:: Algebraic Properties of Homotopies
:: by Adam Grabowski and Artur Korni{\l}owicz
::
:: Received March 18, 2004
:: Copyright (c) 2004-2021 Association of Mizar Users

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;

theorem Th1: :: BORSUK_6:1

the carrier of [:I[01],I[01]:] = [:[.0,1.],[.0,1.]:] by BORSUK_1:40, BORSUK_1:def 2;

theorem Th2: :: BORSUK_6:2

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

proof end;

theorem Th3: :: BORSUK_6:3

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

proof end;

theorem Th4: :: BORSUK_6:4

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

proof end;

theorem Th5: :: BORSUK_6:5

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

proof end;

theorem Th6: :: BORSUK_6:6

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

proof end;

theorem Th8: :: BORSUK_6:8

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

proof end;

theorem Th9: :: BORSUK_6:9

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 Th10: :: BORSUK_6:10

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 Th11: :: BORSUK_6:11

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 Th12: :: BORSUK_6:12

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 Th13: :: BORSUK_6:13

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 :: BORSUK_6:14

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 non empty real-membered for 1-sorted ;

existence
ex b₁ being 1-sorted st
( not b₁ is empty & b₁ is real-membered )

proof end;

cluster non empty TopSpace-like real-membered for TopStruct ;

existence
ex b₁ being TopSpace st
( not b₁ is empty & b₁ is real-membered )

proof end;

end;

registration

let T be real-membered 1-sorted ;

cluster -> real for Element of the carrier of T;

coherence
for b₁ being Element of T holds b₁ is real ;

end;

registration

let T be real-membered TopStruct ;

cluster -> real-membered for SubSpace of T;

coherence
for b₁ being SubSpace of T holds b₁ is real-membered ;

end;

registration

let S, T be non empty real-membered TopSpace;
let p be Element of [:S,T:];

cluster p `1 -> real ;

coherence
p `1 is real

proof end;

cluster p `2 -> real ;

coherence
p `2 is real

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
x `1 is real

proof end;

cluster x `2 -> real ;

coherence
x `2 is real

proof end;

end;

theorem Th15: :: BORSUK_6:15

{ 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 Th16: :: BORSUK_6:16

{ 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 Th17: :: BORSUK_6:17

{ 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 Th18: :: BORSUK_6:18

{ 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 Th19: :: BORSUK_6:19

{ 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 Th20: :: BORSUK_6:20

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 Th21: :: BORSUK_6:21

{ 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 Th22: :: BORSUK_6:22

{ 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 Th23: :: BORSUK_6:23

{ 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 Th24: :: BORSUK_6:24

{ 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 Th25: :: BORSUK_6:25

{ 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 Th26: :: BORSUK_6:26

{ 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 Th27: :: BORSUK_6:27

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 Th28: :: BORSUK_6:28

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 Th29: :: BORSUK_6:29

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 empty compact for Element of bool the carrier of T;

existence
ex b₁ being Subset of T st
( b₁ is empty & b₁ is compact )

proof end;

end;

theorem Th30: :: BORSUK_6:30

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

proof end;

theorem Th31: :: BORSUK_6:31

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

proof end;

theorem :: BORSUK_6:32

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;

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

correctness
coherence
(L[01] (((#) (c,d)),((c,d) (#)))) * (P[01] (a,b,((#) (0,1)),((0,1) (#)))) is Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (c,d));
;

end;

:: deftheorem defines L[01] BORSUK_6:def 1 :
for a, b, c, d being Real holds L[01] (a,b,c,d) = (L[01] (((#) (c,d)),((c,d) (#)))) * (P[01] (a,b,((#) (0,1)),((0,1) (#))));

theorem Th33: :: BORSUK_6:33

for a, b, c, d being Real 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 Th34: :: BORSUK_6:34

for a, b, c, d being Real 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 Th35: :: BORSUK_6:35

