:: BORSUK_6 semantic presentation
scheme :: BORSUK_6:sch 1
s1{ F
1()
-> non
empty set , P
1[
set ], P
2[
set ], P
3[
set ], F
2(
set )
-> set , F
3(
set )
-> set , F
4(
set )
-> set } :
ex b
1 being
Function st
(
dom b
1 = F
1() & ( for b
2 being
Element of F
1() holds
( ( P
1[b
2] implies b
1 . b
2 = F
2(b
2) ) & ( P
2[b
2] implies b
1 . b
2 = F
3(b
2) ) & ( P
3[b
2] implies b
1 . b
2 = F
4(b
2) ) ) ) )
provided
E1:
for b
1 being
Element of F
1() holds
( not ( P
1[b
1] & P
2[b
1] ) & not ( P
1[b
1] & P
3[b
1] ) & not ( P
2[b
1] & P
3[b
1] ) )
and
E2:
for b
1 being
Element of F
1() holds
not ( not P
1[b
1] & not P
2[b
1] & not P
3[b
1] )
theorem Th1: :: BORSUK_6:1
Lemma2:
for b1, b2, b3, b4 being complex number holds (((b4 - b3) / b2) * b1) + b3 = ((1 - (b1 / b2)) * b3) + ((b1 / b2) * b4)
by XCMPLX_1:236;
theorem Th2: :: BORSUK_6:2
canceled;
theorem Th3: :: BORSUK_6:3
canceled;
theorem Th4: :: BORSUK_6:4
canceled;
theorem Th5: :: BORSUK_6:5
theorem Th6: :: BORSUK_6:6
theorem Th7: :: BORSUK_6:7
theorem Th8: :: BORSUK_6:8
theorem Th9: :: BORSUK_6:9
theorem Th10: :: BORSUK_6:10
theorem Th11: :: BORSUK_6:11
canceled;
theorem Th12: :: BORSUK_6:12
theorem Th13: :: BORSUK_6:13
theorem Th14: :: BORSUK_6:14
theorem Th15: :: BORSUK_6:15
theorem Th16: :: BORSUK_6:16
theorem Th17: :: BORSUK_6:17
:: deftheorem Def1 defines real-membered BORSUK_6:def 1 :
theorem Th18: :: BORSUK_6:18
theorem Th19: :: BORSUK_6:19
theorem Th20: :: BORSUK_6:20
theorem Th21: :: BORSUK_6:21
theorem Th22: :: BORSUK_6:22
theorem Th23: :: BORSUK_6:23
theorem Th24: :: BORSUK_6:24
theorem Th25: :: BORSUK_6:25
theorem Th26: :: BORSUK_6:26
theorem Th27: :: BORSUK_6:27
theorem Th28: :: BORSUK_6:28
theorem Th29: :: BORSUK_6:29
theorem Th30: :: BORSUK_6:30
theorem Th31: :: BORSUK_6:31
theorem Th32: :: BORSUK_6:32
for b
1, b
2 being
Subset of
[:I[01] ,I[01] :] holds
( b
1 = [:[.0,(1 / 2).],[.0,1.]:] & b
2 = [:[.(1 / 2),1.],[.0,1.]:] implies
([#] ([:I[01] ,I[01] :] | b1)) \/ ([#] ([:I[01] ,I[01] :] | b2)) = [#] [:I[01] ,I[01] :] )
theorem Th33: :: BORSUK_6:33
for b
1, b
2 being
Subset of
[:I[01] ,I[01] :] holds
( b
1 = [:[.0,(1 / 2).],[.0,1.]:] & b
2 = [:[.(1 / 2),1.],[.0,1.]:] implies
([#] ([:I[01] ,I[01] :] | b1)) /\ ([#] ([:I[01] ,I[01] :] | b2)) = [:{(1 / 2)},[.0,1.]:] )
theorem Th34: :: BORSUK_6:34
theorem Th35: :: BORSUK_6:35
theorem Th36: :: BORSUK_6:36
definition
let c
1, c
2, c
3, c
4 be
real number ;
func L[01] c
1,c
2,c
3,c
4 -> Function of
(Closed-Interval-TSpace a1,a2),
(Closed-Interval-TSpace a3,a4) equals :: BORSUK_6:def 2
(L[01] ((#) a3,a4),(a3,a4 (#) )) * (P[01] a1,a2,((#) 0,1),(0,1 (#) ));
correctness
coherence
(L[01] ((#) c3,c4),(c3,c4 (#) )) * (P[01] c1,c2,((#) 0,1),(0,1 (#) )) is Function of (Closed-Interval-TSpace c1,c2),(Closed-Interval-TSpace c3,c4);
;
end;
:: deftheorem Def2 defines L[01] BORSUK_6:def 2 :
for b
1, b
2, b
3, b
4 being
real number holds
L[01] b
1,b
2,b
3,b
4 = (L[01] ((#) b3,b4),(b3,b4 (#) )) * (P[01] b1,b2,((#) 0,1),(0,1 (#) ));
theorem Th37: :: BORSUK_6:37
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < b
2 & b
3 < b
4 implies (
(L[01] b1,b2,b3,b4) . b
1 = b
3 &
(L[01] b1,b2,b3,b4) . b
2 = b
4 ) )
theorem Th38: :: BORSUK_6:38
theorem Th39: :: BORSUK_6:39
for b
1, b
2, b
3, b
4 being
real number holds
( b
1 < b
2 & b
3 <= b
4 implies for b
5 being
real number holds
( b
1 <= b
5 & b
5 <= b
2 implies
(L[01] b1,b2,b3,b4) . b
