:: BORSUK_6 semantic presentation

scheme :: BORSUK_6:sch 1
ExFunc3CondD{ F₁() -> non empty set , P₁[ set ], P₂[ set ], P₃[ set ], F₂( set ) -> set , F₃( set ) -> set , F₄( set ) -> set } :

ex f being Function st
( dom f = F₁() & ( for c being Element of F₁() holds
( ( P₁[c] implies f . c = F₂(c) ) & ( P₂[c] implies f . c = F₃(c) ) & ( P₃[c] implies f . c = F₄(c) ) ) ) )

provided

A1: for c being Element of F₁() holds
( ( P₁[c] implies not P₂[c] ) & ( P₁[c] implies not P₃[c] ) & ( P₂[c] implies not P₃[c] ) ) and
A2: for c being Element of F₁() holds
( P₁[c] or P₂[c] or P₃[c] )

proof end;

registration

let n be Nat;
cluster TOP-REAL n -> constituted-FinSeqs ;
coherence

proof end;

end;

theorem Th1: :: BORSUK_6:1

the carrier of [:I[01] ,I[01] :] = [:[.0,1.],[.0,1.]:] by BORSUK_1:83, BORSUK_1:def 5;

Lm1: for x, b, c, d being complex number holds (((d - c) / b) * x) + c = ((1 - (x / b)) * c) + ((x / b) * d)
by XCMPLX_1:236;

theorem :: BORSUK_6:2

canceled;

theorem :: BORSUK_6:3

canceled;

theorem :: BORSUK_6:4

canceled;

theorem Th5: :: BORSUK_6:5

for a, b, x being real number st a <= x & x <= b holds
(x - a) / (b - a) in the carrier of (Closed-Interval-TSpace 0,1)

proof end;

theorem Th6: :: BORSUK_6:6

for x being Point of I[01] st x <= 1 / 2 holds
2 * x is Point of I[01]

proof end;

theorem Th7: :: BORSUK_6:7

for x being Point of I[01] st x >= 1 / 2 holds
(2 * x) - 1 is Point of I[01]

proof end;

theorem Th8: :: BORSUK_6:8

for p, q being Point of I[01] holds p * q is Point of I[01]

proof end;

theorem Th9: :: BORSUK_6:9

for x being Point of I[01] holds (1 / 2) * x is Point of I[01]

proof end;

theorem Th10: :: BORSUK_6:10

for x being Point of I[01] st x >= 1 / 2 holds
x - (1 / 4) is Point of I[01]

proof end;

theorem :: BORSUK_6:11

canceled;

theorem Th12: :: BORSUK_6:12

id I[01] is Path of 0[01] , 1[01]

proof end;

theorem Th13: :: BORSUK_6:13

for a, b, c, d being Point of I[01] st a <= b & c <= d holds
[:[.a,b.],[.c,d.]:] is non empty compact Subset of [:I[01] ,I[01] :]

proof end;

theorem Th14: :: BORSUK_6:14

for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1 )) - 1 } & T = { p where p is Point of (TOP-REAL 2) : p `2 <= p `1 } holds
(AffineMap 1,0,(1 / 2),(1 / 2)) .: S = T

proof end;

theorem Th15: :: BORSUK_6:15

for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 >= (2 * (p `1 )) - 1 } & T = { p where p is Point of (TOP-REAL 2) : p `2 >= p `1 } holds
(AffineMap 1,0,(1 / 2),(1 / 2)) .: S = T

proof end;

theorem Th16: :: BORSUK_6:16

for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 >= 1 - (2 * (p `1 )) } & T = { p where p is Point of (TOP-REAL 2) : p `2 >= - (p `1 ) } holds
(AffineMap 1,0,(1 / 2),(- (1 / 2))) .: S = T

proof end;

theorem Th17: :: BORSUK_6:17

for S, T being Subset of (TOP-REAL 2) st S = { p where p is Point of (TOP-REAL 2) : p `2 <= 1 - (2 * (p `1 )) } & T = { p where p is Point of (TOP-REAL 2) : p `2 <= - (p `1 ) } holds
(AffineMap 1,0,(1 / 2),(- (1 / 2))) .: S = T

proof end;

definition

let T be 1-sorted ;
attr T is real-membered means :Def1: :: BORSUK_6:def 1
the carrier of T is real-membered;

end;

