"}
{"_id": "50956", "text": "LIM_AT_INFINITY_WITHIN_POS |- (!f. (!l. (!s. ((((--> f) l) ((within at_infinity) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (((real_lt (real_of_num (NUMERAL _0))) b) /\\ (!x. (((x IN s) /\\ ((real_ge (vector_norm x)) b)) ==> ((real_lt (distance ((f x) , l))) e)))))))))))"}
{"_id": "50957", "text": "COMPLEX_RNEG_UNIQ |- (!x. (!y. ((((vector_add x) y) = (Cx (real_of_num (NUMERAL _0)))) = (y = (vector_neg x)))))"}
{"_id": "50958", "text": "BOUNDED_RELATIVE_INTERIOR |- (!s. ((bounded s) ==> (bounded (relative_interior s))))"}
{"_id": "50959", "text": "Ardbzye.J_SCS_3T3 |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3T3) i)) i1) j) = F)"}
{"_id": "50960", "text": "HOMOTOPY_EQUIVALENT_SING |- (!s. (!a. (((homotopy_equivalent s) (a INSERT EMPTY)) = ((~ (s = EMPTY)) /\\ (contractible s)))))"}
{"_id": "50961", "text": "Packing3.MINIMAL_INTERS_EXISTS |- (!s. (!f. (((FINITE f) /\\ (s = (INTERS f))) ==> (?g. ((g SUBSET f) /\\ ((s = (INTERS g)) /\\ (!g'. ((g' PSUBSET g) ==> (s PSUBSET (INTERS g'))))))))))"}
{"_id": "50962", "text": "FORALL_IN_INTERMEDIATE_CLOSURE_EQ |- (!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": "50963", "text": "DIVISION_OF_UNIONS |- (!f. (((FINITE f) /\\ ((!p. ((p IN f) ==> ((division_of p) (UNIONS p)))) /\\ (!k1. (!k2. (((k1 IN (UNIONS f)) /\\ ((k2 IN (UNIONS f)) /\\ (~ (k1 = k2)))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))))) ==> ((division_of (UNIONS f)) (UNIONS (UNIONS f)))))"}
{"_id": "50964", "text": "SUM_GP_OFFSET |- (!x. (!m. (!n. (((sum (m .. (m + n))) (\\i. ((real_pow x) i))) = (((COND (x = (real_of_num (NUMERAL (BIT1 _0))))) ((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0))))) ((real_mul ((real_pow x) m)) ((real_div ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) ((real_pow x) (SUC n)))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) x))))))))"}
{"_id": "50965", "text": "REAL_CONTINUOUS_AT_RPOW_RIGHT |- (!a. (!x. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((real_continuous (\\x. ((rpow a) x))) (atreal x)))))"}
{"_id": "50966", "text": "CONNECTED_FROM_OPEN_UNION_AND_INTER |- (!s. (!t. (((open s) /\\ ((open t) /\\ ((connected (s UNION t)) /\\ (connected (s INTER t))))) ==> ((connected s) /\\ (connected t)))))"}
{"_id": "50967", "text": "CONTINUOUS_IMP_REAL_MEASURABLE_ON |- (!f. (((real_continuous_on f) UNIV) ==> ((real_measurable_on f) UNIV)))"}
{"_id": "50968", "text": "DIM_NONEMPTY_INTERIOR |- (!s. ((~ ((interior s) = EMPTY)) ==> ((dim s) = (dimindex UNIV))))"}
{"_id": "50969", "text": "SIMPLE_PATH_CONTINUOUS_IMAGE |- (!f. (!g. (((simple_path g) /\\ (((continuous_on f) (path_image g)) /\\ (!x. (!y. (((x IN (path_image g)) /\\ ((y IN (path_image g)) /\\ ((f x) = (f y)))) ==> (x = y)))))) ==> (simple_path (f o g)))))"}
{"_id": "50970", "text": "CONVEX_ON_EPIGRAPH_SLICE_LT |- (!f. (!s. (!a. ((((convex_on f) s) /\\ (convex s)) ==> (convex (GSPEC (\\GEN%PVAR%1705. (?x. (((SETSPEC GEN%PVAR%1705) ((x IN s) /\\ ((real_lt (f x)) a))) x)))))))))"}
{"_id": "50971", "text": "TRANSP_ROWVECTOR |- (!v. ((transp (rowvector v)) = (columnvector v)))"}
{"_id": "50972", "text": "ALL_IMP |- (!P. (!Q. (!l. (((!x. ((((MEM x) l) /\\ (P x)) ==> (Q x))) /\\ ((ALL P) l)) ==> ((ALL Q) l)))))"}
{"_id": "50973", "text": "Ssrbool.andKb |- (!a. (!b. ((a \\/ (b /\\ a)) = a)))"}
{"_id": "50974", "text": "COMMUTING_MATRIX_INV_NORMAL |- (!A. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> (((matrix_mul (matrix_inv A)) A) = ((matrix_mul A) (matrix_inv A)))))"}
{"_id": "50975", "text": "SUBSPACE_EQ_AFFINE |- (!s. ((subspace s) = ((affine s) /\\ ((vec (NUMERAL _0)) IN s))))"}
{"_id": "50976", "text": "EXTREME_POINT_OF_CONVEX_HULL_INSERT_EQ |- (!s. (!a. (!x. (((FINITE s) /\\ (~ (a IN ((hull affine) s)))) ==> (((extreme_point_of x) ((hull convex) (a INSERT s))) = ((x = a) \\/ ((extreme_point_of x) ((hull convex) s))))))))"}
{"_id": "50977", "text": "REAL_INTEGRAL_EQ_0 |- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (real_of_num (NUMERAL _0))))) ==> (((real_integral s) f) = (real_of_num (NUMERAL _0))))))"}
{"_id": "50978", "text": "Miqmcsn.STAB_4M8_13_SCS |- ((((scs_diag (scs_k_v39 scs_4M8_13)) i) j) ==> ((is_scs_v39 (((scs_stab_diag_v39 scs_4M8_13) i) j)) /\\ (scs_basic_v39 (((scs_stab_diag_v39 scs_4M8_13) i) j))))"}
{"_id": "50979", "text": "LINEAR_OPEN_MAPPING |- (!f. (!g. (((linear f) /\\ ((linear g) /\\ ((f o g) = I))) ==> (!s. ((open s) ==> (open ((IMAGE f) s)))))))"}
{"_id": "50980", "text": "Tactics_jordan.strip_lt_lemma |- (!B1. (!B2. (!C. ((((real_lt (real_of_num (NUMERAL _0))) ((real_add B1) B2)) ==> C) ==> (((real_lt (real_of_num (NUMERAL _0))) B2) ==> (((real_le (real_of_num (NUMERAL _0))) B1) ==> C))))))"}
{"_id": "50981", "text": "Lp_ineqs_proofs.sol_y_sym_reflect |- (!y1. (!y3. (!y2. (!y4. (!y6. (!y5. (((((((sol_y y1) y2) y3) y4) y5) y6) = ((((((sol_y y1) y3) y2) y4) y6) y5))))))))"}
{"_id": "50982", "text": "Ssrnat.half_leq |- (!m. (!n. ((m <= n) ==> ((half m) <= (half n)))))"}
{"_id": "50983", "text": "CARD_FINITE_IMAGE |- (!s. ((CARD UNIV) = (dimindex s)))"}
{"_id": "50984", "text": "TAGGED_DIVISION_OF_TRIVIAL |- (!p. (((tagged_division_of p) EMPTY) = (p = EMPTY)))"}
{"_id": "50985", "text": "ROTATE2D_ADD_VECTORS |- (!t. (!w. (!z. (((rotate2d t) ((vector_add w) z)) = ((vector_add ((rotate2d t) w)) ((rotate2d t) z))))))"}
{"_id": "50986", "text": "INTERVAL_IMAGE_STRETCH_INTERVAL |- (!a. (!b. (!m. (?u. (?v. (((IMAGE (\\x. (lambdak. ((real_mul (m k)) (x $ k))))) (closed_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS (u , v)) NIL))))))))"}
{"_id": "50987", "text": "SIMPLEX_IMP_CLOSED |- (!s. (!n. (((simplex n) s) ==> (closed s))))"}
{"_id": "50988", "text": "DIV_MONO_LT |- (!m. (!n. (!p. (((~ (p = (NUMERAL _0))) /\\ ((m + p) <= n)) ==> ((m DIV p) < (n DIV p))))))"}
{"_id": "50989", "text": "LOG_SQRT |- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log (sqrt x)) = ((real_div (log x)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "50990", "text": "Vol1.RADIAL_VS_RADIAL_NORM |- (!x. (!r. (!C. ((((radial r) x) C) = (((radial_norm r) x) C)))))"}
{"_id": "50991", "text": "Seq.mask1 |- (!b. (!x. (((mask ((CONS b) NIL)) ((CONS x) NIL)) = ((nseq (((COND b) (NUMERAL (BIT1 _0))) (NUMERAL _0))) x))))"}
{"_id": "50992", "text": "ALL2_MAP |- (!P. (!f. (!l. ((((ALL2 P) ((MAP f) l)) l) = ((ALL (\\a. ((P (f a)) a))) l)))))"}
{"_id": "50993", "text": "Lp_main_estimate.ineq_tau6_tauVEF |- (!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 (BIT1 (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 (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\\ ((real_le (ye_fan (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) 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 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (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": "50994", "text": "Trigonometry1.VECTOR_NEG_COMPONENT_3 |- (!x. ((((vector_neg x) $ (NUMERAL (BIT1 _0))) = (real_neg (x $ (NUMERAL (BIT1 _0))))) /\\ ((((vector_neg x) $ (NUMERAL (BIT0 (BIT1 _0)))) = (real_neg (x $ (NUMERAL (BIT0 (BIT1 _0)))))) /\\ (((vector_neg x) $ (NUMERAL (BIT1 (BIT1 _0)))) = (real_neg (x $ (NUMERAL (BIT1 (BIT1 _0)))))))))"}
{"_id": "50995", "text": "REAL_MUL_SIN_COS |- (!x. (!y. (((real_mul (sin x)) (cos y)) = ((real_div ((real_add (sin ((real_add x) y))) (sin ((real_sub x) y)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "50996", "text": "POLYTOPE_VERTEX_LOWER_BOUND |- (!p. ((polytope p) ==> ((int_le ((int_add (aff_dim p)) (int_of_num (NUMERAL (BIT1 _0))))) (int_of_num (CARD (GSPEC (\\GEN%PVAR%2456. (?v. (((SETSPEC GEN%PVAR%2456) ((extreme_point_of v) p)) v)))))))))"}
{"_id": "50997", "text": "INT_LT_ADDNEG2 |- (!x. (!y. (!z. (((int_lt ((int_add x) (int_neg y))) z) = ((int_lt x) ((int_add z) y))))))"}
{"_id": "50998", "text": "Hypermap_iso.image_res |- (!f. (!s. (!u. ((s SUBSET u) ==> (((IMAGE f) s) = ((IMAGE ((res f) u)) s))))))"}
{"_id": "50999", "text": "Lp_ineqs_proofs.rhazim3c |- (!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))) ==> (((rhazim3_fan (V , E)) d) = ((rhazim_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))"}
{"_id": "51000", "text": "REAL_INTEGRABLE_INCREASING_PRODUCT |- (!f. (!g. (!a. (!b. ((((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)))))) ==> ((real_integrable_on (\\x. ((real_mul (g x)) (f x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "51001", "text": "FACE_OF_SLICE |- (!f. (!s. (!t. ((((face_of f) s) /\\ (convex t)) ==> ((face_of (f INTER t)) (s INTER t))))))"}
{"_id": "51002", "text": "INT_MIN_LT |- (!x. (!y. (!z. (((int_lt ((int_min x) y)) z) = (((int_lt x) z) \\/ ((int_lt y) z))))))"}
{"_id": "51003", "text": "INT_EXISTS_ABS |- (!P. ((?n. (P (int_of_num n))) = (?x. (P (int_abs x)))))"}
{"_id": "51004", "text": "SUMMABLE_ADD |- (!x. (!y. (!s. ((((summable s) x) /\\ ((summable s) y)) ==> ((summable s) (\\n. ((vector_add (x n)) (y n))))))))"}
{"_id": "51005", "text": "LIM_NULL_COMPLEX_RMUL |- (!net. (!f. (!c. ((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) ==> (((--> (\\x. ((complex_mul (f x)) c))) (Cx (real_of_num (NUMERAL _0)))) net)))))"}
{"_id": "51006", "text": "Ssrnat.succn_inj |- (!n. (!m. (((SUC n) = (SUC m)) ==> (n = m))))"}
{"_id": "51007", "text": "Seq.zip_unzip |- (!s. (((zip (unzip1 s)) (unzip2 s)) = s))"}
{"_id": "51008", "text": "HAS_DERIVATIVE_INVERSE_ON |- (!f. (!s. (((open s) /\\ (!x. ((x IN s) ==> ((((has_derivative f) (f' x)) (at x)) /\\ (((g (f x)) = x) /\\ (((f' x) o (g' x)) = I)))))) ==> (!x. ((x IN s) ==> (((has_derivative g) (g' x)) (at (f x))))))))"}
{"_id": "51009", "text": "REAL_UNIFORMLY_CONTINUOUS_ON_ADD |- (!f. (!g. (!s. ((((real_uniformly_continuous_on f) s) /\\ ((real_uniformly_continuous_on g) s)) ==> ((real_uniformly_continuous_on (\\x. ((real_add (f x)) (g x)))) s)))))"}
{"_id": "51010", "text": "OPEN_INTERVAL_EQ |- ((!a. (!b. ((open (closed_interval ((CONS (a , b)) NIL))) = ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)))) /\\ (!a. (!b. (open (open_interval (a , b))))))"}
{"_id": "51011", "text": "FINITE_FUNSPACE_UNIV |- (((FINITE UNIV) /\\ (FINITE UNIV)) ==> (FINITE UNIV))"}
{"_id": "51012", "text": "INT_MAX |- (!x. (!y. (((int_max x) y) = (((COND ((int_le x) y)) y) x))))"}
{"_id": "51013", "text": "Pack2.ineq_lm5_3_step3 |- (!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_closed S) v)))))))))))"}
{"_id": "51014", "text": "INSIDE_INSIDE_EQ_EMPTY |- (!s. (((closed s) /\\ (connected s)) ==> ((inside (inside s)) = EMPTY)))"}
{"_id": "51015", "text": "INT_MIN_MAX |- (!x. (!y. (((int_min x) y) = (int_neg ((int_max (int_neg x)) (int_neg y))))))"}
{"_id": "51016", "text": "WOSET_FINITE_TOSET |- (!l. (((toset l) /\\ (FINITE (fl l))) ==> (woset l)))"}
{"_id": "51017", "text": "EXISTS_DOUBLE_ARC |- (!g. (!a. (!b. (((simple_path g) /\\ (((pathfinish g) = (pathstart g)) /\\ ((a IN (path_image g)) /\\ ((b IN (path_image g)) /\\ (~ (a = b)))))) ==> (?u. (?d. ((arc u) /\\ ((arc d) /\\ (((pathstart u) = a) /\\ (((pathfinish u) = b) /\\ (((pathstart d) = b) /\\ (((pathfinish d) = a) /\\ ((((path_image u) INTER (path_image d)) = (a INSERT (b INSERT EMPTY))) /\\ (((path_image u) UNION (path_image d)) = (path_image g)))))))))))))))"}
{"_id": "51018", "text": "Lp_ineqs_proofs.y4'_list_def |- (!h. (!V. (!E. (!g. (!L. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((good_list L) ==> (((hyp_iso g) ((hypermap_of_list L) , (hypermap_of_fan (V , E)))) ==> ((!x. ((g x) = ((h (FST x)) , (h (SND x))))) ==> ((ALL (\\d. (((y4'_list (g , (V , E))) d) = ((ye_list g) ((SND d) , (FST ((inverse (f_list_ext L)) d))))))) (list_of_darts L)))))))))))"}
{"_id": "51019", "text": "COUNTABLE_CLOPEN_IN |- (!u. ((compact u) ==> (COUNTABLE (GSPEC (\\GEN%PVAR%1247. (?c. (((SETSPEC GEN%PVAR%1247) (((closed_in ((subtopology euclidean) u)) c) /\\ ((open_in ((subtopology euclidean) u)) c))) c)))))))"}
{"_id": "51020", "text": "OPEN_IN_GDELTA |- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\\ (gdelta t)) ==> (gdelta s))))"}
