fact stringlengths 6 39.6k | type stringclasses 2
values | library stringlengths 5 8 | imports listlengths 0 52 | filename stringlengths 9 12 | symbolic_name stringlengths 7 40 | docstring stringlengths 6 76 |
|---|---|---|---|---|---|---|
for E being non empty set, f being Function of E, E, x being Element of E holds iter(f,0).x = x | theorem | abian | [
"vocabularies NUMBERS, SETFAM_1, FUNCT_1, SUBSET_1, INT_1, RELAT_1, CARD_1,",
"XXREAL_0, ARYTM_3, ARYTM_1, FUNCT_7, XBOOLE_0, TARSKI, ZFMISC_1,",
"FINSET_1, EQREL_1, FUNCOP_1, MCART_1, ABIAN, XCMPLX_0, NAT_1,",
"notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,",
"MCART_1, DOMAIN_... | abian.miz | abian:Th3 | Abian's Fixed Point Theorem |
sE is covering iff union sE = E | theorem | abian | [
"vocabularies NUMBERS, SETFAM_1, FUNCT_1, SUBSET_1, INT_1, RELAT_1, CARD_1,",
"XXREAL_0, ARYTM_3, ARYTM_1, FUNCT_7, XBOOLE_0, TARSKI, ZFMISC_1,",
"FINSET_1, EQREL_1, FUNCOP_1, MCART_1, ABIAN, XCMPLX_0, NAT_1,",
"notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,",
"MCART_1, DOMAIN_... | abian.miz | abian:Th4 | Abian's Fixed Point Theorem |
for E being non empty set, f being Function of E, E, c being Element of Class =_f, e being Element of c holds f.e in c | theorem | abian | [
"vocabularies NUMBERS, SETFAM_1, FUNCT_1, SUBSET_1, INT_1, RELAT_1, CARD_1,",
"XXREAL_0, ARYTM_3, ARYTM_1, FUNCT_7, XBOOLE_0, TARSKI, ZFMISC_1,",
"FINSET_1, EQREL_1, FUNCOP_1, MCART_1, ABIAN, XCMPLX_0, NAT_1,",
"notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,",
"MCART_1, DOMAIN_... | abian.miz | abian:Th6 | Abian's Fixed Point Theorem |
for E being non empty set, f being Function of E, E, c being Element of Class =_f, e being Element of c, n holds iter(f, n).e in c | theorem | abian | [
"vocabularies NUMBERS, SETFAM_1, FUNCT_1, SUBSET_1, INT_1, RELAT_1, CARD_1,",
"XXREAL_0, ARYTM_3, ARYTM_1, FUNCT_7, XBOOLE_0, TARSKI, ZFMISC_1,",
"FINSET_1, EQREL_1, FUNCOP_1, MCART_1, ABIAN, XCMPLX_0, NAT_1,",
"notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,",
"MCART_1, DOMAIN_... | abian.miz | abian:Th7 | Abian's Fixed Point Theorem |
for n being odd Nat holds 1 <= n | theorem | abian | [
"vocabularies NUMBERS, SETFAM_1, FUNCT_1, SUBSET_1, INT_1, RELAT_1, CARD_1,",
"XXREAL_0, ARYTM_3, ARYTM_1, FUNCT_7, XBOOLE_0, TARSKI, ZFMISC_1,",
"FINSET_1, EQREL_1, FUNCOP_1, MCART_1, ABIAN, XCMPLX_0, NAT_1,",
"notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0,",
"MCART_1, DOMAIN_... | abian.miz | abian:Th12 | Abian's Fixed Point Theorem |
x = 0 iff abs x = 0 by Def1,COMPLEX1:47; | theorem | absvalue | [
"vocabularies NUMBERS, XREAL_0, ORDINAL1, COMPLEX1, CARD_1, XXREAL_0, ARYTM_1,",
"ARYTM_3, RELAT_1, SQUARE_1, REAL_1, ABSVALUE, INT_1, NAT_1",
"notations ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1,",
"COMPLEX1, SQUARE_1, XXREAL_0",
"constructors REAL_1, SQUARE_1, COMPLEX1, INT_1",
"registr... | absvalue.miz | absvalue:Th2 | Some Properties of Functions Modul and Signum |
: SQUARE_1:98 0 <= x*y implies sqrt (x*y) = sqrt abs(x)*sqrt abs(y) | theorem | absvalue | [
"vocabularies NUMBERS, XREAL_0, ORDINAL1, COMPLEX1, CARD_1, XXREAL_0, ARYTM_1,",
"ARYTM_3, RELAT_1, SQUARE_1, REAL_1, ABSVALUE, INT_1, NAT_1",
"notations ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1,",
"COMPLEX1, SQUARE_1, XXREAL_0",
"constructors REAL_1, SQUARE_1, COMPLEX1, INT_1",
"registr... | absvalue.miz | absvalue:thm | Some Properties of Functions Modul and Signum |
: SQUARE_1:100 0 < x/y implies sqrt (x/y) = sqrt abs(x) / sqrt abs(y) | theorem | absvalue | [
"vocabularies NUMBERS, XREAL_0, ORDINAL1, COMPLEX1, CARD_1, XXREAL_0, ARYTM_1,",
"ARYTM_3, RELAT_1, SQUARE_1, REAL_1, ABSVALUE, INT_1, NAT_1",