:: deftheorem Def1 defines real-membered BORSUK_6:def 1 :

theorem Th18: :: BORSUK_6:18

for T being non empty 1-sorted holds
( T is real-membered iff for x being Element of T holds x is real )

proof end;

registration

cluster I[01] -> real-membered ;
coherence

proof end;

end;

registration

cluster non empty real-membered 1-sorted ;
existence

proof end;

cluster non empty real-membered TopStruct ;
existence

proof end;

end;

registration

let T be real-membered 1-sorted ;
cluster -> real Element of the carrier of T;
coherence

proof end;

end;

registration

let T be real-membered TopStruct ;
cluster -> real-membered SubSpace of T;
coherence

proof end;

end;

registration

let S, T be non empty real-membered TopSpace;
let p be Element of [:S,T:];
cluster p `1 -> real ;
coherence

proof end;

cluster p `2 -> real ;
coherence

proof end;

end;

registration

let T be non empty SubSpace of [:I[01] ,I[01] :];
let x be Point of T;
cluster x `1 -> real ;
coherence

proof end;

cluster x `2 -> real ;
coherence

proof end;

end;

registration

cluster R^1 -> real-membered ;
coherence

proof end;

end;

theorem Th19: :: BORSUK_6:19

{ p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1 )) - 1 } is closed Subset of (TOP-REAL 2)

proof end;

theorem Th20: :: BORSUK_6:20

{ p where p is Point of (TOP-REAL 2) : p `2 >= (2 * (p `1 )) - 1 } is closed Subset of (TOP-REAL 2)

proof end;

theorem Th21: :: BORSUK_6:21

{ p where p is Point of (TOP-REAL 2) : p `2 <= 1 - (2 * (p `1 )) } is closed Subset of (TOP-REAL 2)

proof end;

theorem Th22: :: BORSUK_6:22

{ p where p is Point of (TOP-REAL 2) : p `2 >= 1 - (2 * (p `1 )) } is closed Subset of (TOP-REAL 2)

proof end;

theorem Th23: :: BORSUK_6:23

{ p where p is Point of (TOP-REAL 2) : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } is closed Subset of (TOP-REAL 2)

proof end;

theorem Th24: :: BORSUK_6:24

ex f being Function of [:R^1 ,R^1 :],(TOP-REAL 2) st
for x, y being Real holds f . [x,y] = <*x,y*>

proof end;

theorem Th25: :: BORSUK_6:25

{ p where p is Point of [:R^1 ,R^1 :] : p `2 <= 1 - (2 * (p `1 )) } is closed Subset of [:R^1 ,R^1 :]

proof end;

theorem Th26: :: BORSUK_6:26

{ p where p is Point of [:R^1 ,R^1 :] : p `2 <= (2 * (p `1 )) - 1 } is closed Subset of [:R^1 ,R^1 :]

proof end;

theorem Th27: :: BORSUK_6:27

{ p where p is Point of [:R^1 ,R^1 :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } is closed Subset of [:R^1 ,R^1 :]

proof end;

theorem Th28: :: BORSUK_6:28

{ p where p is Point of [:I[01] ,I[01] :] : p `2 <= 1 - (2 * (p `1 )) } is non empty closed Subset of [:I[01] ,I[01] :]

proof end;

theorem Th29: :: BORSUK_6:29

{ p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) } is non empty closed Subset of [:I[01] ,I[01] :]

proof end;

theorem Th30: :: BORSUK_6:30

{ p where p is Point of [:I[01] ,I[01] :] : p `2 <= (2 * (p `1 )) - 1 } is non empty closed Subset of [:I[01] ,I[01] :]

proof end;

theorem Th31: :: BORSUK_6:31

for S, T being non empty TopSpace
for p being Point of [:S,T:] holds
( p `1 is Point of S & p `2 is Point of T )

proof end;

theorem Th32: :: BORSUK_6:32

for A, B being Subset of [:I[01] ,I[01] :] st A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] holds
([#] ([:I[01] ,I[01] :] | A)) \/ ([#] ([:I[01] ,I[01] :] | B)) = [#] [:I[01] ,I[01] :]

proof end;

theorem Th33: :: BORSUK_6:33

for A, B being Subset of [:I[01] ,I[01] :] st A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] holds
([#] ([:I[01] ,I[01] :] | A)) /\ ([#] ([:I[01] ,I[01] :] | B)) = [:{(1 / 2)},[.0,1.]:]

proof end;