{"_id": "51021", "text": "REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS |- (!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))))))) ==> ((real_measure (UNIONS f)) = ((sum f) m)))))"}
{"_id": "51022", "text": "CROSS_EQ |- (!s. (!s'. (!t. (!t'. (((s CROSS t) = (s' CROSS t')) = ((((s = EMPTY) \\/ (t = EMPTY)) /\\ ((s' = EMPTY) \\/ (t' = EMPTY))) \\/ ((s = s') /\\ (t = t'))))))))"}
{"_id": "51023", "text": "HOMEOMORPHIC_OPEN_INTERVALS |- (!a. (!b. (!c. (!d. ((((open_interval (a , b)) = EMPTY) = ((open_interval (c , d)) = EMPTY)) ==> ((homeomorphic (open_interval (a , b))) (open_interval (c , d))))))))"}
{"_id": "51024", "text": "SC_RC |- (!R. ((SC (RC R)) = (RC (SC R))))"}
{"_id": "51025", "text": "INT_LT_MAX |- (!x. (!y. (!z. (((int_lt z) ((int_max x) y)) = (((int_lt z) x) \\/ ((int_lt z) y))))))"}
{"_id": "51026", "text": "Seq.all_predT |- (!s. ((all predT) s))"}
{"_id": "51027", "text": "REAL_SUMMABLE_IMP_BOUNDED |- (!f. (!k. (((real_summable k) f) ==> (real_bounded ((IMAGE f) k)))))"}
{"_id": "51028", "text": "Optimize.split_2h0 |- (!a. (!b. (!y. ((((real_le a) y) /\\ ((real_le y) b)) ==> ((((real_le a) y) /\\ ((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) \\/ (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) /\\ ((real_le y) b)))))))"}
{"_id": "51029", "text": "MEASURABLE_COUNTABLE_UNIONS |- (!s. (!B. (((!n. (measurable (s n))) /\\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\\k. (measure (s k))))) B))) ==> (measurable (UNIONS (GSPEC (\\GEN%PVAR%3387. (?n. (((SETSPEC GEN%PVAR%3387) (n IN UNIV)) (s n))))))))))"}
{"_id": "51030", "text": "Tame_lemmas.dart_leads_into_same |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!f. (!v. (!w. (!x. (!y. (((f IN (face_set (hypermap_of_fan (V , E)))) /\\ (((v , w) IN f) /\\ ((x , y) IN f))) ==> ((((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w) = (((((dart_leads_into (vec (NUMERAL _0))) V) E) x) y))))))))))))"}
{"_id": "51031", "text": "MEASURE_INTERVAL_1_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND ((real_lt (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))) /\\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND ((real_lt (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))))"}
{"_id": "51032", "text": "NSUM_PAIR |- (!f. (!m. (!n. (((nsum (((NUMERAL (BIT0 (BIT1 _0))) * m) .. (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))) f) = ((nsum (m .. n)) (\\i. ((f ((NUMERAL (BIT0 (BIT1 _0))) * i)) + (f (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))))"}
{"_id": "51033", "text": "ABSOLUTELY_CONTINUOUS_ON_LIFT_ABS |- (!f. (((absolutely_continuous_on (\\x. (lift (f x)))) s) ==> ((absolutely_continuous_on (\\x. (lift (real_abs (f x))))) s)))"}
{"_id": "51034", "text": "CONTINUOUS_CARD_LT_RANGE_CONSTANT |- (!f. (!s. (((connected s) /\\ (((continuous_on f) s) /\\ (((IMAGE f) s) <_c UNIV))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a)))))))"}
{"_id": "51035", "text": "IN_CBALL_IM |- (!x. (!z. (!e. ((x IN (cball (z , e))) ==> ((real_le (real_abs ((real_sub (Im x)) (Im z)))) e)))))"}
{"_id": "51036", "text": "NUM_GCD |- (!a. (!b. ((int_of_num (num_gcd (a , b))) = (int_gcd ((int_of_num a) , (int_of_num b))))))"}
{"_id": "51037", "text": "COHOMOTOPICALLY_TRIVIAL_ON_COMPONENTS_NULL |- (!s. (!t. (((((locally connected) s) \\/ ((compact s) /\\ (ANR t))) /\\ (path_connected t)) ==> ((!f. ((((continuous_on f) s) /\\ (((IMAGE f) s) SUBSET t)) ==> (?a. ((((homotopic_with (\\x. T)) (s , t)) f) (\\x. a))))) = (!c. ((c IN (components s)) ==> (!f. ((((continuous_on f) c) /\\ (((IMAGE f) c) SUBSET t)) ==> (?a. ((((homotopic_with (\\x. T)) (c , t)) f) (\\x. a)))))))))))"}
{"_id": "51038", "text": "Lp_ineqs_proofs.azim2_hi |- (!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 ((azim2_fan (V , E)) d)) pi)))))))))"}
{"_id": "51039", "text": "REFLECT_ALONG_EQ |- (!v. (!x. (!y. ((((reflect_along v) x) = ((reflect_along v) y)) = (x = y)))))"}
{"_id": "51040", "text": "Rogers.KHEJKCI |- (!V. (!k. (!ul. (((saturated V) /\\ ((packing V) /\\ (((barV V) k) ul))) ==> ((face_of ((voronoi_list V) ul)) ((voronoi_closed V) (HD ul)))))))"}
{"_id": "51041", "text": "HOLOMORPHIC_ON_CATN |- (!s. ((!z. (((z IN s) /\\ ((Re z) = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (Im z))) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((holomorphic_on catn) s)))"}
{"_id": "51042", "text": "SEPARATING_HYPERPLANE_RELATIVE_INTERIORS |- (!s. (!t. (((convex s) /\\ ((convex t) /\\ ((~ (((s = EMPTY) /\\ (t = UNIV)) \\/ ((s = UNIV) /\\ (t = EMPTY)))) /\\ ((DISJOINT (relative_interior s)) (relative_interior 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": "51043", "text": "Localization.localization |- (!V. (!E. (!FF. (((localization (V , E)) FF) = (((v_prime V) FF) , ((e_prime E) FF))))))"}
{"_id": "51044", "text": "LOG_MONO_LE_IMP |- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_le x) y)) ==> ((real_le (log x)) (log y)))))"}
{"_id": "51045", "text": "REAL_DIFFERENTIABLE_WITHIN_TAN |- (!s. (!x. ((~ ((cos x) = (real_of_num (NUMERAL _0)))) ==> ((real_differentiable tan) ((within (atreal x)) s)))))"}
{"_id": "51046", "text": "CCOS_CONVERGES |- (!z. (((sums (\\n. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) n)) ((complex_div ((complex_pow z) ((NUMERAL (BIT0 (BIT1 _0))) * n))) (Cx (real_of_num (FACT ((NUMERAL (BIT0 (BIT1 _0))) * n)))))))) (ccos z)) (from (NUMERAL _0))))"}
{"_id": "51047", "text": "MEASURABLE_TRIANGLE |- (!a. (!b. (!c. (measurable ((hull convex) (a INSERT (b INSERT (c INSERT EMPTY))))))))"}
{"_id": "51048", "text": "DIFFERENTIABLE_ON_NEG |- (!f. (!s. (((differentiable_on f) s) ==> ((differentiable_on (\\z. (vector_neg (f z)))) s))))"}
{"_id": "51049", "text": "FACT_LE |- (!n. ((NUMERAL (BIT1 _0)) <= (FACT n)))"}
{"_id": "51050", "text": "Ssrnat.minn0 |- (!n. (((minn n) (NUMERAL _0)) = (NUMERAL _0)))"}
{"_id": "51051", "text": "REAL_MEASURABLE_COUNTABLE_UNIONS |- (!s. (!B. (((!n. (real_measurable (s n))) /\\ (!n. ((real_le ((sum ((NUMERAL _0) .. n)) (\\k. (real_measure (s k))))) B))) ==> (real_measurable (UNIONS (GSPEC (\\GEN%PVAR%4952. (?n. (((SETSPEC GEN%PVAR%4952) (n IN UNIV)) (s n))))))))))"}
{"_id": "51052", "text": "NEGLIGIBLE_CONVEX_HULL_1 |- (!a. (negligible ((hull convex) (a INSERT EMPTY))))"}
{"_id": "51053", "text": "COMPLEX_INV_II |- ((complex_inv ii) = (vector_neg ii))"}
{"_id": "51054", "text": "ONORM_TRIANGLE_LE |- (!f. (!g. (((linear f) /\\ ((linear g) /\\ ((real_le ((real_add (onorm f)) (onorm g))) e))) ==> ((real_le (onorm (\\x. ((vector_add (f x)) (g x))))) e))))"}
{"_id": "51055", "text": "Seq2.prev_el_rot |- (!x. (!s. (((MEM x) s) ==> (((prev_el s) x) = ((headl x) ((rotr (NUMERAL (BIT1 _0))) ((rot ((indexl x) s)) s)))))))"}
{"_id": "51056", "text": "HAS_REAL_MEASURE_IMP_REAL_MEASURABLE |- (!s. (!m. (((has_real_measure s) m) ==> (real_measurable s))))"}
{"_id": "51057", "text": "RIGHT_POLAR_DECOMPOSITION_INVERTIBLE_UNIQUE |- (!A. (!U. (!V. (!P. (!S. (((invertible A) /\\ ((orthogonal_matrix U) /\\ ((positive_semidefinite P) /\\ ((orthogonal_matrix V) /\\ ((positive_semidefinite S) /\\ ((((matrix_mul U) P) = A) /\\ (((matrix_mul V) S) = A))))))) ==> ((P = S) /\\ (U = V))))))))"}
{"_id": "51058", "text": "HAS_BOUNDED_VARIATION_ON_INDEFINITE_INTEGRAL_LEFT |- (!f. (!a. (!b. (((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((has_bounded_variation_on (\\c. ((integral (closed_interval ((CONS (c , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "51059", "text": "REALLIM_WITHINREAL_LE |- (!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_le (real_abs ((real_sub x) a))) d))) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e))))))))))))"}
{"_id": "51060", "text": "COLLINEAR_DESCALE |- (!a. (!b. (!c. (!x. (!y. (!z. (((~ (a = (real_of_num (NUMERAL _0)))) /\\ ((~ (c = (real_of_num (NUMERAL _0)))) /\\ ((collinear (((% a) x) INSERT (((% b) y) INSERT (((% c) z) INSERT EMPTY)))) /\\ (~ ((vec (NUMERAL _0)) IN ((hull affine) (x INSERT (y INSERT (z INSERT EMPTY))))))))) ==> (collinear (x INSERT (y INSERT (z INSERT EMPTY)))))))))))"}
{"_id": "51061", "text": "COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP |- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\\ ((connected s) /\\ (~ (?a. (s = (a INSERT EMPTY)))))) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\\GEN%PVAR%2081. (?x. (((SETSPEC GEN%PVAR%2081) ((x IN c) /\\ ((p x) = b))) x))))))) = (!t. (((closed_in ((subtopology euclidean) c)) t) ==> ((closed_in ((subtopology euclidean) s)) ((IMAGE p) t)))))))))"}
{"_id": "51062", "text": "REAL_DIFFERENTIABLE_CARATHEODORY_WITHINREAL |- (!f. (!z. (!s. (((real_differentiable 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))))))))"}
{"_id": "51063", "text": "FACET_OF_HALFSPACE_GE |- (!f. (!a. (!b. (((facet_of f) (GSPEC (\\GEN%PVAR%2218. (?x. (((SETSPEC GEN%PVAR%2218) ((real_ge ((dot a) x)) b)) x))))) = ((~ (a = (vec (NUMERAL _0)))) /\\ (f = (GSPEC (\\GEN%PVAR%2219. (?x. (((SETSPEC GEN%PVAR%2219) (((dot a) x) = b)) x))))))))))"}
{"_id": "51064", "text": "HAS_BOUNDED_REAL_VARIATION_ON_EMPTY |- (!f. ((has_bounded_real_variation_on f) EMPTY))"}
{"_id": "51065", "text": "Functional_equation.rotate5_dihatn_x |- ((rotate5 dihatn_x) = dih5atn_x)"}
{"_id": "51066", "text": "INT_LT_NEG |- (!x. (!y. (((int_lt (int_neg x)) (int_neg y)) = ((int_lt y) x))))"}
{"_id": "51067", "text": "NPRODUCT_POS_LT_NUMSEG |- (!f. (!m. (!n. ((!x. (((m <= x) /\\ (x <= n)) ==> ((NUMERAL _0) < (f x)))) ==> ((NUMERAL _0) < ((nproduct (m .. n)) f))))))"}
{"_id": "51068", "text": "Lp_main_estimate.ineq_tau5_diags_std |- (!V. (!E. ((lp_fan (V , E)) ==> ((!V. (!E. (!FF. (((convex_local_fan (V , (E , FF))) /\\ ((packing V) /\\ ((V SUBSET ball_annulus) /\\ (((CARD V) = (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) /\\ ((!v. (!w. (((~ (v = w)) /\\ ((v IN V) /\\ ((w IN V) /\\ (~ ((v INSERT (w INSERT EMPTY)) IN E))))) ==> ((real_le sqrt8) (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 ((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))))))))))))) (((tau_fun V) E) FF)))))) ==> ((E = (ESTD V)) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le sqrt8) ((y4'_fan (V , E)) d)) /\\ (((real_le sqrt8) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) /\\ (((real_le sqrt8) ((y4'_fan (V , E)) (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) d))) /\\ (((real_le sqrt8) ((y4'_fan (V , E)) (((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) d))) /\\ ((real_le sqrt8) ((y4'_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d))))))) ==> ((real_le ((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))))))))))))) ((tau_fan (V , E)) ((face (hypermap_of_fan (V , E))) d)))))))))))"}
{"_id": "51069", "text": "CONVEX_HALFSPACE_RE_GT |- (!b. (convex (GSPEC (\\GEN%PVAR%4769. (?z. (((SETSPEC GEN%PVAR%4769) ((real_gt (Re z)) b)) z))))))"}
{"_id": "51070", "text": "Functional_equation.functional_x1_delta_x |- (x1_delta_x = ((mul6 proj_x1) delta_x))"}
{"_id": "51071", "text": "INT_SGN_INT_SGN |- (!x. ((int_sgn (int_sgn x)) = (int_sgn x)))"}
{"_id": "51072", "text": "REAL_VARIATION_TRANSLATION2 |- (!a. (!f. (!s. (((real_variation ((IMAGE (\\x. ((real_add (real_neg a)) x))) s)) (\\x. (f ((real_add a) x)))) = ((real_variation s) f)))))"}
{"_id": "51073", "text": "INT_ADD_RDISTRIB |- (!x. (!y. (!z. (((int_mul ((int_add x) y)) z) = ((int_add ((int_mul x) z)) ((int_mul y) z))))))"}
{"_id": "51074", "text": "RANK_TRIANGLE |- (!A. (!B. ((rank ((matrix_add A) B)) <= ((rank A) + (rank B)))))"}
{"_id": "51075", "text": "HAS_COMPLEX_DERIVATIVE_SEQUENCE |- (!s. (!f. (!f'. (!g'. (((convex s) /\\ ((!n. (!x. ((x IN s) ==> (((has_complex_derivative (f n)) ((f' n) x)) ((within (at x)) s))))) /\\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (((n >= N) /\\ (x IN s)) ==> ((real_le (vector_norm ((vector_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_complex_derivative g) (g' x)) ((within (at x)) s)))))))))))"}