"notations ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1,",
"COMPLEX1, SQUARE_1, XXREAL_0",
"constructors REAL_1, SQUARE_1, COMPLEX1, INT_1",
"registr... | absvalue.miz | absvalue:thm | Some Properties of Functions Modul and Signum |
sgn x = 0 implies x = 0 | theorem | absvalue | [
"vocabularies NUMBERS, XREAL_0, ORDINAL1, COMPLEX1, CARD_1, XXREAL_0, ARYTM_1,",
"ARYTM_3, RELAT_1, SQUARE_1, REAL_1, ABSVALUE, INT_1, NAT_1",
"notations ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1,",
"COMPLEX1, SQUARE_1, XXREAL_0",
"constructors REAL_1, SQUARE_1, COMPLEX1, INT_1",
"registr... | absvalue.miz | absvalue:Th16 | Some Properties of Functions Modul and Signum |
sgn (x * y) = sgn x * sgn y | theorem | absvalue | [
"vocabularies NUMBERS, XREAL_0, ORDINAL1, COMPLEX1, CARD_1, XXREAL_0, ARYTM_1,",
"ARYTM_3, RELAT_1, SQUARE_1, REAL_1, ABSVALUE, INT_1, NAT_1",
"notations ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1,",
"COMPLEX1, SQUARE_1, XXREAL_0",
"constructors REAL_1, SQUARE_1, COMPLEX1, INT_1",
"registr... | absvalue.miz | absvalue:Th18 | Some Properties of Functions Modul and Signum |
x <> 0 implies sgn x * sgn (1/x) = 1 | theorem | absvalue | [
"vocabularies NUMBERS, XREAL_0, ORDINAL1, COMPLEX1, CARD_1, XXREAL_0, ARYTM_1,",
"ARYTM_3, RELAT_1, SQUARE_1, REAL_1, ABSVALUE, INT_1, NAT_1",
"notations ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1,",
"COMPLEX1, SQUARE_1, XXREAL_0",
"constructors REAL_1, SQUARE_1, COMPLEX1, INT_1",
"registr... | absvalue.miz | absvalue:Th21 | Some Properties of Functions Modul and Signum |
1/(sgn x) = sgn (1/x) | theorem | absvalue | [
"vocabularies NUMBERS, XREAL_0, ORDINAL1, COMPLEX1, CARD_1, XXREAL_0, ARYTM_1,",
"ARYTM_3, RELAT_1, SQUARE_1, REAL_1, ABSVALUE, INT_1, NAT_1",
"notations ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, REAL_1, INT_1, NAT_1,",
"COMPLEX1, SQUARE_1, XXREAL_0",
"constructors REAL_1, SQUARE_1, COMPLEX1, INT_1",
"registr... | absvalue.miz | absvalue:Th22 | Some Properties of Functions Modul and Signum |
for a ex b st a<>b | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th1 | Parallelity and Lines in Affine Spaces |
x,y // y,x & x,y // x,y | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th2 | Parallelity and Lines in Affine Spaces |
x,y // z,z & z,z // x,y | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th3 | Parallelity and Lines in Affine Spaces |
x,y // z,t implies x,y // t,z & y,x // z,t & y,x // t,z & z,t // x,y & z,t // y,x & t,z // x,y & t,z // y,x | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th4 | Parallelity and Lines in Affine Spaces |
a<>b & ( a,b // x,y & a,b // z,t or a,b // x,y & z,t // a,b or x ,y // a,b & z,t // a,b or x,y // a,b & a,b // z,t ) implies x,y // z,t | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th5 | Parallelity and Lines in Affine Spaces |
LIN x,y,z implies LIN x,z,y & LIN y,x,z & LIN y,z,x & LIN z,x,y & LIN z,y,x | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th6 | Parallelity and Lines in Affine Spaces |
LIN x,x,y & LIN x,y,y & LIN x,y,x | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th7 | Parallelity and Lines in Affine Spaces |
x<>y & LIN x,y,z & LIN x,y,t & LIN x,y,u implies LIN z,t,u | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th8 | Parallelity and Lines in Affine Spaces |
x<>y & LIN x,y,z & x,y // z,t implies LIN x,y,t | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th9 | Parallelity and Lines in Affine Spaces |
LIN x,y,z & LIN x,y,t implies x,y // z,t | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th10 | Parallelity and Lines in Affine Spaces |
u<>z & LIN x,y,u & LIN x,y,z & LIN u,z,w implies LIN x,y,w | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th11 | Parallelity and Lines in Affine Spaces |