registration

let T be TopStruct ;
cluster {} T -> compact ;
coherence by COMPTS_1:9;

end;

registration

let T be TopStruct ;
cluster empty compact Element of bool the carrier of T;
existence

proof end;

end;

theorem Th34: :: BORSUK_6:34

for T being TopStruct holds {} is empty compact Subset of T

proof end;

theorem Th35: :: BORSUK_6:35

for T being TopStruct
for a, b being real number st a > b holds
[.a,b.] is empty compact Subset of T

proof end;

theorem :: BORSUK_6:36

for a, b, c, d being Point of I[01] holds [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01] ,I[01] :]

proof end;

definition

let a, b, c, d be real number ;
func L[01] a,b,c,d -> Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace c,d) equals :: BORSUK_6:def 2
(L[01] ((#) c,d),(c,d (#) )) * (P[01] a,b,((#) 0,1),(0,1 (#) ));
correctness ;

end;

:: deftheorem defines L[01] BORSUK_6:def 2 :

theorem Th37: :: BORSUK_6:37

for a, b, c, d being real number st a < b & c < d holds
( (L[01] a,b,c,d) . a = c & (L[01] a,b,c,d) . b = d )

proof end;

theorem Th38: :: BORSUK_6:38

for a, b, c, d being real number st a < b & c <= d holds
L[01] a,b,c,d is continuous Function of (Closed-Interval-TSpace a,b),(Closed-Interval-TSpace c,d)

proof end;

theorem Th39: :: BORSUK_6:39

for a, b, c, d being real number st a < b & c <= d holds
for x being real number st a <= x & x <= b holds
(L[01] a,b,c,d) . x = (((d - c) / (b - a)) * (x - a)) + c

proof end;

theorem Th40: :: BORSUK_6:40

for f1, f2 being Function of [:I[01] ,I[01] :],I[01] st f1 is continuous & f2 is continuous & ( for p being Point of [:I[01] ,I[01] :] holds (f1 . p) * (f2 . p) is Point of I[01] ) holds
ex g being Function of [:I[01] ,I[01] :],I[01] st
( ( for p being Point of [:I[01] ,I[01] :]
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous )

proof end;

theorem Th41: :: BORSUK_6:41

for f1, f2 being Function of [:I[01] ,I[01] :],I[01] st f1 is continuous & f2 is continuous & ( for p being Point of [:I[01] ,I[01] :] holds (f1 . p) + (f2 . p) is Point of I[01] ) holds
ex g being Function of [:I[01] ,I[01] :],I[01] st
( ( for p being Point of [:I[01] ,I[01] :]
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous )

proof end;

theorem :: BORSUK_6:42

for f1, f2 being Function of [:I[01] ,I[01] :],I[01] st f1 is continuous & f2 is continuous & ( for p being Point of [:I[01] ,I[01] :] holds (f1 . p) - (f2 . p) is Point of I[01] ) holds
ex g being Function of [:I[01] ,I[01] :],I[01] st
( ( for p being Point of [:I[01] ,I[01] :]
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous )

proof end;

theorem Th43: :: BORSUK_6:43

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b st P is continuous holds
P * (L[01] (0,1 (#) ),((#) 0,1)) is continuous Function of I[01] ,T

proof end;

theorem Th44: :: BORSUK_6:44

for X being non empty TopStruct
for a, b being Point of X
for P being Path of a,b st P . 0 = a & P . 1 = b holds
( (P * (L[01] (0,1 (#) ),((#) 0,1))) . 0 = b & (P * (L[01] (0,1 (#) ),((#) 0,1))) . 1 = a )

proof end;

theorem Th45: :: BORSUK_6:45

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b st P is continuous & P . 0 = a & P . 1 = b holds
( - P is continuous & (- P) . 0 = b & (- P) . 1 = a )

proof end;

definition

let T be non empty TopSpace;
let a, b be Point of T;
:: original: are_connected
redefine pred a,b are_connected ;
reflexivity

proof end;

symmetry

proof end;

end;

theorem Th46: :: BORSUK_6:46

for T being non empty TopSpace
for a, b, c being Point of T st a,b are_connected & b,c are_connected holds
a,c are_connected

proof end;

theorem :: BORSUK_6:47

canceled;

theorem :: BORSUK_6:48

canceled;

theorem :: BORSUK_6:49

canceled;

theorem Th50: :: BORSUK_6:50

for T being non empty TopSpace
for a, b being Point of T st a,b are_connected holds
for A being Path of a,b holds A = - (- A)

