:: BORSUK_6 semantic presentation
begin
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
A3: for c being set holds
( not c in F1() or P1[c] or P2[c] or P3[c] ) by A2;
A4: for c being set st c in F1() holds
( ( P1[c] implies not P2[c] ) & ( P1[c] implies not P3[c] ) & ( P2[c] implies not P3[c] ) ) by A1;
ex f being Function st
( dom f = F1() & ( for c being set st c in F1() holds
( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) ) ) ) from RECDEF_2:sch_1(A4, A3);
then consider f being Function such that
A5: dom f = F1() and
A6: for c being set st c in F1() holds
( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) ) ;
take f ; ::_thesis: ( 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) ) ) ) )
thus dom f = F1() by A5; ::_thesis: 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) ) )
let c be Element of F1(); ::_thesis: ( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) )
thus ( ( P1[c] implies f . c = F2(c) ) & ( P2[c] implies f . c = F3(c) ) & ( P3[c] implies f . c = F4(c) ) ) by A6; ::_thesis: verum
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 number st a <= x & x <= b holds
(x - a) / (b - a) in the carrier of (Closed-Interval-TSpace (0,1))
proof
let a, b, x be real number ; ::_thesis: ( a <= x & x <= b implies (x - a) / (b - a) in the carrier of (Closed-Interval-TSpace (0,1)) )
assume that
A1: a <= x and
A2: x <= b ; ::_thesis: (x - a) / (b - a) in the carrier of (Closed-Interval-TSpace (0,1))
A3: a <= b by A1, A2, XXREAL_0:2;
A4: x - a <= b - a by A2, XREAL_1:9;
A5: (x - a) / (b - a) <= 1
proof
percases ( b - a = 0 or b - a > 0 ) by A3, XREAL_1:48;
suppose b - a = 0 ; ::_thesis: (x - a) / (b - a) <= 1
hence (x - a) / (b - a) <= 1 by XCMPLX_1:49; ::_thesis: verum
end;
suppose b - a > 0 ; ::_thesis: (x - a) / (b - a) <= 1
hence (x - a) / (b - a) <= 1 by A4, XREAL_1:185; ::_thesis: verum
end;
end;
end;
A6: x - a >= 0 by A1, XREAL_1:48;
b - a >= 0 by A3, XREAL_1:48;
then (x - a) / (b - a) in [.0,1.] by A5, A6, XXREAL_1:1;
hence (x - a) / (b - a) in the carrier of (Closed-Interval-TSpace (0,1)) by TOPMETR:18; ::_thesis: verum
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
let x be Point of I[01]; ::_thesis: ( x <= 1 / 2 implies 2 * x is Point of I[01] )
assume x <= 1 / 2 ; ::_thesis: 2 * x is Point of I[01]
then A1: 2 * x <= 2 * (1 / 2) by XREAL_1:64;
0 <= x by BORSUK_1:43;
hence 2 * x is Point of I[01] by A1, BORSUK_1:43; ::_thesis: verum
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
let x be Point of I[01]; ::_thesis: ( x >= 1 / 2 implies (2 * x) - 1 is Point of I[01] )
x <= 1 by BORSUK_1:43;
then 2 * x <= 2 * 1 by XREAL_1:64;
then A1: (2 * x) - 1 <= 2 - 1 by XREAL_1:9;
assume x >= 1 / 2 ; ::_thesis: (2 * x) - 1 is Point of I[01]
then 2 * x >= 2 * (1 / 2) by XREAL_1:64;
then (2 * x) - 1 >= 1 - 1 by XREAL_1:9;
hence (2 * x) - 1 is Point of I[01] by A1, BORSUK_1:43; ::_thesis: verum
end;
theorem Th5: :: BORSUK_6:5
for p, q being Point of I[01] holds p * q is Point of I[01]
proof
let p, q be Point of I[01]; ::_thesis: p * q is Point of I[01]
A1: 0 <= p by BORSUK_1:43;
( p <= 1 & q <= 1 ) by BORSUK_1:43;