ex x,y,z st not LIN x,y,z | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th12 | Parallelity and Lines in Affine Spaces |
a in Line(a,b) & b in Line(a,b) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th15 | Parallelity and Lines in Affine Spaces |
c in Line(a,b) & d in Line(a,b) & c <>d implies Line(c,d) c= Line(a,b) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th16 | Parallelity and Lines in Affine Spaces |
c in Line(a,b) & d in Line(a,b) & a<>b implies Line(a,b) c= Line (c,d) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th17 | Parallelity and Lines in Affine Spaces |
A is being_line & C is being_line & a in A & b in A & a in C & b in C implies a=b or A=C | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th18 | Parallelity and Lines in Affine Spaces |
A is being_line implies ex a,b st a in A & b in A & a<>b | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th19 | Parallelity and Lines in Affine Spaces |
A is being_line implies ex b st a<>b & b in A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th20 | Parallelity and Lines in Affine Spaces |
LIN a,b,c iff ex A st A is being_line & a in A & b in A & c in A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th21 | Parallelity and Lines in Affine Spaces |
c in Line(a,b) & a<>b implies (d in Line(a,b) iff a,b // c,d) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th22 | Parallelity and Lines in Affine Spaces |
A is being_line & a in A implies (b in A iff a,b // A) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th23 | Parallelity and Lines in Affine Spaces |
A is being_line & a in A & b in A & a<>b & LIN a,b,x implies x in A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th25 | Parallelity and Lines in Affine Spaces |
(ex a,b st a,b // A) implies A is being_line | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th26 | Parallelity and Lines in Affine Spaces |
c in A & d in A & A is being_line & c <>d implies (a,b // A iff a,b // c,d) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th27 | Parallelity and Lines in Affine Spaces |
a,b // A implies ex c,d st c <>d & c in A & d in A & a,b // c,d proof assume a,b // A; then consider c,d such that A1: c <>d and A2: A=Line(c,d) and A3: a,b // c,d by Def4; A4: d in A by A2,Th15; c in A by A2,Th15; hence thesis by A1,A3,A4; end; | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th28 | Parallelity and Lines in Affine Spaces |
a<>b implies a,b // Line(a,b) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th29 | Parallelity and Lines in Affine Spaces |
for A be being_line Subset of AS holds (a,b // A iff ex c,d st c <>d & c in A & d in A & a,b // c,d ) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th30 | Parallelity and Lines in Affine Spaces |
a,b // A & a,b // p,q & a<>b implies p,q // A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th32 | Parallelity and Lines in Affine Spaces |
for A be being_line Subset of AS holds a,a // A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th33 | Parallelity and Lines in Affine Spaces |
a,b // A implies b,a // A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th34 | Parallelity and Lines in Affine Spaces |
A // C implies A is being_line & C is being_line | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th36 | Parallelity and Lines in Affine Spaces |
A // C iff ex a,b,c,d st a<>b & c <>d & a,b // c,d & A=Line(a,b) & C=Line(c,d) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th37 | Parallelity and Lines in Affine Spaces |
for A, C be being_line Subset of AS st a in A & b in A & c in C & d in C & a<>b & c<>d holds (A // C iff a,b // c,d) | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th38 | Parallelity and Lines in Affine Spaces |