proof end;

theorem :: BORSUK_6:51

for T being non empty arcwise_connected TopSpace
for a, b being Point of T
for A being Path of a,b holds A = - (- A)

proof end;

definition

let T be non empty arcwise_connected TopSpace;
let a, b, c be Point of T;
let P be Path of a,b;
let Q be Path of b,c;
canceled;
redefine func P + Q means :: BORSUK_6:def 4
for t being Point of I[01] holds
( ( t <= 1 / 2 implies it . t = P . (2 * t) ) & ( 1 / 2 <= t implies it . t = Q . ((2 * t) - 1) ) );
compatibility

proof end;

end;

:: deftheorem BORSUK_6:def 3 :

:: deftheorem defines + BORSUK_6:def 4 :

definition

let T be non empty arcwise_connected TopSpace;
let a, b be Point of T;
let P be Path of a,b;
redefine func - P means :Def5: :: BORSUK_6:def 5
for t being Point of I[01] holds it . t = P . (1 - t);
compatibility

proof end;

end;

:: deftheorem Def5 defines - BORSUK_6:def 5 :

definition

let T be non empty TopSpace;
let a, b be Point of T;
let P be Path of a,b;
let f be continuous Function of I[01] ,I[01] ;
assume A1: ( f . 0 = 0 & f . 1 = 1 & a,b are_connected ) ;
func RePar P,f -> Path of a,b equals :Def6: :: BORSUK_6:def 6
P * f;
coherence

proof end;

end;

:: deftheorem Def6 defines RePar BORSUK_6:def 6 :

theorem :: BORSUK_6:52

canceled;

theorem Th53: :: BORSUK_6:53

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b
for f being continuous Function of I[01] ,I[01] st f . 0 = 0 & f . 1 = 1 & a,b are_connected holds
RePar P,f,P are_homotopic

proof end;

theorem :: BORSUK_6:54

for T being non empty arcwise_connected TopSpace
for a, b being Point of T
for P being Path of a,b
for f being continuous Function of I[01] ,I[01] st f . 0 = 0 & f . 1 = 1 holds
RePar P,f,P are_homotopic

proof end;

definition

func 1RP -> Function of I[01] ,I[01] means :Def7: :: BORSUK_6:def 7
for t being Point of I[01] holds
( ( t <= 1 / 2 implies it . t = 2 * t ) & ( t > 1 / 2 implies it . t = 1 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def7 defines 1RP BORSUK_6:def 7 :

registration

cluster 1RP -> continuous ;
coherence

proof end;

end;

theorem Th55: :: BORSUK_6:55

( 1RP . 0 = 0 & 1RP . 1 = 1 )

proof end;

definition

func 2RP -> Function of I[01] ,I[01] means :Def8: :: BORSUK_6:def 8
for t being Point of I[01] holds
( ( t <= 1 / 2 implies it . t = 0 ) & ( t > 1 / 2 implies it . t = (2 * t) - 1 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def8 defines 2RP BORSUK_6:def 8 :

registration

cluster 2RP -> continuous ;
coherence

proof end;

end;

theorem Th56: :: BORSUK_6:56

( 2RP . 0 = 0 & 2RP . 1 = 1 )

proof end;

definition

func 3RP -> Function of I[01] ,I[01] means :Def9: :: BORSUK_6:def 9
for x being Point of I[01] holds
( ( x <= 1 / 2 implies it . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies it . x = x - (1 / 4) ) & ( x > 3 / 4 implies it . x = (2 * x) - 1 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines 3RP BORSUK_6:def 9 :

registration

cluster 3RP -> continuous ;
coherence

proof end;

end;

theorem Th57: :: BORSUK_6:57

( 3RP . 0 = 0 & 3RP . 1 = 1 )

proof end;

theorem Th58: :: BORSUK_6:58

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b
for Q being constant Path of b,b st a,b are_connected holds
RePar P,1RP = P + Q

proof end;

theorem Th59: :: BORSUK_6:59

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b
for Q being constant Path of a,a st a,b are_connected holds
RePar P,2RP = Q + P

proof end;

theorem Th60: :: BORSUK_6:60

for T being non empty TopSpace
for a, b, c, d being Point of T
for P being Path of a,b
for Q being Path of b,c
for R being Path of c,d st a,b are_connected & b,c are_connected & c,d are_connected holds
RePar ((P + Q) + R),3RP = P + (Q + R)