{"_id": "51076", "text": "Merge_ineq.nonf_arclength_x_123' |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((arclength_x_123' x1) x2) x3) x4) x5) x6) = (acs ((real_div ((real_add x1) ((real_sub x2) x3))) (sqrt ((real_mul ((real_mul x1) x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))))"}
{"_id": "51077", "text": "REAL_MUL_SIN_SIN |- (!x. (!y. (((real_mul (sin x)) (sin y)) = ((real_div ((real_sub (cos ((real_sub x) y))) (cos ((real_add x) y)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "51078", "text": "INT_POW_ZERO |- (!n. (((int_pow (int_of_num (NUMERAL _0))) n) = (((COND (n = (NUMERAL _0))) (int_of_num (NUMERAL (BIT1 _0)))) (int_of_num (NUMERAL _0)))))"}
{"_id": "51079", "text": "HAS_BOUNDED_VARIATION_INTEGRABLE_NORM_DERIVATIVE |- (!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))) = ((integrable_on (\\x. (lift (vector_norm ((vector_derivative f) (at x)))))) (closed_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "51080", "text": "CONVEX_HULL_REDUNDANT_POINT |- (!s. (!a. ((((hull convex) (s DELETE a)) = ((hull convex) s)) = (~ ((extreme_point_of a) ((hull convex) s))))))"}
{"_id": "51081", "text": "CONTINUOUS_DECREASING_IMAGE_INTERVAL_1 |- (!f. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\\ (((continuous_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 (drop (f y))) (drop (f x)))))))) ==> (((IMAGE f) (closed_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS ((f b) , (f a))) NIL)))))))"}
{"_id": "51082", "text": "MATRIX_NEG_0 |- ((matrix_neg (mat (NUMERAL _0))) = (mat (NUMERAL _0)))"}
{"_id": "51083", "text": "SIN_EQ_1 |- (!x. (((sin x) = (real_of_num (NUMERAL (BIT1 _0)))) = (?n. ((integer n) /\\ (x = ((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": "51084", "text": "COMPACT_ARC_IMAGE |- (!g. ((arc g) ==> (compact (path_image g))))"}
{"_id": "51085", "text": "ENR_DELETE |- (!s. (!a. ((ENR s) ==> (ENR (s DELETE a)))))"}
{"_id": "51086", "text": "Reduction3.size_mapAt |- (!ns. (!xs. (!f. ((sizel (((mapAt ns) f) xs)) = (sizel xs)))))"}
{"_id": "51087", "text": "FRONTIER_FRONTIER_FRONTIER |- (!s. ((frontier (frontier (frontier s))) = (frontier (frontier s))))"}
{"_id": "51088", "text": "INT_EQ_ADD_RCANCEL_0 |- (!x. (!y. ((((int_add x) y) = y) = (x = (int_of_num (NUMERAL _0))))))"}
{"_id": "51089", "text": "REAL_LEBESGUE_MEASURABLE_DIFF |- (!s. (!t. (((real_lebesgue_measurable s) /\\ (real_lebesgue_measurable t)) ==> (real_lebesgue_measurable (s DIFF t)))))"}
{"_id": "51090", "text": "EULER_ROTATION_THEOREM_GEN |- (!A. (((ODD (dimindex UNIV)) /\\ (rotation_matrix A)) ==> (?v. (((vector_norm v) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (((matrix_vector_mul A) v) = v)))))"}
{"_id": "51091", "text": "INDEPENDENT_2 |- (!a. (!b. (!x. (!y. (((independent (a INSERT (b INSERT EMPTY))) /\\ (~ (a = b))) ==> ((((vector_add ((% x) a)) ((% y) b)) = (vec (NUMERAL _0))) = ((x = (real_of_num (NUMERAL _0))) /\\ (y = (real_of_num (NUMERAL _0))))))))))"}
{"_id": "51092", "text": "CONTINUOUS_ON_COMPLEX_DIV |- (!f. (!g. (!s. ((((continuous_on f) s) /\\ (((continuous_on g) s) /\\ (!x. ((x IN s) ==> (~ ((g x) = (Cx (real_of_num (NUMERAL _0))))))))) ==> ((continuous_on (\\x. ((complex_div (f x)) (g x)))) s)))))"}
{"_id": "51093", "text": "CONVEX_HULL_REDUNDANT_SUBSET |- (!s. (!t. (((compact s) /\\ ((t SUBSET s) /\\ ((s DIFF t) SUBSET (interior ((hull convex) s))))) ==> (((hull convex) s) = ((hull convex) t)))))"}
{"_id": "51094", "text": "NORM_SEGMENT_LOWERBOUND |- (!a. (!b. (!x. (!r. (!d. ((((real_lt (real_of_num (NUMERAL _0))) r) /\\ (((vector_norm a) = r) /\\ (((vector_norm b) = r) /\\ ((x IN (closed_segment ((CONS (a , b)) NIL))) /\\ (((dot a) b) = ((real_mul d) ((real_pow r) (NUMERAL (BIT0 (BIT1 _0)))))))))) ==> ((real_le ((real_mul (sqrt ((real_div ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (real_abs d))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) r)) (vector_norm x))))))))"}
{"_id": "51095", "text": "INT_LTE_ANTISYM |- (!x. (!y. (~ (((int_lt x) y) /\\ ((int_le y) x)))))"}
{"_id": "51096", "text": "Functional_equation.functional_delta_x1 |- (delta_x1 = (rotate4 delta_x4))"}
{"_id": "51097", "text": "Functional_equation.nonf_gamma2_x1_div_a |- (!m. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((gamma2_x1_div_a m) x1) x2) x3) x4) x5) x6) = ((gamma2_x_div_azim m) x1)))))))))"}
{"_id": "51098", "text": "CONTINUOUS_ON_COMPOSE_ARG |- (!s. (!p. ((((continuous_on (p o drop)) (closed_interval ((CONS ((vec (NUMERAL _0)) , (lift ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) NIL))) /\\ (((p ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) = (p (real_of_num (NUMERAL _0)))) /\\ (~ ((Cx (real_of_num (NUMERAL _0))) IN s)))) ==> ((continuous_on (\\z. (p (Arg z)))) s))))"}
{"_id": "51099", "text": "Add_triangle.f_split_fan_face_eq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (!d. ((((v , w) IN (dart_of_fan (V , E))) /\\ ((d IN (dart_of_fan (V , E))) /\\ (~ (d IN ((face (hypermap_of_fan (V , E))) (v , w)))))) ==> (((f_fan_pair ((split_fan_face (V , E)) (v , w))) d) = ((f_fan_pair (V , E)) d))))))))))"}
{"_id": "51100", "text": "Functional_equation.taum_x' |- (!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)))))) ==> (((((((taum_x x1) x2) x3) x4) x5) x6) = ((((((taum_x' x1) x2) x3) x4) x5) x6)))))))))"}
{"_id": "51101", "text": "SECOND_MEAN_VALUE_THEOREM_BONNET |- (!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))) /\\ (((integral (closed_interval ((CONS (a , b)) NIL))) (\\x. ((% (g x)) (f x)))) = ((% (g b)) ((integral (closed_interval ((CONS (c , b)) NIL))) f))))))))))"}
{"_id": "51102", "text": "ABSOLUTELY_REAL_INTEGRABLE_DECREASING |- (!f. (!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 f) (closed_real_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "51103", "text": "BOUNDED_ARC_IMAGE |- (!g. ((arc g) ==> (bounded (path_image g))))"}
{"_id": "51104", "text": "CNJ_CSQRT |- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_le (real_of_num (NUMERAL _0))) (Re z))) ==> ((cnj (csqrt z)) = (csqrt (cnj z)))))"}
{"_id": "51105", "text": "Ssrnat.minn_maxr |- (!m. (!n1. (!n2. (((minn m) ((maxn n1) n2)) = ((maxn ((minn m) n1)) ((minn m) n2))))))"}
{"_id": "51106", "text": "HAS_BOUNDED_REAL_VARIATION_ON_NEG |- (!f. (!s. (((has_bounded_real_variation_on f) s) ==> ((has_bounded_real_variation_on (\\x. (real_neg (f x)))) s))))"}
{"_id": "51107", "text": "AFFINE_HULL_EQ_SING |- (!s. (!a. ((((hull affine) s) = (a INSERT EMPTY)) = (s = (a INSERT EMPTY)))))"}
{"_id": "51108", "text": "FUNSPACE_IMP_BOUNDED |- (!s. (!m. (!f. ((f IN (mspace ((funspace s) m))) ==> ((s = EMPTY) \\/ (?c. (?b. (!x. ((x IN s) ==> ((real_le ((mdist m) (c , (f x)))) b))))))))))"}
{"_id": "51109", "text": "DIV_ADD_MOD |- (!a. (!b. (!n. ((~ (n = (NUMERAL _0))) ==> ((((a + b) MOD n) = ((a MOD n) + (b MOD n))) = (((a + b) DIV n) = ((a DIV n) + (b DIV n))))))))"}
{"_id": "51110", "text": "Lp_ineqs_proofs.sol_hi |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le ((sol_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi))))))))))"}
{"_id": "51111", "text": "ORTHOGONAL_TRANSFORMATION_ONTO_SUBSPACE |- (!s. (!t. (((subspace s) /\\ ((subspace t) /\\ ((dim s) = (dim t)))) ==> (?f. ((orthogonal_transformation f) /\\ (((IMAGE f) s) = t))))))"}
{"_id": "51112", "text": "PROPER_LOCAL_HOMEOMORPHISM_GLOBAL |- (!f. (!s. (!t. (((path_connected s) /\\ ((simply_connected t) /\\ (((continuous_on f) s) /\\ ((((IMAGE f) s) = t) /\\ ((!k. (((k SUBSET t) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%2104. (?x. (((SETSPEC GEN%PVAR%2104) ((x IN s) /\\ ((f x) IN k))) x))))))) /\\ (!x. ((x IN s) ==> (?u. (?v. (?q. ((x IN u) /\\ (((open_in ((subtopology euclidean) s)) u) /\\ (((open_in ((subtopology euclidean) t)) v) /\\ ((homeomorphism (u , v)) (f , q))))))))))))))) ==> (?g. ((homeomorphism (s , t)) (f , g)))))))"}
{"_id": "51113", "text": "FCONS_UNDO |- (!f. (f = ((FCONS (f (NUMERAL _0))) (f o SUC))))"}
{"_id": "51114", "text": "ISOMETRY_IMP_AFFINITY |- (!f. ((!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y))))) ==> (?h. ((linear h) /\\ (!x. ((f x) = ((vector_add (f (vec (NUMERAL _0)))) (h x))))))))"}
{"_id": "51115", "text": "IS_AFFINE_HULL |- (!s. ((affine s) = (?t. (s = ((hull affine) t)))))"}
{"_id": "51116", "text": "Fnjlbxs.closedS |- (!k. (!S. ((((NUMERAL (BIT1 (BIT1 _0))) * k) <= (dimindex UNIV)) ==> ((!l. ((l IN S) = (((set_of_list l) SUBSET ball_annulus) /\\ ((packing (set_of_list l)) /\\ (((LENGTH l) = k) /\\ (uniq l)))))) ==> (closed ((IMAGE (vector_of_list (NUMERAL (BIT1 (BIT1 _0))))) S))))))"}
{"_id": "51117", "text": "REAL_MEASURABLE_TRANSLATION |- (!s. ((real_measurable ((IMAGE (\\x. ((real_add a) x))) s)) = (real_measurable s)))"}
{"_id": "51118", "text": "ZERO_AE_DERIVATIVE_IMP_CONSTANT |- (!f. (!a. (!b. (!s. ((((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((negligible 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": "51119", "text": "CLOSURE_OF_EMPTY |- (!top. (((closure_of top) EMPTY) = EMPTY))"}
{"_id": "51120", "text": "BETWEEN_COLLINEAR_DIST_EQ |- (!a. (!b. (!x. (((between x) (a , b)) = ((collinear (a INSERT (x INSERT (b INSERT EMPTY)))) /\\ (((real_le (distance (x , a))) (distance (a , b))) /\\ ((real_le (distance (x , b))) (distance (a , b)))))))))"}
{"_id": "51121", "text": "EXPOSED_POINT_OF_INTER_SUPPORTING_HYPERPLANE_GE |- (!s. (!a. (!b. (!c. (((!x. ((x IN s) ==> ((real_ge ((dot a) x)) b))) /\\ ((s INTER (GSPEC (\\GEN%PVAR%2203. (?x. (((SETSPEC GEN%PVAR%2203) (((dot a) x) = b)) x))))) = (c INSERT EMPTY))) ==> ((exposed_face_of (c INSERT EMPTY)) s))))))"}
{"_id": "51122", "text": "Ssrbool.addbT |- (!b. (((XOR b) T) = (~ b)))"}
{"_id": "51123", "text": "Miqmcsn.SCS_4M8_13_ARROW_3T7_OPP |- ((scs_arrow_v39 (scs_4M8_02 INSERT EMPTY)) ((scs_opp_v39 scs_3T7) INSERT EMPTY))"}
{"_id": "51124", "text": "Seq.constantP |- (!s. ((?x. (s = ((nseq (sizel s)) x))) = (constant s)))"}
{"_id": "51125", "text": "OPEN_IN_FSIGMA |- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\\ (fsigma t)) ==> (fsigma s))))"}
{"_id": "51126", "text": "LIPSCHITZ_VECTOR_POLYNOMIAL_FUNCTION |- (!f. (!s. (((vector_polynomial_function f) /\\ (bounded s)) ==> (?B. (((real_lt (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))))))))))))"}
{"_id": "51127", "text": "CONVEX_MAX |- (!f. (!g. (!s. ((((convex_on f) s) /\\ ((convex_on g) s)) ==> ((convex_on (\\x. ((real_max (f x)) (g x)))) s)))))"}
{"_id": "51128", "text": "Dih2k_hypermap.EE_OF_HYP_HAS_ORDERS |- (((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)))))) ==> ((has_orders (ee_of_hyp ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l)))))) (NUMERAL (BIT0 (BIT1 _0)))))"}
{"_id": "51129", "text": "EXPLICIT_SUBSET_INTERIOR_CONVEX_HULL_MINIMAL |- (!s. (((FINITE s) /\\ (((hull affine) s) = UNIV)) ==> ((GSPEC (\\GEN%PVAR%1795. (?y. (((SETSPEC GEN%PVAR%1795) (?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 (interior ((hull convex) s)))))"}
{"_id": "51130", "text": "INT_LE_ADD |- (!x. (!y. ((((int_le (int_of_num (NUMERAL _0))) x) /\\ ((int_le (int_of_num (NUMERAL _0))) y)) ==> ((int_le (int_of_num (NUMERAL _0))) ((int_add x) y)))))"}
{"_id": "51131", "text": "Add_triangle.in_face_lemma |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> (((v , w) IN ((face (hypermap_of_fan (V , E))) (v , w))) /\\ ((((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v) IN ((face (hypermap_of_fan (V , E))) (v , w))) /\\ (((f_fan_pair (V , E)) (v , w)) IN ((face (hypermap_of_fan (V , E))) (v , w)))))))))))"}
{"_id": "51132", "text": "Hypermap.lemma_map_loop_map |- (!H. (!NF. (!L. (!x. ((((is_normal H) NF) /\\ ((L IN NF) /\\ (((in_dart_of_loop x) L) /\\ (((loop_map L) x) IN (((atom H) L) x))))) ==> (((loop_map L) x) = ((inverse (node_map H)) x)))))))"}
{"_id": "51133", "text": "BETWEEN_NORM_LT |- (!r. (!a. (!b. (!x. ((((real_lt (vector_norm a)) r) /\\ (((real_lt (vector_norm b)) r) /\\ ((between x) (a , b)))) ==> ((real_lt (vector_norm x)) r))))))"}