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

proof end;

theorem Th36: :: BORSUK_6:36

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 st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous )

proof end;

theorem Th37: :: BORSUK_6:37

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 st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous )

proof end;

theorem :: BORSUK_6:38

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 st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous )

proof end;

theorem Th39: :: BORSUK_6:39

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 Th40: :: BORSUK_6:40

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 Th41: :: BORSUK_6:41

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
for a being Point of T holds R62(T,b₁,b₁)

proof end;

symmetry
for a, b being Point of T st R62(T,b₁,b₂) holds
R62(T,b₂,b₁)

proof end;

end;

theorem Th42: :: BORSUK_6:42

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 Th43: :: BORSUK_6:43

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_6:44

for T being non empty pathwise_connected TopSpace
for a, b being Point of T
for A being Path of a,b holds A = - (- A) by Th43, BORSUK_2:def 3;

definition

let T be non empty pathwise_connected TopSpace;
let a, b, c be Point of T;
let P be Path of a,b;
let Q be Path of b,c;

redefine func P + Q means :: BORSUK_6:def 2
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
for b₁ being Path of a,c holds
( b₁ = P + Q iff for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₁ . t = P . (2 * t) ) & ( 1 / 2 <= t implies b₁ . t = Q . ((2 * t) - 1) ) ) )

proof end;

end;

:: deftheorem defines + BORSUK_6:def 2 :
for T being non empty pathwise_connected TopSpace
for a, b, c being Point of T
for P being Path of a,b
for Q being Path of b,c
for b₇ being Path of a,c holds
( b₇ = P + Q iff for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₇ . t = P . (2 * t) ) & ( 1 / 2 <= t implies b₇ . t = Q . ((2 * t) - 1) ) ) );

definition

let T be non empty pathwise_connected TopSpace;
let a, b be Point of T;
let P be Path of a,b;

redefine func - P means :Def3: :: BORSUK_6:def 3
for t being Point of I[01] holds it . t = P . (1 - t);

compatibility
for b₁ being Path of b,a holds
( b₁ = - P iff for t being Point of I[01] holds b₁ . t = P . (1 - t) )

proof end;

end;

:: deftheorem Def3 defines - BORSUK_6:def 3 :
for T being non empty pathwise_connected TopSpace
for a, b being Point of T
for P being Path of a,b
for b₅ being Path of b,a holds
( b₅ = - P iff for t being Point of I[01] holds b₅ . t = P . (1 - t) );

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 that
A1: f . 0 = 0 and
A2: f . 1 = 1 and
A3: a,b are_connected ;

func RePar (P,f) -> Path of a,b equals :Def4: :: BORSUK_6:def 4
P * f;

coherence
P * f is Path of a,b

proof end;

end;

:: deftheorem Def4 defines RePar BORSUK_6:def 4 :
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 * f;

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
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_6:46

for T being non empty pathwise_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 by Th45, BORSUK_2:def 3;

definition

func 1RP -> Function of I[01],I[01] means :Def5: :: BORSUK_6:def 5
for t being Point of I[01] holds
( ( t <= 1 / 2 implies it . t = 2 * t ) & ( t > 1 / 2 implies it . t = 1 ) );

existence
ex b₁ being Function of I[01],I[01] st
for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₁ . t = 2 * t ) & ( t > 1 / 2 implies b₁ . t = 1 ) )

proof end;

uniqueness
for b₁, b₂ being Function of I[01],I[01] st ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₁ . t = 2 * t ) & ( t > 1 / 2 implies b₁ . t = 1 ) ) ) & ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₂ . t = 2 * t ) & ( t > 1 / 2 implies b₂ . t = 1 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines 1RP BORSUK_6:def 5 :
for b₁ being Function of I[01],I[01] holds
( b₁ = 1RP iff for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₁ . t = 2 * t ) & ( t > 1 / 2 implies b₁ . t = 1 ) ) );

registration

cluster 1RP -> continuous ;

coherence
1RP is continuous

proof end;

end;

theorem Th47: :: BORSUK_6:47

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

proof end;

definition

func 2RP -> Function of I[01],I[01] means :Def6: :: BORSUK_6:def 6
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
ex b₁ being Function of I[01],I[01] st
for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₁ . t = 0 ) & ( t > 1 / 2 implies b₁ . t = (2 * t) - 1 ) )