proof end;

definition

func LowerLeftUnitTriangle -> Subset of [:I[01] ,I[01] :] means :Def10: :: BORSUK_6:def 10
for x being set holds
( x in it iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def10 defines LowerLeftUnitTriangle BORSUK_6:def 10 :

notation

synonym IAA for LowerLeftUnitTriangle ;

end;

definition

func UpperUnitTriangle -> Subset of [:I[01] ,I[01] :] means :Def11: :: BORSUK_6:def 11
for x being set holds
( x in it iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines UpperUnitTriangle BORSUK_6:def 11 :

notation

synonym IBB for UpperUnitTriangle ;

end;

definition

func LowerRightUnitTriangle -> Subset of [:I[01] ,I[01] :] means :Def12: :: BORSUK_6:def 12
for x being set holds
( x in it iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def12 defines LowerRightUnitTriangle BORSUK_6:def 12 :

notation

synonym ICC for LowerRightUnitTriangle ;

end;

theorem Th61: :: BORSUK_6:61

IAA = { p where p is Point of [:I[01] ,I[01] :] : p `2 <= 1 - (2 * (p `1 )) }

proof end;

theorem Th62: :: BORSUK_6:62

IBB = { p where p is Point of [:I[01] ,I[01] :] : ( p `2 >= 1 - (2 * (p `1 )) & p `2 >= (2 * (p `1 )) - 1 ) }

proof end;

theorem Th63: :: BORSUK_6:63

ICC = { p where p is Point of [:I[01] ,I[01] :] : p `2 <= (2 * (p `1 )) - 1 }

proof end;

registration

cluster LowerLeftUnitTriangle -> non empty closed ;
coherence by Th28, Th61;
cluster UpperUnitTriangle -> non empty closed ;
coherence by Th29, Th62;
cluster LowerRightUnitTriangle -> non empty closed ;
coherence by Th30, Th63;

end;

theorem Th64: :: BORSUK_6:64

(IAA \/ IBB ) \/ ICC = [:[.0,1.],[.0,1.]:]

proof end;

theorem Th65: :: BORSUK_6:65

IAA /\ IBB = { p where p is Point of [:I[01] ,I[01] :] : p `2 = 1 - (2 * (p `1 )) }

proof end;

theorem Th66: :: BORSUK_6:66

ICC /\ IBB = { p where p is Point of [:I[01] ,I[01] :] : p `2 = (2 * (p `1 )) - 1 }

proof end;

theorem Th67: :: BORSUK_6:67

for x being Point of [:I[01] ,I[01] :] st x in IAA holds
x `1 <= 1 / 2

proof end;

theorem Th68: :: BORSUK_6:68

for x being Point of [:I[01] ,I[01] :] st x in ICC holds
x `1 >= 1 / 2

proof end;

theorem Th69: :: BORSUK_6:69

for x being Point of I[01] holds [0,x] in IAA

proof end;

theorem Th70: :: BORSUK_6:70

for s being set st [0,s] in IBB holds
s = 1

proof end;

theorem Th71: :: BORSUK_6:71

for s being set st [s,1] in ICC holds
s = 1

proof end;

theorem Th72: :: BORSUK_6:72

[0,1] in IBB

proof end;

theorem Th73: :: BORSUK_6:73

for x being Point of I[01] holds [x,1] in IBB

proof end;

theorem Th74: :: BORSUK_6:74

( [(1 / 2),0] in ICC & [1,1] in ICC )

proof end;

theorem Th75: :: BORSUK_6:75

[(1 / 2),0] in IBB

proof end;

theorem Th76: :: BORSUK_6:76

for x being Point of I[01] holds [1,x] in ICC

proof end;

theorem Th77: :: BORSUK_6:77

for x being Point of I[01] st x >= 1 / 2 holds
[x,0] in ICC

proof end;

theorem Th78: :: BORSUK_6:78

for x being Point of I[01] st x <= 1 / 2 holds
[x,0] in IAA

proof end;

theorem Th79: :: BORSUK_6:79

for x being Point of I[01] st x < 1 / 2 holds
( not [x,0] in IBB & not [x,0] in ICC )

proof end;

theorem Th80: :: BORSUK_6:80

IAA /\ ICC = {[(1 / 2),0]}

proof end;