{"_id": "51134", "text": "Collect_geom2.c_coef |- (!x12. (!x13. (!x14. (!x23. (!x24. (((((((b_coef x12) x13) x14) x23) x24) = ((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)))))))))) /\\ ((((((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": "51135", "text": "CROSS_JACOBI |- (!x. (!y. (!z. (((vector_add ((cross x) ((cross y) z))) ((vector_add ((cross y) ((cross z) x))) ((cross z) ((cross x) y)))) = (vec (NUMERAL _0))))))"}
{"_id": "51136", "text": "COVERING_SPACE_COMPACT |- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> ((compact c) = ((compact s) /\\ (!b. ((b IN s) ==> (FINITE (GSPEC (\\GEN%PVAR%2085. (?x. (((SETSPEC GEN%PVAR%2085) ((x IN c) /\\ ((p x) = b))) x)))))))))))))"}
{"_id": "51137", "text": "BOREL_UNION |- (!s. (!t. (((borel s) /\\ (borel t)) ==> (borel (s UNION t)))))"}
{"_id": "51138", "text": "REAL_DIFFERENTIABLE_EQ |- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((real_differentiable_on f) s) = ((real_differentiable_on g) s))))))"}
{"_id": "51139", "text": "Calc_derivative.imp_eq |- (!p. (!p'. (!x. (!a. (!a'. (!b. (!b'. (((x = (real_of_num (NUMERAL _0))) /\\ (((((ratform p) x) a) b) /\\ ((p = p') /\\ ((a = a') /\\ (b = b'))))) ==> ((p' /\\ (~ (b' = (real_of_num (NUMERAL _0))))) ==> (a' = (real_of_num (NUMERAL _0))))))))))))"}
{"_id": "51140", "text": "Nonlinear_lemma.gcy_low_hminus |- (!y. (((real_le y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) hminus)) ==> ((gcy y) = (gchi y))))"}
{"_id": "51141", "text": "REAL_SECOND_MEAN_VALUE_THEOREM |- (!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))) /\\ (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\\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)))))))))))"}
{"_id": "51142", "text": "UPPER_HEMICONTINUOUS |- (!f. (!t. (!s. ((!x. ((x IN s) ==> ((f x) SUBSET t))) ==> ((!u. (((open_in ((subtopology euclidean) t)) u) ==> ((open_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%880. (?x. (((SETSPEC GEN%PVAR%880) ((x IN s) /\\ ((f x) SUBSET u))) x))))))) = (!u. (((closed_in ((subtopology euclidean) t)) u) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\\GEN%PVAR%881. (?x. (((SETSPEC GEN%PVAR%881) ((x IN s) /\\ (~ (((f x) INTER u) = EMPTY)))) x))))))))))))"}
{"_id": "51143", "text": "CONNECTED_OPEN_DIFF_CARD_LT |- (!s. (!t. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\\ ((open s) /\\ ((connected s) /\\ (t <_c UNIV)))) ==> (connected (s DIFF t)))))"}
{"_id": "51144", "text": "OPEN_PATH_CONNECTED_COMPONENT |- (!s. (!x. ((open s) ==> (((path_component s) x) = ((connected_component s) x)))))"}
{"_id": "51145", "text": "HOMEOMORPHIC_SIMPLE_PATH_ARC_EQ |- (!g. (!h. (((simple_path g) /\\ ((simple_path h) /\\ ((homeomorphic (path_image g)) (path_image h)))) ==> ((arc g) = (arc h)))))"}
{"_id": "51146", "text": "COUNTABLE_ENR_PATH_COMPONENTS |- (!s. ((ENR s) ==> (COUNTABLE (GSPEC (\\GEN%PVAR%2725. (?x. (((SETSPEC GEN%PVAR%2725) (x IN s)) ((path_component s) x))))))))"}
{"_id": "51147", "text": "Lp_gen_theory.card_face_dart_iso |- (!H. (!g. (!G. (((hyp_iso g) (G , H)) ==> (!d. ((d IN (dart G)) ==> ((CARD ((IMAGE g) ((face G) d))) = (CARD ((face G) d)))))))))"}
{"_id": "51148", "text": "SETDIST_UNIV |- ((!s. ((setdist (s , UNIV)) = (real_of_num (NUMERAL _0)))) /\\ (!t. ((setdist (UNIV , t)) = (real_of_num (NUMERAL _0)))))"}
{"_id": "51149", "text": "RELATIVE_FRONTIER_OF_TRIANGLE |- (!a. (!b. (!c. ((~ (collinear (a INSERT (b INSERT (c INSERT EMPTY))))) ==> ((relative_frontier ((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": "51150", "text": "Geomdetail.IN_SET4 |- ((x IN (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))) = ((x = a) \\/ ((x = b) \\/ ((x = c) \\/ (x = d)))))"}
{"_id": "51151", "text": "CARD_EQ_PERFECT_SET |- (!s. (((closed s) /\\ ((!x. ((x IN s) ==> ((limit_point_of x) s))) /\\ (~ (s = EMPTY)))) ==> (s =_c UNIV)))"}
{"_id": "51152", "text": "HAUSDIST_STILL_NONEMPTY_INTERIOR |- (!s. (((bounded s) /\\ ((convex s) /\\ (~ ((interior s) = EMPTY)))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (!s'. (((bounded s') /\\ ((convex s') /\\ ((~ (s' = EMPTY)) /\\ ((real_lt (hausdist (s , s'))) e)))) ==> (~ ((interior s') = EMPTY))))))))"}
{"_id": "51153", "text": "STARLIKE_IMP_CONNECTED |- (!s. ((starlike s) ==> (connected s)))"}
{"_id": "51154", "text": "EXP_MONO_EQ |- (!x. (!y. (!n. (((x EXP n) = (y EXP n)) = ((x = y) \\/ (n = (NUMERAL _0)))))))"}
{"_id": "51155", "text": "CONVEX_HAS_BOUNDED_VARIATION |- (!f. (!a. (!b. (((convex_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((has_bounded_variation_on (lift o f)) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "51156", "text": "UNION_WITH_OUTSIDE |- (!s. ((s UNION (outside s)) = (UNIV DIFF (inside s))))"}
{"_id": "51157", "text": "REAL_MEASURABLE_ON_MAX |- (!f. (!g. (!s. ((((real_measurable_on f) s) /\\ ((real_measurable_on g) s)) ==> ((real_measurable_on (\\x. ((real_max (f x)) (g x)))) s)))))"}
{"_id": "51158", "text": "UNBOUNDED_HALFSPACE_COMPONENT_GT |- (!a. (!k. (~ (bounded (GSPEC (\\GEN%PVAR%839. (?x. (((SETSPEC GEN%PVAR%839) ((real_gt (x $ k)) a)) x))))))))"}
{"_id": "51159", "text": "REAL_SUMMABLE_ALTERNATING_SERIES |- (!a. (!m. (((!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))))) /\\ (((---> a) (real_of_num (NUMERAL _0))) sequentially)))) ==> ((real_summable (from m)) a))))"}
{"_id": "51160", "text": "Seq.eqseq_rcons |- (!s1. (!s2. (!x1. (!x2. ((((rcons s1) x1) = ((rcons s2) x2)) = ((s1 = s2) /\\ (x1 = x2)))))))"}
{"_id": "51161", "text": "REAL_CONTINUOUS_ON_DIV |- (!f. (!g. (!s. ((((real_continuous_on f) s) /\\ (((real_continuous_on g) s) /\\ (!x. ((x IN s) ==> (~ ((g x) = (real_of_num (NUMERAL _0)))))))) ==> ((real_continuous_on (\\x. ((real_div (f x)) (g x)))) s)))))"}
{"_id": "51162", "text": "Asfutbf.affine_hull_neg |- (!v. (((IMAGE vector_neg) ((hull affine) ((vec (NUMERAL _0)) INSERT (v INSERT EMPTY)))) = ((hull affine) ((vec (NUMERAL _0)) INSERT ((vector_neg v) INSERT EMPTY)))))"}
{"_id": "51163", "text": "INTERIOR_STD_SIMPLEX |- ((interior ((hull convex) ((vec (NUMERAL _0)) INSERT (GSPEC (\\GEN%PVAR%1807. (?i. (((SETSPEC GEN%PVAR%1807) (((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV)))) (basis i)))))))) = (GSPEC (\\GEN%PVAR%1808. (?x. (((SETSPEC GEN%PVAR%1808) ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_lt (real_of_num (NUMERAL _0))) (x $ i)))) /\\ ((real_lt ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\\i. (x $ i)))) (real_of_num (NUMERAL (BIT1 _0)))))) x)))))"}
{"_id": "51164", "text": "HIGHER_COMPLEX_DERIVATIVE_1 |- (!f. (!z. ((((higher_complex_derivative (NUMERAL (BIT1 _0))) f) z) = ((complex_derivative f) z))))"}
{"_id": "51165", "text": "CROSS_ROTATION_MATRIX |- (!A. (!x. (!y. ((rotation_matrix A) ==> (((cross ((matrix_vector_mul A) x)) ((matrix_vector_mul A) y)) = ((matrix_vector_mul A) ((cross x) y)))))))"}
{"_id": "51166", "text": "Lp_ineqs_proofs.sol_sum |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((sum f) (azim_dart (V , E))) = ((real_add ((sol_fan (V , E)) f)) ((real_mul (real_of_num ((CARD f) - (NUMERAL (BIT0 (BIT1 _0)))))) pi)))))))))))"}
{"_id": "51167", "text": "EVEN_EXP |- (!m. (!n. ((EVEN (m EXP n)) = ((EVEN m) /\\ (~ (n = (NUMERAL _0)))))))"}
{"_id": "51168", "text": "LINSEG_WOSET |- (!l. (!a. ((woset l) ==> (woset ((linseg l) a)))))"}
{"_id": "51169", "text": "REAL_SGN_SQRT |- (!x. ((real_sgn (sqrt x)) = (real_sgn x)))"}
{"_id": "51170", "text": "Conforming.lemma_card_node_eq_set_of_orbits |- (!x. (!V. (!E. (!f. (!y. (((FAN (x , (V , E))) /\\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\\ ((f IN (node_set (hypermap1_of_fanx (x , (V , E))))) /\\ (y IN f)))) ==> ((CARD (GSPEC (\\GEN%PVAR%5670. (?z. (((SETSPEC GEN%PVAR%5670) (?i. ((i >= (NUMERAL _0)) /\\ (z = (x , ((pr2 y) , ((((((((power_map_points sigma_fan) x) V) E) (pr2 y)) (pr3 y)) i) , (((((((power_map_points sigma_fan) x) V) E) (pr2 y)) (pr3 y)) (SUC i))))))))) z))))) = (CARD (((((set_of_orbits_points_fan x) V) E) (pr2 y)) (pr3 y))))))))))"}
{"_id": "51171", "text": "NSUM_UNION_LZERO |- (!f. (!u. (!v. (((FINITE v) /\\ (!x. (((x IN u) /\\ (~ (x IN v))) ==> ((f x) = (NUMERAL _0))))) ==> (((nsum (u UNION v)) f) = ((nsum v) f))))))"}
{"_id": "51172", "text": "Lp_gen_theory.iso_dart_pairs_trans |- (!H. (!G. (!g. (((hyp_iso g) (G , H)) ==> (!P. ((!p. ((p IN ((dart H) CROSS (dart H))) ==> (P p))) ==> (!p. ((p IN ((dart G) CROSS (dart G))) ==> (P ((g (FST p)) , (g (SND p))))))))))))"}
{"_id": "51173", "text": "EVEN_AND_ODD |- (!n. (~ ((EVEN n) /\\ (ODD n))))"}
{"_id": "51174", "text": "EXISTS_TRIPLED_THM |- (!P. ((? (GABS (\\f. (!x. (!y. (!z. ((GEQ (f (x , (y , z)))) (((P x) y) z)))))))) = (?x. (?y. (?z. (((P x) y) z))))))"}
{"_id": "51175", "text": "COMPLEX_DERIVATIVE_TRANSFORM_WITHIN_OPEN |- (!f. (!g. (!s. (!z. (((open s) /\\ (((holomorphic_on f) s) /\\ (((holomorphic_on g) s) /\\ ((z IN s) /\\ (!w. ((w IN s) ==> ((f w) = (g w)))))))) ==> (((complex_derivative f) z) = ((complex_derivative g) z)))))))"}
{"_id": "51176", "text": "REAL_DIFFERENTIABLE_WITHIN_LOG |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable log) ((within (atreal x)) s)))))"}
{"_id": "51177", "text": "FRECHET_DERIVATIVE_WITHIN_CLOSED_INTERVAL |- (!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)))))) ==> (((frechet_derivative f) ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) = f')))))))"}
{"_id": "51178", "text": "RESTRICTION_IDEMP |- (!s. (!f. (((RESTRICTION s) ((RESTRICTION s) f)) = ((RESTRICTION s) f))))"}
{"_id": "51179", "text": "STC_INC_SC |- (!R. (!x. (!y. ((((SC R) x) y) ==> (((STC R) x) y)))))"}
{"_id": "51180", "text": "REAL_STEINHAUS |- (!s. (((real_measurable s) /\\ ((real_lt (real_of_num (NUMERAL _0))) (real_measure s))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ ((open_real_interval ((real_neg d) , d)) SUBSET (GSPEC (\\GEN%PVAR%4987. (?x. (?y. (((SETSPEC GEN%PVAR%4987) ((x IN s) /\\ (y IN s))) ((real_sub x) y)))))))))))"}
{"_id": "51181", "text": "REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS_IMAGE |- (!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)))))))) ==> ((real_measure (UNIONS ((IMAGE f) s))) = ((sum s) (\\x. (real_measure (f x))))))))"}
{"_id": "51182", "text": "Seq2.all_subseq |- (!s1. (!s2. (!P. ((((subseq s1) s2) /\\ ((all P) s2)) ==> ((all P) s1)))))"}
{"_id": "51183", "text": "Trigonometry.UIVNNRR1 |- (!V. (((aff_ge V) EMPTY) = ((aff_gt V) EMPTY)))"}
{"_id": "51184", "text": "SYMMETRIC_LINEAR_IMAGE |- (!f. (!s. (((!x. ((x IN s) ==> ((vector_neg x) IN s))) /\\ (linear f)) ==> (!x. ((x IN ((IMAGE f) s)) ==> ((vector_neg x) IN ((IMAGE f) s)))))))"}
{"_id": "51185", "text": "Oxl_def.BIJ_NUMSEG |- (!s. (!a. (!b. (((BIJ (\\j. (j + s))) (a .. b)) ((a + s) .. (b + s))))))"}
{"_id": "51186", "text": "NORM_CSIN_POW_2 |- (!z. (((real_pow (vector_norm (csin z))) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_div ((real_add (exp ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Im z)))) ((real_sub (real_inv (exp ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Im z))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (cos ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (Re z))))))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))"}
{"_id": "51187", "text": "HAS_REAL_DERIVATIVE_ZERO_UNIQUE_STRONG_INTERVAL |- (!f. (!a. (!b. (!k. (!y. (((COUNTABLE k) /\\ (((real_continuous_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\\ (((f a) = y) /\\ (!x. ((x IN ((closed_real_interval ((CONS (a , b)) NIL)) DIFF k)) ==> (((has_real_derivative f) (real_of_num (NUMERAL _0))) ((within (atreal x)) (closed_real_interval ((CONS (a , b)) NIL))))))))) ==> (!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> ((f x) = y)))))))))"}
{"_id": "51188", "text": "REAL_INTEGRABLE_DECREASING_PRODUCT_UNIV |- (!f. (!g. (!B. ((((real_integrable_on f) UNIV) /\\ ((!x. (!y. (((real_le x) y) ==> ((real_le (g y)) (g x))))) /\\ (!x. ((real_le (real_abs (g x))) B)))) ==> ((real_integrable_on (\\x. ((real_mul (g x)) (f x)))) UNIV)))))"}