5 = (((b4 - b3) / (b2 - b1)) * (b5 - b1)) + b
3 ) )
theorem Th40: :: BORSUK_6:40
theorem Th41: :: BORSUK_6:41
theorem Th42: :: BORSUK_6:42
Lemma39:
Closed-Interval-TSpace 0,1 = I[01]
by TOPMETR:27;
theorem Th43: :: BORSUK_6:43
theorem Th44: :: BORSUK_6:44
theorem Th45: :: BORSUK_6:45
theorem Th46: :: BORSUK_6:46
theorem Th47: :: BORSUK_6:47
canceled;
theorem Th48: :: BORSUK_6:48
canceled;
theorem Th49: :: BORSUK_6:49
canceled;
theorem Th50: :: BORSUK_6:50
theorem Th51: :: BORSUK_6:51
:: deftheorem Def3 BORSUK_6:def 3 :
canceled;
:: deftheorem Def4 defines + BORSUK_6:def 4 :
:: deftheorem Def5 defines - BORSUK_6:def 5 :
:: deftheorem Def6 defines RePar BORSUK_6:def 6 :
theorem Th52: :: BORSUK_6:52
canceled;
theorem Th53: :: BORSUK_6:53
theorem Th54: :: BORSUK_6:54
:: deftheorem Def7 defines 1RP BORSUK_6:def 7 :
theorem Th55: :: BORSUK_6:55
:: deftheorem Def8 defines 2RP BORSUK_6:def 8 :
theorem Th56: :: BORSUK_6:56
:: deftheorem Def9 defines 3RP BORSUK_6:def 9 :
theorem Th57: :: BORSUK_6:57
theorem Th58: :: BORSUK_6:58
theorem Th59: :: BORSUK_6:59
theorem Th60: :: BORSUK_6:60
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5 being
Point of b
1for b
6 being
Path of b
2,b
3for b
7 being
Path of b
3,b
4for b
8 being
Path of b
4,b
5 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
4,b
5 are_connected implies
RePar ((b6 + b7) + b8),
3RP = b
6 + (b7 + b8) )
definition
func LowerLeftUnitTriangle -> Subset of
[:I[01] ,I[01] :] means :
Def10:
:: BORSUK_6:def 10
for b
1 being
set holds
( b
1 in a
1 iff ex b
2, b
3 being
Point of
I[01] st
( b
1 = [b2,b3] & b
3 <= 1
- (2 * b2) ) );
existence
ex b1 being Subset of [:I[01] ,I[01] :] st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being Point of I[01] st
( b2 = [b3,b4] & b4 <= 1 - (2 * b3) ) )
uniqueness
for b1, b2 being Subset of [:I[01] ,I[01] :] holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being Point of I[01] st
( b3 = [b4,b5] & b5 <= 1 - (2 * b4) ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being Point of I[01] st
( b3 = [b4,b5] & b5 <= 1 - (2 * b4) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def10 defines LowerLeftUnitTriangle BORSUK_6:def 10 :
definition
func UpperUnitTriangle -> Subset of
[:I[01] ,I[01] :] means :
Def11:
:: BORSUK_6:def 11
for b
1 being
set holds
( b
1 in a
1 iff ex b
2, b
3 being
Point of
I[01] st
( b
1 = [b2,b3] & b
3 >= 1
- (2 * b2) & b
3 >= (2 * b2) - 1 ) );
existence
ex b1 being Subset of [:I[01] ,I[01] :] st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being Point of I[01] st
( b2 = [b3,b4] & b4 >= 1 - (2 * b3) & b4 >= (2 * b3) - 1 ) )
uniqueness
for b1, b2 being Subset of [:I[01] ,I[01] :] holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being Point of I[01] st
( b3 = [b4,b5] & b5 >= 1 - (2 * b4) & b5 >= (2 * b4) - 1 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being Point of I[01] st
( b3 = [b4,b5] & b5 >= 1 - (2 * b4) & b5 >= (2 * b4) - 1 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def11 defines UpperUnitTriangle BORSUK_6:def 11 :
definition
func LowerRightUnitTriangle -> Subset of
[:I[01] ,I[01] :] means :
Def12:
:: BORSUK_6:def 12
for b
1 being
set holds
( b
1 in a
1 iff ex b
2, b
3 being
Point of
I[01] st
( b
1 = [b2,b3] & b
3 <= (2 * b2) - 1 ) );
existence
ex b1 being Subset of [:I[01] ,I[01] :] st
for b2 being set holds
( b2 in b1 iff ex b3, b4 being Point of I[01] st
( b2 = [b3,b4] & b4 <= (2 * b3) - 1 ) )