proof end;

uniqueness
for b₁, b₂ being Function of I[01],I[01] st ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₁ . t = 0 ) & ( t > 1 / 2 implies b₁ . t = (2 * t) - 1 ) ) ) & ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₂ . t = 0 ) & ( t > 1 / 2 implies b₂ . t = (2 * t) - 1 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def6 defines 2RP BORSUK_6:def 6 :
for b₁ being Function of I[01],I[01] holds
( b₁ = 2RP iff for t being Point of I[01] holds
( ( t <= 1 / 2 implies b₁ . t = 0 ) & ( t > 1 / 2 implies b₁ . t = (2 * t) - 1 ) ) );

registration

cluster 2RP -> continuous ;

coherence
2RP is continuous

proof end;

end;

theorem Th48: :: BORSUK_6:48

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

proof end;

definition

func 3RP -> Function of I[01],I[01] means :Def7: :: BORSUK_6:def 7
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
ex b₁ being Function of I[01],I[01] st
for x being Point of I[01] holds
( ( x <= 1 / 2 implies b₁ . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies b₁ . x = x - (1 / 4) ) & ( x > 3 / 4 implies b₁ . x = (2 * x) - 1 ) )

proof end;

uniqueness
for b₁, b₂ being Function of I[01],I[01] st ( for x being Point of I[01] holds
( ( x <= 1 / 2 implies b₁ . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies b₁ . x = x - (1 / 4) ) & ( x > 3 / 4 implies b₁ . x = (2 * x) - 1 ) ) ) & ( for x being Point of I[01] holds
( ( x <= 1 / 2 implies b₂ . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies b₂ . x = x - (1 / 4) ) & ( x > 3 / 4 implies b₂ . x = (2 * x) - 1 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def7 defines 3RP BORSUK_6:def 7 :
for b₁ being Function of I[01],I[01] holds
( b₁ = 3RP iff for x being Point of I[01] holds
( ( x <= 1 / 2 implies b₁ . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies b₁ . x = x - (1 / 4) ) & ( x > 3 / 4 implies b₁ . x = (2 * x) - 1 ) ) );

registration

cluster 3RP -> continuous ;

coherence
3RP is continuous

proof end;

end;

theorem Th49: :: BORSUK_6:49

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

proof end;

theorem Th50: :: BORSUK_6:50

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 Th51: :: BORSUK_6:51

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 Th52: :: BORSUK_6:52

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 :Def8: :: BORSUK_6:def 8
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
ex b₁ being Subset of [:I[01],I[01]:] st
for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) )

proof end;

uniqueness
for b₁, b₂ being Subset of [:I[01],I[01]:] st ( for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) ) & ( for x being set holds
( x in b₂ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def8 defines LowerLeftUnitTriangle BORSUK_6:def 8 :
for b₁ being Subset of [:I[01],I[01]:] holds
( b₁ = LowerLeftUnitTriangle iff for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) );

notation

synonym IAA for LowerLeftUnitTriangle ;

end;

definition

func UpperUnitTriangle -> Subset of [:I[01],I[01]:] means :Def9: :: BORSUK_6:def 9
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
ex b₁ being Subset of [:I[01],I[01]:] st
for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) )

proof end;

uniqueness
for b₁, b₂ being Subset of [:I[01],I[01]:] st ( for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) ) & ( for x being set holds
( x in b₂ iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def9 defines UpperUnitTriangle BORSUK_6:def 9 :
for b₁ being Subset of [:I[01],I[01]:] holds
( b₁ = UpperUnitTriangle iff for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) );

notation

synonym IBB for UpperUnitTriangle ;

end;

definition

func LowerRightUnitTriangle -> 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 <= (2 * a) - 1 ) );

existence
ex b₁ being Subset of [:I[01],I[01]:] st
for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) )

proof end;

uniqueness
for b₁, b₂ being Subset of [:I[01],I[01]:] st ( for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) ) & ( for x being set holds
( x in b₂ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines LowerRightUnitTriangle BORSUK_6:def 10 :
for b₁ being Subset of [:I[01],I[01]:] holds
( b₁ = LowerRightUnitTriangle iff for x being set holds
( x in b₁ iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) );