Lm2: for x, y being Point of I[01] holds [x,y] in the carrier of [:I[01] ,I[01] :]
;

theorem Th81: :: BORSUK_6:81

proof end;

theorem :: BORSUK_6:82

for X being non empty arcwise_connected TopSpace
for a1, b1, c1, d1 being Point of X
for P being Path of a1,b1
for Q being Path of b1,c1
for R being Path of c1,d1 holds (P + Q) + R,P + (Q + R) are_homotopic

proof end;

theorem Th83: :: BORSUK_6:83

for T being non empty TopSpace
for a, b, c being Point of T
for P1, P2 being Path of a,b
for Q1, Q2 being Path of b,c st a,b are_connected & b,c are_connected & P1,P2 are_homotopic & Q1,Q2 are_homotopic holds
P1 + Q1,P2 + Q2 are_homotopic

proof end;

theorem :: BORSUK_6:84

for X being non empty arcwise_connected TopSpace
for a1, b1, c1 being Point of X
for P1, P2 being Path of a1,b1
for Q1, Q2 being Path of b1,c1 st P1,P2 are_homotopic & Q1,Q2 are_homotopic holds
P1 + Q1,P2 + Q2 are_homotopic

proof end;

theorem Th85: :: BORSUK_6:85

for T being non empty TopSpace
for a, b being Point of T
for P, Q being Path of a,b st a,b are_connected & P,Q are_homotopic holds
- P, - Q are_homotopic

proof end;

theorem :: BORSUK_6:86

for X being non empty arcwise_connected TopSpace
for a1, b1 being Point of X
for P, Q being Path of a1,b1 st P,Q are_homotopic holds
- P, - Q are_homotopic

proof end;

theorem :: BORSUK_6:87

for T being non empty TopSpace
for a, b being Point of T
for P, Q, R being Path of a,b st P,Q are_homotopic & Q,R are_homotopic holds
P,R are_homotopic

proof end;

theorem Th88: :: BORSUK_6:88

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b
for Q being constant Path of b,b st a,b are_connected holds
P + Q,P are_homotopic

proof end;

theorem :: BORSUK_6:89

for X being non empty arcwise_connected TopSpace
for a1, b1 being Point of X
for P being Path of a1,b1
for Q being constant Path of b1,b1 holds P + Q,P are_homotopic

proof end;

theorem Th90: :: BORSUK_6:90

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b
for Q being constant Path of a,a st a,b are_connected holds
Q + P,P are_homotopic

proof end;

theorem :: BORSUK_6:91

for X being non empty arcwise_connected TopSpace
for a1, b1 being Point of X
for P being Path of a1,b1
for Q being constant Path of a1,a1 holds Q + P,P are_homotopic

proof end;

theorem Th92: :: BORSUK_6:92

for T being non empty TopSpace
for a, b being Point of T
for P being Path of a,b
for Q being constant Path of a,a st a,b are_connected holds
P + (- P),Q are_homotopic

proof end;

theorem :: BORSUK_6:93

for X being non empty arcwise_connected TopSpace
for a1, b1 being Point of X
for P being Path of a1,b1
for Q being constant Path of a1,a1 holds P + (- P),Q are_homotopic

proof end;

theorem Th94: :: BORSUK_6:94

for T being non empty TopSpace
for b, a being Point of T
for P being Path of b,a
for Q being constant Path of a,a st b,a are_connected holds
(- P) + P,Q are_homotopic

proof end;

theorem :: BORSUK_6:95

for X being non empty arcwise_connected TopSpace
for b1, a1 being Point of X
for P being Path of b1,a1
for Q being constant Path of a1,a1 holds (- P) + P,Q are_homotopic

proof end;

theorem :: BORSUK_6:96

for T being non empty TopSpace
for a being Point of T
for P, Q being constant Path of a,a holds P,Q are_homotopic

proof end;

definition

let T be non empty TopSpace;
let a, b be Point of T;
let P, Q be Path of a,b;
assume A1: P,Q are_homotopic ;
mode Homotopy of P,Q -> Function of [:I[01] ,I[01] :],T means :: BORSUK_6:def 13
( it is continuous & ( for s being Point of I[01] holds
( it . s,0 = P . s & it . s,1 = Q . s & ( for t being Point of I[01] holds
( it . 0,t = a & it . 1,t = b ) ) ) ) );
existence by A1, BORSUK_2:def 7;

end;

:: deftheorem defines Homotopy BORSUK_6:def 13 :