uniqueness
for b1, b2 being Subset of [:I[01] ,I[01] :] holds
( ( for b3 being set holds
( b3 in b1 iff ex b4, b5 being Point of I[01] st
( b3 = [b4,b5] & b5 <= (2 * b4) - 1 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ex b4, b5 being Point of I[01] st
( b3 = [b4,b5] & b5 <= (2 * b4) - 1 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines LowerRightUnitTriangle BORSUK_6:def 12 :
theorem Th61: :: BORSUK_6:61
theorem Th62: :: BORSUK_6:62
theorem Th63: :: BORSUK_6:63
theorem Th64: :: BORSUK_6:64
theorem Th65: :: BORSUK_6:65
theorem Th66: :: BORSUK_6:66
theorem Th67: :: BORSUK_6:67
theorem Th68: :: BORSUK_6:68
theorem Th69: :: BORSUK_6:69
theorem Th70: :: BORSUK_6:70
for b
1 being
set holds
(
[0,b1] in IBB implies b
1 = 1 )
theorem Th71: :: BORSUK_6:71
for b
1 being
set holds
(
[b1,1] in ICC implies b
1 = 1 )
theorem Th72: :: BORSUK_6:72
theorem Th73: :: BORSUK_6:73
theorem Th74: :: BORSUK_6:74
theorem Th75: :: BORSUK_6:75
theorem Th76: :: BORSUK_6:76
theorem Th77: :: BORSUK_6:77
theorem Th78: :: BORSUK_6:78
theorem Th79: :: BORSUK_6:79
theorem Th80: :: BORSUK_6:80
Lemma80:
for b1, b2 being Point of I[01] holds [b1,b2] in the carrier of [:I[01] ,I[01] :]
;
theorem Th81: :: BORSUK_6:81
for b
1 being non
empty TopSpacefor b
2, b
3, b
4, b
5 being
Point of b
1for b
6 being
Path of b
2,b
3for b
7 being
Path of b
3,b
4for b
8 being
Path of b
4,b
5 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
4,b
5 are_connected implies
(b6 + b7) + b
8,b
6 + (b7 + b8) are_homotopic )
theorem Th82: :: BORSUK_6:82
theorem Th83: :: BORSUK_6:83
for b
1 being non
empty TopSpacefor b
2, b
3, b
4 being
Point of b
1for b
5, b
6 being
Path of b
2,b
3for b
7, b
8 being
Path of b
3,b
4 holds
( b
2,b
3 are_connected & b
3,b
4 are_connected & b
5,b
6 are_homotopic & b
7,b
8 are_homotopic implies b
5 + b
7,b
6 + b
8 are_homotopic )
theorem Th84: :: BORSUK_6:84
theorem Th85: :: BORSUK_6:85
theorem Th86: :: BORSUK_6:86
theorem Th87: :: BORSUK_6:87
theorem Th88: :: BORSUK_6:88
theorem Th89: :: BORSUK_6:89
theorem Th90: :: BORSUK_6:90
theorem Th91: :: BORSUK_6:91
theorem Th92: :: BORSUK_6:92
theorem Th93: :: BORSUK_6:93
theorem Th94: :: BORSUK_6:94
theorem Th95: :: BORSUK_6:95
theorem Th96: :: BORSUK_6:96
definition
let c
1 be non
empty TopSpace;
let c
2, c
3 be
Point of c
1;
let c
4, c
5 be
Path of c
2,c
3;
assume E88:
c
4,c
5 are_homotopic
;
mode Homotopy of c
4,c
5 -> Function of
[:I[01] ,I[01] :],a
1 means :: BORSUK_6:def 13
( a
6 is
continuous & ( for b
1 being
Point of
I[01] holds
( a
6 . b
1,0
= a
4 . b
1 & a
6 . b
1,1
= a
5 . b
1 & ( for b
2 being
Point of
I[01] holds
( a
6 . 0,b
2 = a
2 & a
6 . 1,b
2 = a
3 ) ) ) ) );
existence
ex b1 being Function of [:I[01] ,I[01] :],c1 st
( b1 is continuous & ( for b2 being Point of I[01] holds
( b1 . b2,0 = c4 . b2 & b1 . b2,1 = c5 . b2 & ( for b3 being Point of I[01] holds
( b1 . 0,b3 = c2 & b1 . 1,b3 = c3 ) ) ) ) )
by E88, BORSUK_2:def 7;
end;
:: deftheorem Def13 defines Homotopy BORSUK_6:def 13 :
for b
1 being non
empty TopSpacefor b
2, b
3 being
Point of b
1for b
4, b
5 being
Path of b
2,b
3 holds
( b
4,b
5 are_homotopic implies for b
6 being
Function of
[:I[01] ,I[01] :],b
1 holds
( b
6 is
Homotopy of b
4,b
5 iff ( b
6 is
continuous & ( for b
7 being
Point of
I[01] holds
( b
6 . b
7,0
= b
4 . b
7 & b
6 . b
7,1
= b
5 . b
7 & ( for b
8 being
Point of
I[01] holds
( b
6 . 0,b
8 = b
2 & b
6 . 1,b
8 = b
3 ) ) ) ) ) ) );