{"_id": "51189", "text": "OPEN_IN_TRANS_EQ |- (!s. (!t. ((!u. (((open_in ((subtopology euclidean) t)) u) ==> ((open_in ((subtopology euclidean) s)) t))) = ((open_in ((subtopology euclidean) s)) t))))"}
{"_id": "51190", "text": "CONIC_IMP_CONNECTED |- (!s. ((conic s) ==> (connected s)))"}
{"_id": "51191", "text": "OUTSIDE_CONNECTED_COMPONENT_LE |- (!s. ((outside s) = (GSPEC (\\GEN%PVAR%2012. (?x. (((SETSPEC GEN%PVAR%2012) (!B. (?y. (((real_le B) (vector_norm y)) /\\ (((connected_component (UNIV DIFF s)) x) y))))) x))))))"}
{"_id": "51192", "text": "HOLOMORPHIC_ON_IMP_DIFFERENTIABLE_WITHIN |- (!f. (!s. (!x. ((((holomorphic_on f) s) /\\ (x IN s)) ==> ((complex_differentiable f) ((within (at x)) s))))))"}
{"_id": "51193", "text": "INT_LE_ADDR |- (!x. (!y. (((int_le x) ((int_add x) y)) = ((int_le (int_of_num (NUMERAL _0))) y))))"}
{"_id": "51194", "text": "Seq2.take_belast |- (!s. (!x. (!n. (((n < (sizel s)) /\\ ((NUMERAL _0) < n)) ==> (((take n) ((belast x) s)) = ((CONS x) ((take (n - (NUMERAL (BIT1 _0)))) s)))))))"}
{"_id": "51195", "text": "CLOSED_UNION_COMPLEMENT_COMPONENTS |- (!s. (!c. (((closed s) /\\ (c SUBSET (components (UNIV DIFF s)))) ==> (closed (s UNION (UNIONS c))))))"}
{"_id": "51196", "text": "POSITIVE_DEFINITE_MUL |- (!A. (!B. (((positive_definite A) /\\ ((positive_definite B) /\\ (((matrix_mul A) B) = ((matrix_mul B) A)))) ==> (positive_definite ((matrix_mul A) B)))))"}
{"_id": "51197", "text": "AR_PCROSS_EQ |- (!s. (!t. ((AR (s PCROSS t)) = ((AR s) /\\ ((AR t) /\\ ((~ (s = EMPTY)) /\\ (~ (t = EMPTY))))))))"}
{"_id": "51198", "text": "Odxlstc.DEFORMATION_DIST_LE_ALL_COM_LE3 |- (((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)))))))))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (!t. (!i. ((((real_lt (real_of_num (NUMERAL _0))) t) /\\ (((real_lt t) e) /\\ (~ ((i MOD k) = (l MOD k))))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) (w l)) , (w i))))))))))"}
{"_id": "51199", "text": "NPRODUCT_SUPPORT |- (!f. (!s. (((nproduct (((support *) f) s)) f) = ((nproduct s) f))))"}
{"_id": "51200", "text": "SEPARATING_HYPERPLANE_AFFINE_HULLS |- (!s. (!t. (((~ (s = EMPTY)) /\\ ((~ (t = EMPTY)) /\\ ((DISJOINT ((hull affine) s)) ((hull affine) 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": "51201", "text": "EVENTUALLY_FALSE |- (!net. (((eventually (\\x. F)) net) = (trivial_limit net)))"}
{"_id": "51202", "text": "PATH_COMPONENT_EMPTY |- (!x. (((path_component EMPTY) x) = EMPTY))"}
{"_id": "51203", "text": "INFSUM_EQ |- (!f. (!g. (!k. ((((summable k) f) /\\ (((summable k) g) /\\ (!x. ((x IN k) ==> ((f x) = (g x)))))) ==> (((infsum k) f) = ((infsum k) g))))))"}
{"_id": "51204", "text": "NPRODUCT_CONST_NUMSEG |- (!c. (!m. (!n. (((nproduct (m .. n)) (\\x. c)) = (c EXP ((n + (NUMERAL (BIT1 _0))) - m))))))"}
{"_id": "51205", "text": "MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_BOREL_EQ |- (!f. (!s. ((((measurable_on f) s) /\\ (lebesgue_measurable s)) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\\GEN%PVAR%3662. (?x. (((SETSPEC GEN%PVAR%3662) ((x IN s) /\\ ((f x) IN t))) x))))))))))"}
{"_id": "51206", "text": "REAL_INTEGRABLE_REFLECT_GEN |- (!f. (!s. (((real_integrable_on (\\x. (f (real_neg x)))) s) = ((real_integrable_on f) ((IMAGE real_neg) s)))))"}
{"_id": "51207", "text": "REALLIM_1_OVER_POW |- (!k. (((NUMERAL (BIT1 _0)) <= k) ==> (((---> (\\n. (real_inv ((real_pow (real_of_num n)) k)))) (real_of_num (NUMERAL _0))) sequentially)))"}
{"_id": "51208", "text": "COMPLEX_STONE_WEIERSTRASS |- (!P. (!s. (((compact s) /\\ ((!f. ((P f) ==> ((continuous_on f) 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) /\\ (~ ((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": "51209", "text": "Nonlinear_lemma.lmdih5_0' |- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y5) ==> (((((((lmdih5_x_div_sqrtdelta_posbranch ((real_mul y1) y1)) ((real_mul y2) y2)) ((real_mul y3) y3)) ((real_mul y4) y4)) ((real_mul y5) y5)) ((real_mul y6) y6)) = (real_of_num (NUMERAL _0))))))))))"}
{"_id": "51210", "text": "WF_EREC |- ((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": "51211", "text": "Functional_equation.functional_asn797k |- (asn797k = ((mul6 proj_x1) (uni (asn , ((mul6 (constant6 cos797)) (uni (sin , ((div6 (constant6 pi)) proj_x1))))))))"}
{"_id": "51212", "text": "ITSET_EQ |- (!s. (!f. (!g. (!b. (((FINITE s) /\\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\\ ((!x. (!y. (!s. ((~ (x = y)) ==> (((f x) ((f y) s)) = ((f y) ((f x) s))))))) /\\ (!x. (!y. (!s. ((~ (x = y)) ==> (((g x) ((g y) s)) = ((g y) ((g x) s)))))))))) ==> ((((ITSET f) s) b) = (((ITSET g) s) b)))))))"}
{"_id": "51213", "text": "CLOSED_SIMPLE_PATH_IMAGE |- (!g. ((simple_path g) ==> (closed (path_image g))))"}
{"_id": "51214", "text": "RSTC_RULES |- (!R. ((!x. (!y. (((R x) y) ==> (((RSTC R) x) y)))) /\\ ((!x. (((RSTC R) x) x)) /\\ ((!x. (!y. ((((RSTC R) x) y) ==> (((RSTC R) y) x)))) /\\ (!x. (!y. (!z. (((((RSTC R) x) y) /\\ (((RSTC R) y) z)) ==> (((RSTC R) x) z)))))))))"}
{"_id": "51215", "text": "Lp_ineqs_proofs.rho_fan_eq_rho |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((rho_fan v) = (rho (vector_norm v)))))))))))"}
{"_id": "51216", "text": "MEASURE_DISJOINT_UNIONS |- (!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)))))) ==> ((measure (UNIONS f)) = ((sum f) m)))))"}
{"_id": "51217", "text": "CONVEX_HULL_INTERIOR_SUBSET |- (!s. (((hull convex) (interior s)) SUBSET (interior ((hull convex) s))))"}
{"_id": "51218", "text": "Hypermap_iso.fan_hypermaps_iso2 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((iso (hypermap_of_fan (V , E))) (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E)))))))"}
{"_id": "51219", "text": "LOG_DIV |- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((log ((real_div x) y)) = ((real_sub (log x)) (log y))))))"}
{"_id": "51220", "text": "ROOT_MONO_LT_EQ |- (!n. (!x. (!y. ((~ (n = (NUMERAL _0))) ==> (((real_lt ((root n) x)) ((root n) y)) = ((real_lt x) y))))))"}
{"_id": "51221", "text": "CONTINUOUS_FROM_CLOSED_GRAPH |- (!f. (!s. (!t. (((compact t) /\\ ((((IMAGE f) s) SUBSET t) /\\ (closed (GSPEC (\\GEN%PVAR%967. (?x. (((SETSPEC GEN%PVAR%967) (x IN s)) ((pastecart x) (f x))))))))) ==> ((continuous_on f) s)))))"}
{"_id": "51222", "text": "FINITE_CBALL |- (!a. (!r. ((FINITE (cball (a , r))) = ((real_le r) (real_of_num (NUMERAL _0))))))"}
{"_id": "51223", "text": "Polar_fan.AZIM_CYCLE_SING |- (!x. (!u. (!v. (((((azim_cycle (v INSERT EMPTY)) x) u) v) = v))))"}
{"_id": "51224", "text": "CONTINUOUS_WITHIN_CNJ |- (!s. (!z. ((continuous cnj) ((within (at z)) s))))"}
{"_id": "51225", "text": "ODD_MOD |- (!n. ((ODD n) = ((n MOD (NUMERAL (BIT0 (BIT1 _0)))) = (NUMERAL (BIT1 _0)))))"}
{"_id": "51226", "text": "LIPSCHITZ_IMP_RECTIFIABLE_PATH |- (!g. (!b. ((!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)))) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))) ==> (rectifiable_path g))))"}
{"_id": "51227", "text": "INT_MAX_LE |- (!x. (!y. (!z. (((int_le ((int_max x) y)) z) = (((int_le x) z) /\\ ((int_le y) z))))))"}
{"_id": "51228", "text": "MATRIX_ADD_LNEG |- (!A. (((matrix_add (matrix_neg A)) A) = (mat (NUMERAL _0))))"}
{"_id": "51229", "text": "MEASURE_LINEAR_IMAGE_SAME |- (!f. (!s. (((linear f) /\\ ((measurable s) /\\ ((real_abs (det (matrix f))) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((measure ((IMAGE f) s)) = (measure s)))))"}
{"_id": "51230", "text": "UNIFORMLY_CONTINUOUS_ON_MUL |- (!f. (!g. (!s. ((((uniformly_continuous_on (lift o f)) s) /\\ (((uniformly_continuous_on g) s) /\\ ((bounded ((IMAGE (lift o f)) s)) /\\ (bounded ((IMAGE g) s))))) ==> ((uniformly_continuous_on (\\x. ((% (f x)) (g x)))) s)))))"}
{"_id": "51231", "text": "Tame_list.mapAt_ALT |- (((((mapAt NIL) f) xs) = xs) /\\ ((((mapAt ((CONS n) ns)) f) xs) = (((COND (n < (LENGTH xs))) (((mapAt ns) f) (((list_update xs) n) (f ((#!# xs) n))))) (((mapAt ns) f) xs))))"}
{"_id": "51232", "text": "COUNTABLE_OPEN_CONNECTED_COMPONENTS |- (!s. (!t. ((open s) ==> (COUNTABLE (GSPEC (\\GEN%PVAR%1917. (?x. (((SETSPEC GEN%PVAR%1917) (x IN t)) ((connected_component s) x)))))))))"}
{"_id": "51233", "text": "WOSET_INSEG_ORDINAL |- (woset (GABS (\\f. (!x. (!y. ((GEQ (f (x , y))) ((ordinal x) /\\ ((ordinal y) /\\ ((inseg x) y)))))))))"}
{"_id": "51234", "text": "IMAGE_STRETCH_REAL_INTERVAL |- (!a. (!b. (!m. (((IMAGE (\\x. ((real_mul m) x))) (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_mul m) a) , ((real_mul m) b))) NIL))) (closed_real_interval ((CONS (((real_mul m) b) , ((real_mul m) a))) NIL))))))))"}
{"_id": "51235", "text": "Merge_ineq.delta_pos_4_784 |- (!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)) /\\ (((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))))))))))))))"}
{"_id": "51236", "text": "INT_POW_EQ_0 |- (!x. (!n. ((((int_pow x) n) = (int_of_num (NUMERAL _0))) = ((x = (int_of_num (NUMERAL _0))) /\\ (~ (n = (NUMERAL _0)))))))"}
{"_id": "51237", "text": "FACE_OF_HALFSPACE_GE |- (!f. (!a. (!b. (((face_of f) (GSPEC (\\GEN%PVAR%2156. (?x. (((SETSPEC GEN%PVAR%2156) ((real_ge ((dot a) x)) b)) x))))) = ((f = EMPTY) \\/ ((f = (GSPEC (\\GEN%PVAR%2157. (?x. (((SETSPEC GEN%PVAR%2157) (((dot a) x) = b)) x))))) \\/ (f = (GSPEC (\\GEN%PVAR%2158. (?x. (((SETSPEC GEN%PVAR%2158) ((real_ge ((dot a) x)) b)) x)))))))))))"}
{"_id": "51238", "text": "MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_1D |- (!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))))))))))) ==> ((convex_on f) s))))"}
{"_id": "51239", "text": "CONTINUOUS_ON_LIFT_PRODUCT |- (!f. (!s. (!t. (((FINITE t) /\\ (!i. ((i IN t) ==> ((continuous_on (\\x. (lift ((f x) i)))) s)))) ==> ((continuous_on (\\x. (lift ((product t) (f x))))) s)))))"}
{"_id": "51240", "text": "CONNECTED_NEST_GEN |- (!s. (((!n. ((closed (s n)) /\\ (connected (s n)))) /\\ ((?n. (compact (s n))) /\\ (!m. (!n. ((m <= n) ==> ((s n) SUBSET (s m))))))) ==> (connected (INTERS (GSPEC (\\GEN%PVAR%1155. (?n. (((SETSPEC GEN%PVAR%1155) (n IN UNIV)) (s n)))))))))"}
{"_id": "51241", "text": "REALLIM_ATREAL_ID |- (((---> (\\x. x)) a) (atreal a))"}
{"_id": "51242", "text": "ABSOLUTELY_CONTINUOUS_ISOMETRIC_COMPOSE |- (!f. (!g. (!s. ((!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y))))) ==> (((absolutely_continuous_on (f o g)) s) = ((absolutely_continuous_on g) s))))))"}
{"_id": "51243", "text": "MEASURABLE_SCALING_EQ |- (!s. (!c. ((~ (c = (real_of_num (NUMERAL _0)))) ==> ((measurable ((IMAGE (\\x. ((% c) x))) s)) = (measurable s)))))"}
{"_id": "51244", "text": "NEGATIONS_CBALL |- (!r. (((IMAGE vector_neg) (cball ((vec (NUMERAL _0)) , r))) = (cball ((vec (NUMERAL _0)) , r))))"}
{"_id": "51245", "text": "TAN_SUB |- (!x. (!y. (((~ ((cos x) = (real_of_num (NUMERAL _0)))) /\\ ((~ ((cos y) = (real_of_num (NUMERAL _0)))) /\\ (~ ((cos ((real_sub x) y)) = (real_of_num (NUMERAL _0)))))) ==> ((tan ((real_sub x) y)) = ((real_div ((real_sub (tan x)) (tan y))) ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul (tan x)) (tan y))))))))"}
{"_id": "51246", "text": "MEASURABLE_TETRAHEDRON |- (!a. (!b. (!c. (!d. (measurable ((hull convex) (a INSERT (b INSERT (c INSERT (d INSERT EMPTY))))))))))"}
{"_id": "51247", "text": "SUBSET_CONVEX_HULL_FRONTIER |- (!s. ((bounded s) ==> (s SUBSET ((hull convex) (frontier s)))))"}
{"_id": "51248", "text": "RTC_NE_IMP_TC |- (!R. (!x. (!y. (((((RTC R) x) y) /\\ (~ (x = y))) ==> (((TC R) x) y)))))"}
{"_id": "51249", "text": "NPRODUCT_CLOSED |- (!P. (!f. (!s. (((P (NUMERAL (BIT1 _0))) /\\ ((!x. (!y. (((P x) /\\ (P y)) ==> (P (x * y))))) /\\ (!a. ((a IN s) ==> (P (f a)))))) ==> (P ((nproduct s) f))))))"}
{"_id": "51250", "text": "CX_COSH |- ((Cx ((real_div ((real_add (exp x)) (real_inv (exp x)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (ccos ((complex_mul ii) (Cx x))))"}
{"_id": "51251", "text": "ITERATE_DIFF_GEN |- (!op. ((monoidal op) ==> (!f. (!s. (!t. (((FINITE (((support op) f) s)) /\\ ((((support op) f) t) SUBSET (((support op) f) s))) ==> (((op (((iterate op) (s DIFF t)) f)) (((iterate op) t) f)) = (((iterate op) s) f))))))))"}