a in A & b in A & c in C & d in C & A // C implies a,b // c,d | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th39 | Parallelity and Lines in Affine Spaces |
for A being being_line Subset of AS holds A // A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th41 | Parallelity and Lines in Affine Spaces |
A // C implies C // A | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th42 | Parallelity and Lines in Affine Spaces |
a,b // A & A // C implies a,b // C | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th43 | Parallelity and Lines in Affine Spaces |
A // C & p in A & p in C implies A=C | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th45 | Parallelity and Lines in Affine Spaces |
not LIN o,a,b & LIN o,a,a9 & LIN o,b,b9 & a9=b9 implies a9=o & b9=o | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th54 | Parallelity and Lines in Affine Spaces |
not LIN o,a,b & LIN o,b,b9 & a,b // a9,b9 & a9=o implies b9=o | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th55 | Parallelity and Lines in Affine Spaces |
A is being_line & C is being_line & not A // C implies ex x st x in A & x in C | theorem | aff_1 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, INCSP_1, STRUCT_0, TARSKI, AFF_1",
"notations TARSKI, STRUCT_0, ANALOAF, DIRAF",
"constructors DIRAF",
"registrations STRUCT_0",
"requirements SUBSET, BOOLE",
"theorems DIRAF, TARSKI, XBOOLE_0",
"schemes SUBSET_1"
] | aff_1.miz | aff_1:Th58 | Parallelity and Lines in Affine Spaces |
AP is Moufangian implies AP is satisfying_TDES_1 | theorem | aff_2 | [
"vocabularies DIRAF, SUBSET_1, AFF_1, ANALOAF, INCSP_1, AFF_2",
"notations STRUCT_0, ANALOAF, DIRAF, AFF_1",
"constructors AFF_1",
"registrations STRUCT_0",
"theorems AFF_1"
] | aff_2.miz | aff_2:Th3 | Classical Configurations in Affine Planes |
AP is translational iff AP is satisfying_des_1 | theorem | aff_2 | [
"vocabularies DIRAF, SUBSET_1, AFF_1, ANALOAF, INCSP_1, AFF_2",
"notations STRUCT_0, ANALOAF, DIRAF, AFF_1",
"constructors AFF_1",
"registrations STRUCT_0",
"theorems AFF_1"
] | aff_2.miz | aff_2:Th7 | Classical Configurations in Affine Planes |
AP is Pappian implies AP is satisfying_pap | theorem | aff_2 | [
"vocabularies DIRAF, SUBSET_1, AFF_1, ANALOAF, INCSP_1, AFF_2",
"notations STRUCT_0, ANALOAF, DIRAF, AFF_1",
"constructors AFF_1",
"registrations STRUCT_0",
"theorems AFF_1"
] | aff_2.miz | aff_2:Th9 | Classical Configurations in Affine Planes |
AP is satisfying_PPAP iff AP is Pappian & AP is satisfying_pap | theorem | aff_2 | [
"vocabularies DIRAF, SUBSET_1, AFF_1, ANALOAF, INCSP_1, AFF_2",
"notations STRUCT_0, ANALOAF, DIRAF, AFF_1",
"constructors AFF_1",
"registrations STRUCT_0",
"theorems AFF_1"
] | aff_2.miz | aff_2:Th10 | Classical Configurations in Affine Planes |
AP is satisfying_TDES_1 implies AP is satisfying_des_1 | theorem | aff_2 | [
"vocabularies DIRAF, SUBSET_1, AFF_1, ANALOAF, INCSP_1, AFF_2",
"notations STRUCT_0, ANALOAF, DIRAF, AFF_1",
"constructors AFF_1",
"registrations STRUCT_0",
"theorems AFF_1"
] | aff_2.miz | aff_2:Th13 | Classical Configurations in Affine Planes |
(LIN p,a,a9 or LIN p,a9,a) & p<>a implies ex b9 st LIN p,b,b9 & a ,b // a9,b9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th1 | Planes in Affine Spaces |
(a,b // A or b,a // A) & a in A implies b in A | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th2 | Planes in Affine Spaces |
(a,b // A or b,a // A) & A // K implies a,b // K & b,a // K | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th3 | Planes in Affine Spaces |
(a,b // A or b,a // A) & (a,b // c,d or c,d // a,b) & a<>b implies c,d // A & d,c // A | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th4 | Planes in Affine Spaces |