notation

synonym ICC for LowerRightUnitTriangle ;

end;

theorem Th53: :: BORSUK_6:53

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

proof end;

theorem Th54: :: BORSUK_6:54

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 Th55: :: BORSUK_6:55

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
( IAA is closed & not IAA is empty ) by Th24, Th53;

cluster UpperUnitTriangle -> non empty closed ;

coherence
( IBB is closed & not IBB is empty ) by Th25, Th54;

cluster LowerRightUnitTriangle -> non empty closed ;

coherence
( ICC is closed & not ICC is empty ) by Th26, Th55;

end;

theorem Th56: :: BORSUK_6:56

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

proof end;

theorem Th57: :: BORSUK_6:57

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

proof end;

theorem Th58: :: BORSUK_6:58

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

proof end;

theorem Th59: :: BORSUK_6:59

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

proof end;

theorem Th60: :: BORSUK_6:60

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

proof end;

theorem Th61: :: BORSUK_6:61

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

proof end;

theorem Th62: :: BORSUK_6:62

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

proof end;

theorem Th63: :: BORSUK_6:63

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

proof end;

theorem Th64: :: BORSUK_6:64

[0,1] in IBB

proof end;

theorem Th65: :: BORSUK_6:65

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

proof end;

theorem Th66: :: BORSUK_6:66

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

proof end;

theorem Th67: :: BORSUK_6:67

[(1 / 2),0] in IBB

proof end;

theorem Th68: :: BORSUK_6:68

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

proof end;

theorem Th69: :: BORSUK_6:69

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

proof end;

theorem Th70: :: BORSUK_6:70

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

proof end;

theorem Th71: :: BORSUK_6:71

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 Th72: :: BORSUK_6:72

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

proof end;

Lm1: for x, y being Point of I[01] holds [x,y] in the carrier of [:I[01],I[01]:]
;

theorem Th73: :: BORSUK_6:73

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
(P + Q) + R,P + (Q + R) are_homotopic

proof end;

theorem :: BORSUK_6:74

for X being non empty pathwise_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 Th75: :: BORSUK_6:75

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_6:76

for X being non empty pathwise_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 Th77: :: BORSUK_6:77

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_6:78

for X being non empty pathwise_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 by Th77, BORSUK_2:def 3;

theorem :: BORSUK_6:79

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 Th80: :: BORSUK_6:80

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_6:81

for X being non empty pathwise_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 by Th80, BORSUK_2:def 3;

theorem Th82: :: BORSUK_6:82

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_6:83

for X being non empty pathwise_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 by Th82, BORSUK_2:def 3;

theorem Th84: :: BORSUK_6:84

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_6:85

for X being non empty pathwise_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 by Th84, BORSUK_2:def 3;

theorem Th86: :: BORSUK_6:86

for T being non empty TopSpace
for a, b 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_6:87

for X being non empty pathwise_connected TopSpace
for a1, b1 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 by Th86, BORSUK_2:def 3;

theorem :: BORSUK_6:88

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 11
( it is continuous & ( for t being Point of I[01] holds
( it . (t,0) = P . t & it . (t,1) = Q . t & it . (0,t) = a & it . (1,t) = b ) ) );

existence
ex b₁ being Function of [:I[01],I[01]:],T st
( b₁ is continuous & ( for t being Point of I[01] holds
( b₁ . (t,0) = P . t & b₁ . (t,1) = Q . t & b₁ . (0,t) = a & b₁ . (1,t) = b ) ) ) by A1;

end;

:: deftheorem defines Homotopy BORSUK_6:def 11 :
for T being non empty TopSpace
for a, b being Point of T
for P, Q being Path of a,b st P,Q are_homotopic holds
for b₆ being Function of [:I[01],I[01]:],T holds
( b₆ is Homotopy of P,Q iff ( b₆ is continuous & ( for t being Point of I[01] holds
( b₆ . (t,0) = P . t & b₆ . (t,1) = Q . t & b₆ . (0,t) = a & b₆ . (1,t) = b ) ) ) );