{"_id": "51252", "text": "POSITIVE_DEFINITE_2 |- (!A. ((positive_definite A) = ((((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT1 _0))) = ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\\ (((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_pow ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0)))))))))))"}
{"_id": "51253", "text": "Misc_defs_and_lemmas.UNIONS_SUBSET |- (!U. (!X. ((!A. ((A IN U) ==> (A SUBSET X))) ==> ((UNIONS U) SUBSET X))))"}
{"_id": "51254", "text": "CARD_LT_REFL |- (!s. (~ (s <_c s)))"}
{"_id": "51255", "text": "REAL_INTEGRAL_SUBSTITUTION_SIMPLE |- (!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))))))) ==> (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\\x. ((real_mul (f (g x))) (g' x)))) = ((real_integral (closed_real_interval ((CONS ((g a) , (g b))) NIL))) f))))))))))"}
{"_id": "51256", "text": "REAL_INTEGRAL_LBOUND |- (!f. (!a. (!b. ((((real_le a) b) /\\ (((real_integrable_on f) (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))) ((real_integral (closed_real_interval ((CONS (a , b)) NIL))) f))))))"}
{"_id": "51257", "text": "REAL_OPEN_RATIONAL |- (!P. (((real_open (GSPEC (\\GEN%PVAR%4978. (?x. (((SETSPEC GEN%PVAR%4978) (P x)) x))))) /\\ (?x. (P x))) ==> (?x. ((rational x) /\\ (P x)))))"}
{"_id": "51258", "text": "REAL_INTEGRABLE_BY_PARTS_EQ |- (!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": "51259", "text": "Packing3.INTERS_INTER_INTERS_ALT |- (!f. (!g. (!P. (((INTERS (GSPEC (\\GEN%PVAR%6019. (?x. (((SETSPEC GEN%PVAR%6019) (x IN f)) (P x)))))) INTER (INTERS (GSPEC (\\GEN%PVAR%6020. (?y. (((SETSPEC GEN%PVAR%6020) (y IN g)) (P y))))))) = (INTERS (GSPEC (\\GEN%PVAR%6021. (?u. (((SETSPEC GEN%PVAR%6021) (u IN (f UNION g))) (P u))))))))))"}
{"_id": "51260", "text": "Trigonometry.ISRTTNZ |- (((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)) /\\ (!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": "51261", "text": "ANR_COVERING_SPACE |- (!p. (!s. (!c. ((((covering_space (c , p)) s) /\\ (ANR c)) ==> (ANR s)))))"}
{"_id": "51262", "text": "Miqmcsn.mod_prep |- (((i + (NUMERAL (BIT1 (BIT1 _0)))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT1 (BIT1 _0)))) + i)) /\\ (((i + (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) + i)) /\\ (((i + (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) + i)) /\\ (((i + (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) = (((NUMERAL (BIT1 _0)) * (NUMERAL (BIT0 (BIT1 (BIT1 _0))))) + i)) /\\ ((SUC (((NUMERAL (BIT1 _0)) * k) + i)) = (((NUMERAL (BIT1 _0)) * k) + (SUC i)))))))"}
{"_id": "51263", "text": "DIFFERENTIAL_ZERO_MAXMIN_COMPONENT |- (!f. (!x. (!e. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\\ ((k <= (dimindex UNIV)) /\\ (((real_lt (real_of_num (NUMERAL _0))) e) /\\ (((!y. ((y IN (ball (x , e))) ==> ((real_le ((f y) $ k)) ((f x) $ k)))) \\/ (!y. ((y IN (ball (x , e))) ==> ((real_le ((f x) $ k)) ((f y) $ k))))) /\\ ((differentiable f) (at x)))))) ==> ((((jacobian f) (at x)) $ k) = (vec (NUMERAL _0))))))))"}
{"_id": "51264", "text": "IN_EPIGRAPH |- (!x. (!y. ((((pastecart x) (lift y)) IN ((epigraph s) f)) = ((x IN s) /\\ ((real_le (f x)) y)))))"}
{"_id": "51265", "text": "HAS_INTEGRAL_SPIKE_INTERIOR_EQ |- (!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": "51266", "text": "Trigonometry2.WHEN_K_DIFF0_ARCV |- ((~ (k = (real_of_num (NUMERAL _0)))) ==> ((((arcV (vec (NUMERAL _0))) u) v) = (((arcV (vec (NUMERAL _0))) u) ((% (real_abs k)) v))))"}
{"_id": "51267", "text": "TC_RC |- (!R. ((TC (RC R)) = (RC (TC R))))"}
{"_id": "51268", "text": "INT_ABS_SUB |- (!x. (!y. ((int_abs ((int_sub x) y)) = (int_abs ((int_sub y) x)))))"}
{"_id": "51269", "text": "UNION_FRONTIER |- (!s. (!t. (((frontier s) UNION (frontier t)) = ((frontier (s UNION t)) UNION ((frontier (s INTER t)) UNION ((frontier s) INTER (frontier t)))))))"}
{"_id": "51270", "text": "REAL_CONVEX_ON_CONTINUOUS |- (!f. (!s. (((real_open s) /\\ ((real_convex_on f) s)) ==> ((real_continuous_on f) s))))"}
{"_id": "51271", "text": "Ssrnat.max0n |- (!n. (((maxn (NUMERAL _0)) n) = n))"}
{"_id": "51272", "text": "REAL_SECOND_MEAN_VALUE_THEOREM_BONNET |- (!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))) /\\ (((real_integral (closed_real_interval ((CONS (a , b)) NIL))) (\\x. ((real_mul (g x)) (f x)))) = ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f))))))))))"}
{"_id": "51273", "text": "INT_MUL_AC |- ((((int_mul m) n) = ((int_mul n) m)) /\\ ((((int_mul ((int_mul m) n)) p) = ((int_mul m) ((int_mul n) p))) /\\ (((int_mul m) ((int_mul n) p)) = ((int_mul n) ((int_mul m) p)))))"}
{"_id": "51274", "text": "Conforming.NEGLIGIBLE_AFF_3_INTER_BALL |- (!x. (!v. (!u. (!r. (negligible ((aff (x INSERT (v INSERT (u INSERT EMPTY)))) INTER ((normball x) r)))))))"}
{"_id": "51275", "text": "REAL_INTEGRABLE_ON_SUPERSET |- (!f. (!s. (!t. (((!x. ((~ (x IN s)) ==> ((f x) = (real_of_num (NUMERAL _0))))) /\\ ((s SUBSET t) /\\ ((real_integrable_on f) s))) ==> ((real_integrable_on f) t)))))"}
{"_id": "51276", "text": "REAL_COMPACT_IMP_CLOSED |- (!s. ((real_compact s) ==> (real_closed s)))"}
{"_id": "51277", "text": "ARZELA_ASCOLI_LIPSCHITZ |- (!f. (!s. (!t. (!b. (((compact s) /\\ ((bounded t) /\\ ((!n. (!x. ((x IN s) ==> (~ ((((IMAGE (f n)) s) INTER t) = EMPTY))))) /\\ (!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))))))))))) ==> (?g. ((!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))))))) /\\ (?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": "51278", "text": "REALLIM_CONST_EQ |- (!net. (!c. (!d. ((((---> (\\x. c)) d) net) = ((trivial_limit net) \\/ (c = d))))))"}
{"_id": "51279", "text": "MOD_MOD_EXP_MIN |- (!x. (!p. (!m. (!n. ((~ (p = (NUMERAL _0))) ==> (((x MOD (p EXP m)) MOD (p EXP n)) = (x MOD (p EXP ((MIN m) n)))))))))"}
{"_id": "51280", "text": "CONIC_POSITIVE_ORTHANT |- (conic (GSPEC (\\GEN%PVAR%1467. (?x. (((SETSPEC GEN%PVAR%1467) (!i. ((((NUMERAL (BIT1 _0)) <= i) /\\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i))))) x)))))"}
{"_id": "51281", "text": "INTEGRABLE_BOUNDED_VARIATION_COMPLEX_RMUL |- (!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 (f x)) (g x)))) (closed_interval ((CONS (a , b)) NIL))))))))"}
{"_id": "51282", "text": "FRONTIER_UNION |- (!s. (!t. ((((closure s) INTER (closure t)) = EMPTY) ==> ((frontier (s UNION t)) = ((frontier s) UNION (frontier t))))))"}
{"_id": "51283", "text": "LOG_CONVEX_ON_SUBSET |- (!f. (!s. (!t. ((((log_convex_on f) t) /\\ (s SUBSET t)) ==> ((log_convex_on f) s)))))"}
{"_id": "51284", "text": "Ssrnat.distn_eq1 |- (!m. (!n. ((((distn m) n) = (NUMERAL (BIT1 _0))) = (((COND (m < n)) ((SUC m) = n)) (m = (SUC n))))))"}
{"_id": "51285", "text": "LOCALLY_CONNECTED_PCROSS_EQ |- (!s. (!t. (((locally connected) (s PCROSS t)) = ((s = EMPTY) \\/ ((t = EMPTY) \\/ (((locally connected) s) /\\ ((locally connected) t)))))))"}
{"_id": "51286", "text": "Seq.size_behead |- (!s. ((sizel (behead s)) = ((sizel s) - (NUMERAL (BIT1 _0)))))"}
{"_id": "51287", "text": "LIM_CSIN_OVER_X |- (((--> (\\z. ((complex_div (csin z)) z))) (Cx (real_of_num (NUMERAL (BIT1 _0))))) (at (Cx (real_of_num (NUMERAL _0)))))"}
{"_id": "51288", "text": "Functional_equation.functional_proj_x3 |- (proj_x3 = (rotate3 proj_x1))"}
{"_id": "51289", "text": "Ssrbool.addbI |- (!x. (!y1. (!y2. ((((XOR x) y1) = ((XOR x) y2)) ==> (y1 = y2)))))"}
{"_id": "51290", "text": "CONVEX_ON_IMP_MIDPOINT_CONVEX |- (!f. (!s. (!x. (!y. ((((convex_on f) s) /\\ ((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)))))))))))"}
{"_id": "51291", "text": "CARD_EQ_COMPACT_SETS |- ((GSPEC (\\GEN%PVAR%1256. (?s. (((SETSPEC GEN%PVAR%1256) (compact s)) s)))) =_c UNIV)"}
{"_id": "51292", "text": "HAUSDIST_ALT |- (!s. (!t. (((bounded s) /\\ ((bounded t) /\\ ((~ (s = EMPTY)) /\\ (~ (t = EMPTY))))) ==> ((hausdist (s , t)) = (sup (GSPEC (\\GEN%PVAR%1203. (?x. (((SETSPEC GEN%PVAR%1203) (x IN UNIV)) (real_abs ((real_sub (setdist ((x INSERT EMPTY) , s))) (setdist ((x INSERT EMPTY) , t)))))))))))))"}
{"_id": "51293", "text": "RELATIVE_BOUNDARY_RETRACT_OF_PUNCTURED_AFFINE_HULL |- (!s. (!a. (((convex s) /\\ ((compact s) /\\ (a IN (relative_interior s)))) ==> ((retract_of (s DIFF (relative_interior s))) (((hull affine) s) DELETE a)))))"}
{"_id": "51294", "text": "FCCOVERABLE_INTERMEDIATE_CLOSURE |- (!s. (!t. (((s SUBSET t) /\\ ((t SUBSET (closure s)) /\\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?c. ((FINITE c) /\\ (((UNIONS c) = s) /\\ (!t. ((t IN c) ==> ((connected t) /\\ ((bounded t) /\\ ((real_le (diameter t)) e)))))))))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?c. ((FINITE c) /\\ (((UNIONS c) = t) /\\ (!t. ((t IN c) ==> ((connected t) /\\ ((bounded t) /\\ ((real_le (diameter t)) e)))))))))))))"}
{"_id": "51295", "text": "DENSE_LIMIT_POINTS |- (!x. (((GSPEC (\\GEN%PVAR%583. (?x. (((SETSPEC GEN%PVAR%583) ((limit_point_of x) s)) x)))) = UNIV) = ((closure s) = UNIV)))"}
{"_id": "51296", "text": "Rogers.QXSKIIT |- (!vf. (!b. (((FINITE ((IMAGE vf) UNIV)) /\\ ((~ (affine_dependent ((IMAGE vf) UNIV))) /\\ (!i. (!j. (((vf i) = (vf j)) ==> ((b i) = (b j))))))) ==> (?!p. ((p IN ((hull affine) ((IMAGE vf) UNIV))) /\\ (!i. (!j. (((dot p) ((vector_sub (vf i)) (vf j))) = ((real_sub (b i)) (b j))))))))))"}
{"_id": "51297", "text": "SECOND_MEAN_VALUE_THEOREM_GEN |- (!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))) /\\ (((integral (closed_interval ((CONS (a , b)) NIL))) (\\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)))))))))))))"}
{"_id": "51298", "text": "EXTREME_POINT_OF_CONIC |- (!s. (!x. (((conic s) /\\ ((extreme_point_of x) s)) ==> (x = (vec (NUMERAL _0))))))"}
{"_id": "51299", "text": "REAL_DIFFERENTIABLE_WITHIN_SQRT |- (!s. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable sqrt) ((within (atreal x)) s)))))"}
{"_id": "51300", "text": "IS_REALINTERVAL_CONTINUOUS_IMAGE |- (!s. ((((real_continuous_on f) s) /\\ (is_realinterval s)) ==> (is_realinterval ((IMAGE f) s))))"}
{"_id": "51301", "text": "Seq.count_predUI |- (!a1. (!a2. (!s. ((((count ((predU a1) a2)) s) + ((count ((predI a1) a2)) s)) = (((count a1) s) + ((count a2) s))))))"}
{"_id": "51302", "text": "CAUCHY_CONTINUOUS_IMP_CONTINUOUS |- (!f. (!s. ((!x. (((cauchy x) /\\ (!n. ((x n) IN s))) ==> (cauchy (f o x)))) ==> ((continuous_on f) s))))"}
{"_id": "51303", "text": "ANR_FROM_UNION_AND_INTER |- (!s. (!t. (((closed s) /\\ ((closed t) /\\ ((ANR (s UNION t)) /\\ (ANR (s INTER t))))) ==> ((ANR s) /\\ (ANR t)))))"}
{"_id": "51304", "text": "LIEB |- (!f. (!g. (!s. (!t. (((!n. ((absolutely_integrable_on (f n)) s)) /\\ (((absolutely_integrable_on g) s) /\\ ((negligible t) /\\ (!x. ((x IN (s DIFF t)) ==> (((--> (\\n. ((f n) x))) (g x)) sequentially)))))) ==> (((--> (\\n. ((vector_sub ((integral s) (\\x. (lift (vector_norm ((vector_sub ((f n) x)) (g x))))))) ((vector_sub ((integral s) (\\x. (lift (vector_norm ((f n) x)))))) ((integral s) (\\x. (lift (vector_norm (g x))))))))) (vec (NUMERAL _0))) sequentially))))))"}
{"_id": "51305", "text": "RADON |- (!c. ((affine_dependent c) ==> (?m. (?p. ((m SUBSET c) /\\ ((p SUBSET c) /\\ (((DISJOINT m) p) /\\ (~ ((DISJOINT ((hull convex) m)) ((hull convex) p))))))))))"}
{"_id": "51306", "text": "Ssrbool.andb_orl |- (!b. (!c. (!p. (((b \\/ c) /\\ p) = ((b /\\ p) \\/ (c /\\ p))))))"}
{"_id": "51307", "text": "Ssrbool.orb_idl |- (!a. (!b. ((a ==> b) ==> ((a \\/ b) = b))))"}
{"_id": "51308", "text": "REAL_ADD_SIN |- (!x. (!y. (((real_add (sin x)) (sin y)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (sin ((real_div ((real_add x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) (cos ((real_div ((real_sub x) y)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))"}
{"_id": "51309", "text": "Seq.reshapeKr |- (!sh. (!s. (((sizel s) <= (sumn sh)) ==> ((flatten ((reshape sh) s)) = s))))"}
{"_id": "51310", "text": "WF_REFL |- (!x. ((WF <<) ==> (~ ((<< x) x))))"}