(A // C or C // A) & a<>b & (a,b // c,d or c,d // a,b) & a in A & b in A & c in C implies d in C | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th7 | Planes in Affine Spaces |
q in M & q in N & a in M & b in N & b9 in N & q<>a & q<>b & M<>N & (a,b // a9,b9 or b,a // b9,a9) & M is being_line & N is being_line & q=a9 implies q=b9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th8 | Planes in Affine Spaces |
q in M & q in N & a in M & a9 in M & b in N & b9 in N & q<>a & q <>b & M<>N & (a,b // a9,b9 or b,a // b9,a9) & M is being_line & N is being_line & a=a9 implies b=b9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th9 | Planes in Affine Spaces |
(M // N or N // M) & a in M & b in N & b9 in N & M<>N & (a,b // a9,b9 or b,a // b9,a9) & a=a9 implies b=b9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th10 | Planes in Affine Spaces |
ex A st a in A & b in A & A is being_line | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th11 | Planes in Affine Spaces |
A is being_line implies ex q st not q in A | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th12 | Planes in Affine Spaces |
K is being_line implies P c= Plane(K,P) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th14 | Planes in Affine Spaces |
K // M implies Plane(K,P) = Plane(M,P) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th16 | Planes in Affine Spaces |
a in M & b in M & a9 in N & b9 in N & a in P & a9 in P & b in Q & b9 in Q & M<>N & M // N & P is being_line & Q is being_line implies (P // Q or ex q st q in P & q in Q) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th18 | Planes in Affine Spaces |
X is being_plane & a in X & b in X & a<>b implies Line(a,b) c= X | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th19 | Planes in Affine Spaces |
K is being_line & P is being_line & Q is being_line & not K // Q & Q c= Plane(K,P) implies Plane(K,Q) = Plane(K,P) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th20 | Planes in Affine Spaces |
K is being_line & P is being_line & Q is being_line & Q c= Plane (K,P) implies P // Q or ex q st q in P & q in Q | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th21 | Planes in Affine Spaces |
X is being_plane & M is being_line & N is being_line & M c= X & N c= X implies M // N or ex q st q in M & q in N | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th22 | Planes in Affine Spaces |
X is being_plane & a in X & M c= X & a in N & (M // N or N // M) implies N c= X | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th23 | Planes in Affine Spaces |
X is being_plane & Y is being_plane & a in X & b in X & a in Y & b in Y & X<>Y & a<>b implies X /\ Y is being_line | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th24 | Planes in Affine Spaces |
X is being_plane & Y is being_plane & a in X & b in X & c in X & a in Y & b in Y & c in Y & not LIN a,b,c implies X = Y | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th25 | Planes in Affine Spaces |
X is being_plane & Y is being_plane & M is being_line & N is being_line & M c= X & N c= X & M c= Y & N c= Y & M<>N implies X = Y | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th26 | Planes in Affine Spaces |
A is being_line implies a*A is being_line | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th27 | Planes in Affine Spaces |
X is being_plane & M is being_line & a in X & M c= X implies a*M c= X | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th28 | Planes in Affine Spaces |
X is being_plane & a in X & b in X & c in X & a,b // c,d & a<>b implies d in X | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th29 | Planes in Affine Spaces |