{"_id": "51311", "text": "INCREASING_RIGHT_LIMIT |- (!f. (!a. (!b. (!c. (((!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))))) /\\ (c IN (closed_real_interval ((CONS (a , b)) NIL)))) ==> (?l. (((---> f) l) ((within (atreal c)) (closed_real_interval ((CONS (c , b)) NIL))))))))))"}
{"_id": "51312", "text": "SETDIST_ZERO_STRONG |- (!s. (!t. ((~ ((DISJOINT (closure s)) (closure t))) ==> ((setdist (s , t)) = (real_of_num (NUMERAL _0))))))"}
{"_id": "51313", "text": "REAL_CONTINUOUS_TRANSFORM_WITHIN_SET_IMP |- (!f. (!a. (!s. (!t. ((((eventually (\\x. ((x IN t) ==> (x IN s)))) (at a)) /\\ ((real_continuous f) ((within (at a)) s))) ==> ((real_continuous f) ((within (at a)) t)))))))"}
{"_id": "51314", "text": "Nonlinear_lemma.quartic_has_real_derivative |- (!x. (!c0. (!c1. (!c2. (!c3. (!c4. (((has_real_derivative (\\x. ((real_add ((real_mul c0) (real_of_num (NUMERAL (BIT1 _0))))) ((real_add ((real_mul c1) ((real_pow x) (NUMERAL (BIT1 _0))))) ((real_add ((real_mul c2) ((real_pow x) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul c3) ((real_pow x) (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul c4) ((real_pow x) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))))) ((real_add ((real_mul c0) (real_of_num (NUMERAL _0)))) ((real_add ((real_mul c1) ((real_mul (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul ((real_pow x) ((NUMERAL (BIT1 _0)) - (NUMERAL (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 _0))))))) ((real_add ((real_mul c2) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow x) ((NUMERAL (BIT0 (BIT1 _0))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))) ((real_add ((real_mul c3) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) ((real_pow x) ((NUMERAL (BIT1 (BIT1 _0))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))) ((real_mul c4) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow x) ((NUMERAL (BIT0 (BIT0 (BIT1 _0)))) - (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL (BIT1 _0)))))))))) (atreal x))))))))"}
{"_id": "51315", "text": "LIM_RE_UBOUND |- (!net. (!f. (!l. (!b. (((~ (trivial_limit net)) /\\ ((((--> f) l) net) /\\ ((eventually (\\x. ((real_le (Re (f x))) b))) net))) ==> ((real_le (Re l)) b))))))"}
{"_id": "51316", "text": "CONVEX_HALFSPACE_IM_LE |- (!b. (convex (GSPEC (\\GEN%PVAR%4774. (?z. (((SETSPEC GEN%PVAR%4774) ((real_le (Im z)) b)) z))))))"}
{"_id": "51317", "text": "TRACE_MUL_POSITIVE_SEMIDEFINITE_LE |- (!A. (!B. (((positive_semidefinite A) /\\ (positive_semidefinite B)) ==> ((real_le (trace ((matrix_mul A) B))) ((real_mul (trace A)) (trace B))))))"}
{"_id": "51318", "text": "REAL_INTEGRABLE_ON_LITTLE_SUBINTERVALS |- (!f. (!a. (!b. ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\\ (!u. (!v. (((x IN (closed_real_interval ((CONS (u , v)) NIL))) /\\ (!y. ((y IN (closed_real_interval ((CONS (u , v)) NIL))) ==> (((real_lt (real_abs ((real_sub y) x))) d) /\\ (y IN (closed_real_interval ((CONS (a , b)) NIL))))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (u , v)) NIL)))))))))) ==> ((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "51319", "text": "INT_POW_NEG |- (!x. (!n. (((int_pow (int_neg x)) n) = (((COND (EVEN n)) ((int_pow x) n)) (int_neg ((int_pow x) n))))))"}
{"_id": "51320", "text": "CLOSURE_IMAGE_BOUNDED |- (!f. (!s. ((((continuous_on f) (closure s)) /\\ (bounded s)) ==> ((closure ((IMAGE f) s)) = ((IMAGE f) (closure s))))))"}
{"_id": "51321", "text": "PATH_COMPOSE_JOIN |- (!f. (!p. (!q. ((f o ((++ p) q)) = ((++ (f o p)) (f o q))))))"}
{"_id": "51322", "text": "REAL_INFSUM_SUB |- (!x. (!y. (!s. ((((real_summable s) x) /\\ ((real_summable s) y)) ==> (((real_infsum s) (\\i. ((real_sub (x i)) (y i)))) = ((real_sub ((real_infsum s) x)) ((real_infsum s) y)))))))"}
{"_id": "51323", "text": "NADD_LE_RMUL |- (!x. (!y. (!z. (((nadd_le x) y) ==> ((nadd_le ((nadd_mul x) z)) ((nadd_mul y) z))))))"}
{"_id": "51324", "text": "NEGLIGIBLE_TRANSLATION |- (!s. (!a. ((negligible s) ==> (negligible ((IMAGE (\\x. ((vector_add a) x))) s)))))"}
{"_id": "51325", "text": "Seq2.list_sum_EL_alt |- (!s. (!f. (((list_sum s) f) = ((sum ((NUMERAL (BIT1 _0)) .. (sizel s))) (\\i. (f ((EL (i - (NUMERAL (BIT1 _0)))) s)))))))"}
{"_id": "51326", "text": "Functional_equation.functional_proj_y6 |- (proj_y6 = (rotate6 proj_y1))"}
{"_id": "51327", "text": "AFFINE_INDEPENDENT_CONVEX_AFFINE_HULL |- (!s. (!t. (((~ (affine_dependent s)) /\\ (t SUBSET s)) ==> (((hull convex) t) = (((hull affine) t) INTER ((hull convex) s))))))"}
{"_id": "51328", "text": "RECTIFIABLE_PATH_SYM |- (!p. (!q. ((((pathfinish p) = (pathstart q)) /\\ ((pathfinish q) = (pathstart p))) ==> ((rectifiable_path ((++ p) q)) = (rectifiable_path ((++ q) p))))))"}
{"_id": "51329", "text": "Lp_ineqs_proofs.tau_sum4 |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((lp_tau (V , E)) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((CARD f) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ==> (((sum f) (rhazim_fan (V , E))) = ((real_add ((tau_fan (V , E)) f)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_add pi) sol0))))))))))))))"}
{"_id": "51330", "text": "LINEAR_IMP_HAS_BOUNDED_VARIATION |- (!f. (((linear f) /\\ (bounded s)) ==> ((has_bounded_variation_on f) s)))"}
{"_id": "51331", "text": "Fnjlbxs.scriptL_eq |- (!S. (!k. ((((NUMERAL (BIT1 (BIT1 _0))) * k) <= (dimindex UNIV)) ==> ((!l. ((l IN S) = (((set_of_list l) SUBSET ball_annulus) /\\ ((packing (set_of_list l)) /\\ (((LENGTH l) = k) /\\ (uniq l)))))) ==> (!v. ((v IN ((IMAGE (vector_of_list (NUMERAL (BIT1 (BIT1 _0))))) S)) ==> ((scriptL (set_of_list (((list_of_vector (NUMERAL (BIT1 (BIT1 _0)))) k) v))) = ((list_sum (((list_of_vector (NUMERAL (BIT1 (BIT1 _0)))) k) v)) (\\v. (lmfun ((real_div (vector_norm v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))"}
{"_id": "51332", "text": "RCONE_GT_NEG |- (!v0. (!v1. (!a. ((((rcone_gt v0) v1) (real_neg a)) = ((IMAGE (\\x. ((vector_sub ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) v0)) x))) (UNIV DIFF (((rcone_ge v0) v1) a)))))))"}
{"_id": "51333", "text": "RE_CACS_BOUND |- (!z. ((real_le (real_abs (Re (cacs z)))) pi))"}
{"_id": "51334", "text": "COMPLEX_NEG_MINUS1 |- (!x. ((vector_neg x) = ((complex_mul (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) x)))"}
{"_id": "51335", "text": "AFFINE_NEGATIONS |- (!s. ((affine s) ==> (affine ((IMAGE vector_neg) s))))"}
{"_id": "51336", "text": "DIAMETER_SCALING |- (!a. (!s. ((bounded s) ==> ((diameter ((IMAGE (\\x. ((% a) x))) s)) = ((real_mul (real_abs a)) (diameter s))))))"}
{"_id": "51337", "text": "Constants_approx.INTERVAL_SQRT |- ((((interval_arith x) (a , b)) /\\ (((real_le ((real_mul c) c)) a) /\\ ((real_le b) ((real_mul d) d)))) ==> ((interval_arith (sqrt x)) ((real_abs c) , (real_abs d))))"}
{"_id": "51338", "text": "INT_LT_MUL |- (!x. (!y. ((((int_lt (int_of_num (NUMERAL _0))) x) /\\ ((int_lt (int_of_num (NUMERAL _0))) y)) ==> ((int_lt (int_of_num (NUMERAL _0))) ((int_mul x) y)))))"}
{"_id": "51339", "text": "MEASURE_INTERVAL_2_ALT |- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ ((real_lt (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_lt (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\\ ((real_lt (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": "51340", "text": "ALL2_MAP2 |- (!l. (!m. ((((ALL2 P) ((MAP f) l)) ((MAP g) m)) = (((ALL2 (\\x. (\\y. ((P (f x)) (g y))))) l) m))))"}
{"_id": "51341", "text": "ENR_PCROSS |- (!s. (!t. (((ENR s) /\\ (ENR t)) ==> (ENR (s PCROSS t)))))"}
{"_id": "51342", "text": "REAL_CONTINUOUS_WITHIN_SUBSET |- (!f. (!s. (!t. ((((real_continuous f) ((within (at x)) s)) /\\ (t SUBSET s)) ==> ((real_continuous f) ((within (at x)) t))))))"}
{"_id": "51343", "text": "ANGLE_EQ_0_DIST_ABS |- (!A. (!B. (!C. (((angle (A , (B , C))) = (real_of_num (NUMERAL _0))) = ((~ (A = B)) /\\ ((~ (C = B)) /\\ ((distance (A , C)) = (real_abs ((real_sub (distance (A , B))) (distance (C , B)))))))))))"}
{"_id": "51344", "text": "Pack2.finite_voronoi2 |- (!v. (!S. ((packing S) ==> (FINITE (GSPEC (\\GEN%PVAR%5977. (?x. (?y. (((SETSPEC GEN%PVAR%5977) ((y IN (S DELETE v)) /\\ ((x = v) /\\ (y IN (ball (v , (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))) ((bis_le x) y))))))))))"}
{"_id": "51345", "text": "INT_LE_DOUBLE |- (!x. (((int_le (int_of_num (NUMERAL _0))) ((int_add x) x)) = ((int_le (int_of_num (NUMERAL _0))) x)))"}
{"_id": "51346", "text": "Functional_equation.domain6_ineq |- (!c1. (!c2. (!P. (!F1. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!a1. (!a2. (!a3. (!a4. (!a5. (!a6. (!b1. (!b2. (!b3. (!b4. (!b5. (!b6. ((((domain6 P) c2) c1) ==> (((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) ((((((P x1) x2) x3) x4) x5) x6)) ==> (((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) (F1 ((((((c2 x1) x2) x3) x4) x5) x6))) = ((ineq ((CONS (a1 , (x1 , b1))) ((CONS (a2 , (x2 , b2))) ((CONS (a3 , (x3 , b3))) ((CONS (a4 , (x4 , b4))) ((CONS (a5 , (x5 , b5))) ((CONS (a6 , (x6 , b6))) NIL))))))) (F1 ((((((c1 x1) x2) x3) x4) x5) x6))))))))))))))))))))))))))))"}
{"_id": "51347", "text": "HAS_BOUNDED_REAL_VARIATION_ON_MAX |- (!f. (!g. (!s. ((((has_bounded_real_variation_on f) s) /\\ ((has_bounded_real_variation_on g) s)) ==> ((has_bounded_real_variation_on (\\x. ((real_max (f x)) (g x)))) s)))))"}
{"_id": "51348", "text": "HOLOMORPHIC_ON_CSIN |- (!s. ((holomorphic_on csin) s))"}
{"_id": "51349", "text": "LOCALLY_PATH_CONNECTED_RIGHT_INVERTIBLE_IMAGE |- (!f. (!g. (!s. ((((continuous_on f) s) /\\ (((continuous_on g) ((IMAGE f) s)) /\\ ((((IMAGE g) ((IMAGE f) s)) SUBSET s) /\\ ((!x. ((x IN ((IMAGE f) s)) ==> ((f (g x)) = x))) /\\ ((locally path_connected) s))))) ==> ((locally path_connected) ((IMAGE f) s))))))"}
{"_id": "51350", "text": "ABSOLUTELY_REAL_INTEGRABLE_SUM |- (!f. (!s. (!t. (((FINITE t) /\\ (!a. ((a IN t) ==> ((absolutely_real_integrable_on (f a)) s)))) ==> ((absolutely_real_integrable_on (\\x. ((sum t) (\\a. ((f a) x))))) s)))))"}
{"_id": "51351", "text": "CPOW_REAL_REAL |- (!w. (!z. (((real w) /\\ ((real z) /\\ ((real_lt (real_of_num (NUMERAL _0))) (Re w)))) ==> (((cpow w) z) = (Cx (exp ((real_mul (Re z)) (log (Re w)))))))))"}
{"_id": "51352", "text": "EXISTS_CNJ |- ((?z. (P (cnj z))) = (?z. (P z)))"}
{"_id": "51353", "text": "LEFT_POLAR_DECOMPOSITION_INVERTIBLE |- (!A. ((invertible A) = (?U. (?P. ((orthogonal_matrix U) /\\ ((positive_definite P) /\\ (((matrix_mul P) U) = A)))))))"}
{"_id": "51354", "text": "BANACH_FIX_ITER |- (!f. (!s. (!c. (!n. (((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 ((((ITER n) f) x) , (((ITER n) f) y)))) ((real_mul c) (distance (x , y)))))))))))) ==> (?!x. ((x IN s) /\\ ((f x) = x))))))))"}
{"_id": "51355", "text": "Bump.MCELL2_CELL_PARAMETERS_EXIST |- (!V. (!ul. (!X. (((packing V) /\\ ((saturated V) /\\ ((X = ((mcell2 V) ul)) /\\ ((ul IN ((barV V) (NUMERAL (BIT1 (BIT1 _0))))) /\\ (~ (negligible X)))))) ==> ((FST ((cell_params V) X)) = (NUMERAL (BIT0 (BIT1 _0))))))))"}
{"_id": "51356", "text": "NEGLIGIBLE_AFFINE_HULL_2 |- (!a. (!b. (negligible ((hull affine) (a INSERT (b INSERT EMPTY))))))"}
{"_id": "51357", "text": "Functional_equation.functional_rotate2 |- (!f. ((rotate2 f) = (((((((compose6 f) proj_x2) proj_x3) proj_x1) proj_x5) proj_x6) proj_x4)))"}
{"_id": "51358", "text": "ROOT_EQ_0 |- (!n. (!x. ((((root n) x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))))"}
{"_id": "51359", "text": "ABSOLUTELY_REAL_INTEGRABLE_ABSOLUTELY_REAL_INTEGRABLE_UBOUND |- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((real_le (f x)) (g x)))) /\\ (((real_integrable_on f) s) /\\ ((absolutely_real_integrable_on g) s))) ==> ((absolutely_real_integrable_on g) s)))))"}
{"_id": "51360", "text": "PROPER_LOCALLY_INJECTIVE_OPEN_IMP_HOMEOMORPHISM |- (!f. (!s. (!t. (((path_connected s) /\\ ((simply_connected t) /\\ (((continuous_on f) s) /\\ ((((IMAGE f) s) = t) /\\ ((!k. (((k SUBSET t) /\\ (compact k)) ==> (compact (GSPEC (\\GEN%PVAR%2105. (?x. (((SETSPEC GEN%PVAR%2105) ((x IN s) /\\ ((f x) IN k))) x))))))) /\\ ((!u. (((open_in ((subtopology euclidean) s)) u) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) u)))) /\\ (!x. ((x IN s) ==> (?u. ((x IN u) /\\ (((open_in ((subtopology euclidean) s)) u) /\\ (!y. (!z. (((y IN u) /\\ ((z IN u) /\\ ((f y) = (f z)))) ==> (y = z))))))))))))))) ==> (?g. ((homeomorphism (s , t)) (f , g)))))))"}
{"_id": "51361", "text": "REAL_MEASURE_REAL_NEGLIGIBLE_UNION |- (!s. (!t. (((real_measurable s) /\\ ((real_measurable t) /\\ (real_negligible (s INTER t)))) ==> ((real_measure (s UNION t)) = ((real_add (real_measure s)) (real_measure t))))))"}
{"_id": "51362", "text": "COMPLEX_TAYLOR_MVT |- (!f. (!w. (!z. (!n. ((!i. (!x. (((x IN (closed_segment ((CONS (w , z)) NIL))) /\\ (i <= n)) ==> (((has_complex_derivative (f i)) ((f (i + (NUMERAL (BIT1 _0)))) x)) (at x))))) ==> (?u. ((u IN (closed_segment ((CONS (w , z)) NIL))) /\\ ((Re ((f (NUMERAL _0)) z)) = (Re ((vector_add ((vsum ((NUMERAL _0) .. n)) (\\i. ((complex_mul ((f i) w)) ((complex_div ((complex_pow ((vector_sub z) w)) i)) (Cx (real_of_num (FACT i)))))))) ((complex_mul ((complex_mul ((f (n + (NUMERAL (BIT1 _0)))) u)) ((complex_div ((complex_pow ((vector_sub z) u)) n)) (Cx (real_of_num (FACT n)))))) ((vector_sub z) w))))))))))))"}
{"_id": "51363", "text": "Lp_main_estimate.v_prime_eq |- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. (!k. ((d IN ((darts_k k) (hypermap_of_fan (V , E)))) ==> (((v_prime V) ((face (hypermap_of_fan (V , E))) d)) = (GSPEC (\\GEN%PVAR%8731. (?i. (((SETSPEC GEN%PVAR%8731) (i < k)) (FST (((POWER (f_fan_pair_ext (V , E))) i) d)))))))))))))"}
{"_id": "51364", "text": "COMPLEX_NOT_ROOT_UNITY |- (!n. (((NUMERAL (BIT1 _0)) <= n) ==> (?u. (((vector_norm u) = (real_of_num (NUMERAL (BIT1 _0)))) /\\ (~ (((complex_pow u) n) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))))))"}
{"_id": "51365", "text": "POWER_REAL_SERIES_CONV_IMP_ABSCONV_WEAK |- (!a. (!k. (!w. (!z. ((((real_summable k) (\\n. ((real_mul (a n)) ((real_pow z) n)))) /\\ ((real_lt (real_abs w)) (real_abs z))) ==> ((real_summable k) (\\n. ((real_mul (real_abs (a n))) ((real_pow w) n)))))))))"}
{"_id": "51366", "text": "Geomdetail.aff |- (aff = (hull affine))"}
{"_id": "51367", "text": "TRACE_MATRIX_INV_RMUL |- (!A. ((trace ((matrix_mul A) (matrix_inv A))) = (real_of_num (rank A))))"}
{"_id": "51368", "text": "Lp_ineqs_proofs.ln_fan_ly |- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. ((v IN V) ==> ((ln_fan v) = (ly (vector_norm v)))))))))))"}
{"_id": "51369", "text": "REALLIM_AT_ZERO |- (!f. (!l. (!a. ((((---> f) l) (at a)) = (((---> (\\x. (f ((vector_add a) x)))) l) (at (vec (NUMERAL _0))))))))"}
{"_id": "51370", "text": "VECTOR_EXPAND_4 |- (!x. (x = (vector ((CONS (x $ (NUMERAL (BIT1 _0)))) ((CONS (x $ (NUMERAL (BIT0 (BIT1 _0))))) ((CONS (x $ (NUMERAL (BIT1 (BIT1 _0))))) ((CONS (x $ (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) NIL)))))))"}
{"_id": "51371", "text": "Pwssrat.rotate_to_hd |- (((rotate_to ((CONS h) t)) h) = ((CONS h) t))"}
{"_id": "51372", "text": "Seq.eq_all_r |- (!s1. (!s2. ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> (!a. (((all a) s1) = ((all a) s2))))))"}
{"_id": "51373", "text": "SUBMETRIC_UNIV |- (((submetric m) UNIV) = m)"}
{"_id": "51374", "text": "OPEN_NON_PATH_COMPONENT |- (!s. (!x. ((open s) ==> (open (s DIFF ((path_component s) x))))))"}
{"_id": "51375", "text": "HAS_VECTOR_DERIVATIVE_VECTOR_POLYNOMIAL_FUNCTION |- (!p. ((vector_polynomial_function p) ==> (?p'. ((vector_polynomial_function p') /\\ (!x. (((has_vector_derivative p) (p' x)) (at x)))))))"}
{"_id": "51376", "text": "Pack2.measure_unions_sum_voronoi |- (!S. (!p. (!r. ((((real_le (real_of_num (NUMERAL _0))) r) /\\ ((packing S) /\\ (saturated S))) ==> ((measure (UNIONS (GSPEC (\\GEN%PVAR%5985. (?w. (?v. (((SETSPEC GEN%PVAR%5985) ((v IN (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) /\\ (w = S))) ((voronoi_closed w) v)))))))) = ((sum (S INTER (ball (p , ((real_add r) (real_of_num (NUMERAL (BIT1 _0)))))))) (\\v. (measure ((voronoi_closed S) v)))))))))"}
{"_id": "51377", "text": "DIFFERENTIABLE_ON_COMPOSE |- (!f. (!g. (!s. ((((differentiable_on f) s) /\\ ((differentiable_on g) ((IMAGE f) s))) ==> ((differentiable_on (g o f)) s)))))"}
{"_id": "51378", "text": "INT_SUB_LDISTRIB |- (!x. (!y. (!z. (((int_mul x) ((int_sub y) z)) = ((int_sub ((int_mul x) y)) ((int_mul x) z))))))"}
{"_id": "51379", "text": "INF_INSERT |- (!x. (!s. ((bounded ((IMAGE lift) s)) ==> ((inf (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_min x) (inf s)))))))"}
{"_id": "51380", "text": "Functional_equation.delta_all_atn |- (!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)) /\\ ((((((((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)) /\\ ((((((((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))) /\\ ((((((((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)) /\\ ((((((((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": "51381", "text": "REAL_CONTINUOUS_AT_AZIM |- (!v. (!w. (!w1. (!w2. ((~ (coplanar (v INSERT (w INSERT (w1 INSERT (w2 INSERT EMPTY)))))) ==> ((real_continuous (((azim v) w) w1)) (at w2)))))))"}
{"_id": "51382", "text": "BARYCENTRE_NOT_IN_SET |- (!s. (((~ (affine_dependent s)) /\\ (~ (?a. (s = (a INSERT EMPTY))))) ==> (~ ((barycentre s) IN s))))"}
{"_id": "51383", "text": "ISOMETRY_SUBSET_SUBSPACE |- (!s. (!t. (((subspace s) /\\ ((subspace t) /\\ ((dim s) <= (dim t)))) ==> (?f. ((linear f) /\\ ((((IMAGE f) s) SUBSET t) /\\ (!x. ((x IN s) ==> ((vector_norm (f x)) = (vector_norm x))))))))))"}
{"_id": "51384", "text": "EXISTS_SUBPATH_OF_PATH |- (!g. (!a. (!b. (((path g) /\\ ((a IN (path_image g)) /\\ (b IN (path_image g)))) ==> (?h. ((path h) /\\ (((pathstart h) = a) /\\ (((pathfinish h) = b) /\\ ((path_image h) SUBSET (path_image g))))))))))"}
{"_id": "51385", "text": "Constants_approx.INTERVAL_INV_NEG |- ((((interval_arith x) (a , b)) /\\ ((real_lt b) (real_of_num (NUMERAL _0)))) ==> ((interval_arith (real_inv x)) ((real_inv b) , (real_inv a))))"}
{"_id": "51386", "text": "HOMEOMORPHISM_CLOSED_IN_EQ |- (!f. (!g. (!s. (!t. (!u. (!v. ((((homeomorphism (s , t)) (f , g)) /\\ ((u SUBSET s) /\\ (v SUBSET s))) ==> (((closed_in ((subtopology euclidean) ((IMAGE f) u))) ((IMAGE f) v)) = ((closed_in ((subtopology euclidean) u)) v)))))))))"}
{"_id": "51387", "text": "JOINABLE_COMPONENTS_EQ |- (!s. (!t. (!c1. (!c2. (((connected t) /\\ ((t SUBSET s) /\\ ((c1 IN (components s)) /\\ ((c2 IN (components s)) /\\ ((~ ((c1 INTER t) = EMPTY)) /\\ (~ ((c2 INTER t) = EMPTY))))))) ==> (c1 = c2))))))"}
{"_id": "51388", "text": "REALLIM_ZERO_NEGINFINITY |- (!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": "51389", "text": "Miqmcsn.J_SCS_3M1_prime |- ((((scs_J_v39 ((scs_prop_equ_v39 scs_3M1_prime) i)) i1) j) = F)"}
{"_id": "51390", "text": "MEASURABLE_INTER_HALFSPACE_GE |- (!s. (!a. (!i. ((measurable s) ==> (measurable (s INTER (GSPEC (\\GEN%PVAR%3589. (?x. (((SETSPEC GEN%PVAR%3589) ((real_ge (x $ i)) a)) x))))))))))"}
{"_id": "51391", "text": "CONIC_HALFSPACE_GE |- (!a. (conic (GSPEC (\\GEN%PVAR%1465. (?x. (((SETSPEC GEN%PVAR%1465) ((real_ge ((dot a) x)) (real_of_num (NUMERAL _0)))) x))))))"}
{"_id": "51392", "text": "REAL_MEASURE_DIFF_SUBSET |- (!s. (!t. (((real_measurable s) /\\ ((real_measurable t) /\\ (t SUBSET s))) ==> ((real_measure (s DIFF t)) = ((real_sub (real_measure s)) (real_measure t))))))"}
{"_id": "51393", "text": "MEASURABLE_ON_MEASURABLE_PREIMAGE_OPEN_INTERVAL |- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!a. (!b. (measurable (GSPEC (\\GEN%PVAR%3653. (?x. (((SETSPEC GEN%PVAR%3653) ((x IN s) /\\ ((f x) IN (open_interval (a , b))))) x)))))))))))"}
{"_id": "51394", "text": "MBOUNDED_MBALL |- (!m. (!c. (!b. ((mbounded m) ((mball m) (c , b))))))"}
{"_id": "51395", "text": "PATH_CONNECTED_PUNCTURED_BALL |- (!a. (!r. (((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) ==> (path_connected ((ball (a , r)) DELETE a)))))"}
{"_id": "51396", "text": "BOUNDED_RECTIFIABLE_PATH_IMAGE |- (!g. ((rectifiable_path g) ==> (bounded (path_image g))))"}
{"_id": "51397", "text": "Functional_equation.functional_acs_sqrt_x1_d4 |- (acs_sqrt_x1_d4 = (uni (acs , ((scalar6 proj_y1) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))"}
{"_id": "51398", "text": "REAL_DERIVATIVE_POS_RIGHT_MAXIMUM |- (!f. (!f'. (!a. (!b. (!e. ((((real_lt a) b) /\\ (((real_lt (real_of_num (NUMERAL _0))) e) /\\ ((((has_real_derivative f) f') ((within (atreal b)) (closed_real_interval ((CONS (a , b)) NIL)))) /\\ (!x. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\\ ((real_lt (real_abs ((real_sub x) b))) e)) ==> ((real_le (f x)) (f b))))))) ==> ((real_le (real_of_num (NUMERAL _0))) f')))))))"}
{"_id": "51399", "text": "TAGGED_DIVISION_OF_ANOTHER |- (!p. (!s. (!s'. ((((tagged_partial_division_of p) s') /\\ ((!t. (!k. (((t , k) IN p) ==> (k SUBSET s)))) /\\ (!x. ((x IN s) ==> (?t. (?k. (((t , k) IN p) /\\ (x IN k)))))))) ==> ((tagged_division_of p) s)))))"}
{"_id": "51400", "text": "NSUM_ADD_GEN |- (!f. (!g. (!s. (((FINITE (GSPEC (\\GEN%PVAR%172. (?x. (((SETSPEC GEN%PVAR%172) ((x IN s) /\\ (~ ((f x) = (NUMERAL _0))))) x))))) /\\ (FINITE (GSPEC (\\GEN%PVAR%173. (?x. (((SETSPEC GEN%PVAR%173) ((x IN s) /\\ (~ ((g x) = (NUMERAL _0))))) x)))))) ==> (((nsum s) (\\x. ((f x) + (g x)))) = (((nsum s) f) + ((nsum s) g)))))))"}
{"_id": "51401", "text": "Imjxphr.DEFORMATION_DIST_LE_V3_DEFOR_A |- (((is_scs_v39 s) /\\ (((MMs_v39 s) w) /\\ (((scs_k_v39 s) = k) /\\ (((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) /\\ ((a = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) /\\ (!i. (((~ ((l MOD k) = (i MOD k))) /\\ (~ ((l MOD k) = ((SUC i) MOD k)))) ==> ((real_lt (((scs_a_v39 s) l) i)) (distance (v2 , (w i)))))))))))))))) ==> (!i. (((~ ((l MOD k) = (i MOD k))) /\\ (~ ((l MOD k) = ((SUC i) 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 (((((((((v3_defor_v1 a) v1) v2) x1) x2) x6) x6) ((real_sub x2) t)) , (w i)))))))))))"}
{"_id": "51402", "text": "CBALL_INTERVAL_0 |- (!e. ((cball ((vec (NUMERAL _0)) , e)) = (closed_interval ((CONS ((vector_neg (lift e)) , (lift e))) NIL))))"}
{"_id": "51403", "text": "ABSOLUTELY_CONTINUOUS_ON_DIVISION |- (!f. (!a. (!b. (!d. (((division_of d) (closed_interval ((CONS (a , b)) NIL))) ==> ((!k. ((k IN d) ==> ((absolutely_continuous_on f) k))) = ((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))))"}
{"_id": "51404", "text": "Merge_ineq.eta135_x_bd |- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!c. ((~ ((real_lt ((real_add ((real_mul ((((((unit6 x1) x2) x3) x4) x5) x6)) c)) ((real_mul ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) (real_neg (real_of_num (NUMERAL (BIT1 _0))))))) (real_of_num (NUMERAL _0)))) ==> (((real_lt c) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ==> ((real_lt ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))))"}
{"_id": "51405", "text": "ABSOLUTELY_CONTINUOUS_ON_TRANSLATION |- (!f. (!s. (!a. (((absolutely_continuous_on (\\x. ((vector_add a) (f x)))) s) = ((absolutely_continuous_on f) s)))))"}
{"_id": "51406", "text": "RE_CSQRT |- (!z. ((real_le (real_of_num (NUMERAL _0))) (Re (csqrt z))))"}
{"_id": "51407", "text": "RSC_INC_RC |- (!R. (!x. (!y. ((((RC R) x) y) ==> (((RSC R) x) y)))))"}
{"_id": "51408", "text": "RE_COMPLEX_DIV_LE_0 |- (!a. (!b. (((real_le (Re ((complex_div a) b))) (real_of_num (NUMERAL _0))) = ((real_le (Re ((complex_mul a) (cnj b)))) (real_of_num (NUMERAL _0))))))"}
{"_id": "51409", "text": "AFFINE_AFFINITY |- (!s. (!a. (!c. ((affine s) ==> (affine ((IMAGE (\\x. ((vector_add a) ((% c) x)))) s))))))"}
{"_id": "51410", "text": "ABSOLUTELY_CONTINUOUS_DIFFERENTIABLE_BV |- (!f. (!a. (!b. ((((differentiable_on f) (closed_interval ((CONS (a , b)) NIL))) /\\ ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL)))) ==> ((absolutely_continuous_on f) (closed_interval ((CONS (a , b)) NIL)))))))"}
{"_id": "51411", "text": "SUBSPACE_ORTHOGONAL_TO_VECTORS |- (!s. (subspace (GSPEC (\\GEN%PVAR%163. (?y. (((SETSPEC GEN%PVAR%163) (!x. ((x IN s) ==> ((orthogonal x) y)))) y))))))"}