A is being_line implies a*A = a*(q*A) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th31 | Planes in Affine Spaces |
K // M implies a*K=a*M | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th32 | Planes in Affine Spaces |
X c= Y & ( X is being_line & Y is being_line or X is being_plane & Y is being_plane ) implies X=Y | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th33 | Planes in Affine Spaces |
X is being_plane implies ex a,b,c st a in X & b in X & c in X & not LIN a,b,c | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th34 | Planes in Affine Spaces |
for a,A st A is being_line ex X st a in X & A c= X & X is being_plane | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th36 | Planes in Affine Spaces |
ex X st a in X & b in X & c in X & X is being_plane | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th37 | Planes in Affine Spaces |
q in M & q in N & M is being_line & N is being_line implies ex X st M c= X & N c= X & X is being_plane | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th38 | Planes in Affine Spaces |
M // N implies ex X st M c= X & N c= X & X is being_plane | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th39 | Planes in Affine Spaces |
M is being_line & X is being_plane implies (M '||' X iff ex N st N c= X & (M // N or N // M) ) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th41 | Planes in Affine Spaces |
K,M,N is_coplanar implies K,N,M is_coplanar & M,K,N is_coplanar & M,N,K is_coplanar & N,K,M is_coplanar & N,M,K is_coplanar | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th44 | Planes in Affine Spaces |
K is being_line & M is being_line & X is being_plane & K c= X & M c= X & K<>M implies (K,M,A is_coplanar iff A c= X) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th46 | Planes in Affine Spaces |
q in K & q in M & K is being_line & M is being_line implies K,M, M is_coplanar & M,K,M is_coplanar & M,M,K is_coplanar | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th47 | Planes in Affine Spaces |
AS is not AffinPlane & X is being_plane implies ex q st not q in X | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th48 | Planes in Affine Spaces |
AS is not AffinPlane & q in A & q in P & q in C & q<>a & q<>b & q<>c & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A<>P & A<>C & a,b // a9,b9 & a,c // a9,c9 implies b,c // b9,c9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th49 | Planes in Affine Spaces |
AS is not AffinPlane & A // P & A // C & a in A & a9 in A & b in P & b9 in P & c in C & c9 in C & A is being_line & P is being_line & C is being_line & A<>P & A<>C & a,b // a9,b9 & a,c // a9,c9 implies b,c // b9,c9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th51 | Planes in Affine Spaces |
AS is AffinPlane & not LIN a,b,c implies ex c9 st a,c // a9,c9 & b,c // b9,c9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th53 | Planes in Affine Spaces |
not LIN a,b,c & a9<>b9 & a,b // a9,b9 implies ex c9 st a,c // a9 ,c9 & b,c // b9,c9 | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th54 | Planes in Affine Spaces |
X is being_plane & Y is being_plane implies (X '||' Y iff ex A,P ,M,N st not A // P & A c= X & P c= X & M c= Y & N c= Y & (A // M or M // A) & ( P // N or N // P) ) | theorem | aff_4 | [
"vocabularies DIRAF, SUBSET_1, ANALOAF, AFF_1, INCSP_1, STRUCT_0, XBOOLE_0,",
"TARSKI, RELAT_1, PARSP_1, AFF_2, ARYTM_3, AFF_4",
"notations TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, ANALOAF, DIRAF, AFF_1, AFF_2",
"constructors AFF_1, AFF_2",
"registrations XBOOLE_0, STRUCT_0",
"requirements SUBSET, BOOLE",
... | aff_4.miz | aff_4:Th55 | Planes in Affine Spaces |
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