then ( 0 <= q & p * q <= 1 ) by A1, BORSUK_1:43, XREAL_1:160;
hence p * q is Point of I[01] by A1, BORSUK_1:43; ::_thesis: verum
end;
theorem Th6: :: BORSUK_6:6
for x being Point of I[01] holds (1 / 2) * x is Point of I[01]
proof
let x be Point of I[01]; ::_thesis: (1 / 2) * x is Point of I[01]
x <= 1 by BORSUK_1:43;
then (1 / 2) * x <= (1 / 2) * 1 by XREAL_1:64;
then ( x >= 0 & (1 / 2) * x <= 1 ) by BORSUK_1:43, XXREAL_0:2;
hence (1 / 2) * x is Point of I[01] by BORSUK_1:43; ::_thesis: verum
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
let x be Point of I[01]; ::_thesis: ( x >= 1 / 2 implies x - (1 / 4) is Point of I[01] )
x <= 1 by BORSUK_1:43;
then x <= 1 + (1 / 4) by XXREAL_0:2;
then A1: x - (1 / 4) <= 1 by XREAL_1:20;
assume x >= 1 / 2 ; ::_thesis: x - (1 / 4) is Point of I[01]
then x >= (1 / 4) + 0 by XXREAL_0:2;
then x - (1 / 4) >= 0 by XREAL_1:19;
hence x - (1 / 4) is Point of I[01] by A1, BORSUK_1:43; ::_thesis: verum
end;
theorem Th8: :: BORSUK_6:8
id I[01] is Path of 0[01] , 1[01]
proof
set f = id I[01];
( (id I[01]) . 0 = 0[01] & (id I[01]) . 1 = 1[01] ) by BORSUK_1:def_14, BORSUK_1:def_15, FUNCT_1:18;
hence id I[01] is Path of 0[01] , 1[01] by BORSUK_2:def_4; ::_thesis: verum
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
let a, b, c, d be Point of I[01]; ::_thesis: ( a <= b & c <= d implies [:[.a,b.],[.c,d.]:] is non empty compact Subset of [:I[01],I[01]:] )
( [.a,b.] is Subset of I[01] & [.c,d.] is Subset of I[01] ) by BORSUK_4:18;
then A1: [:[.a,b.],[.c,d.]:] c= [: the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:96;
assume A2: ( a <= b & c <= d ) ; ::_thesis: [:[.a,b.],[.c,d.]:] is non empty compact Subset of [:I[01],I[01]:]
then ( a in [.a,b.] & c in [.c,d.] ) by XXREAL_1:1;
then reconsider Ewa = [:[.a,b.],[.c,d.]:] as non empty Subset of [:I[01],I[01]:] by A1, BORSUK_1:def_2;
( [.a,b.] is compact Subset of I[01] & [.c,d.] is compact Subset of I[01] ) by A2, BORSUK_4:24;
then Ewa is compact Subset of [:I[01],I[01]:] by BORSUK_3:23;
hence [:[.a,b.],[.c,d.]:] is non empty compact Subset of [:I[01],I[01]:] ; ::_thesis: verum
end;
begin
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
set f = AffineMap (1,0,(1 / 2),(1 / 2));
set A = 1;
set B = 0 ;
set C = 1 / 2;
set D = 1 / 2;
let S, T be Subset of (TOP-REAL 2); ::_thesis: ( 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 } implies (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T )
assume that
A1: S = { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1)) - 1 } and
A2: T = { p where p is Point of (TOP-REAL 2) : p `2 <= p `1 } ; ::_thesis: (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
(AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
proof
thus (AffineMap (1,0,(1 / 2),(1 / 2))) .: S c= T :: according to XBOOLE_0:def_10 ::_thesis: T c= (AffineMap (1,0,(1 / 2),(1 / 2))) .: S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S or x in T )
assume x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S ; ::_thesis: x in T
then consider y being set such that
y in dom (AffineMap (1,0,(1 / 2),(1 / 2))) and
A3: y in S and
A4: x = (AffineMap (1,0,(1 / 2),(1 / 2))) . y by FUNCT_1:def_6;
consider p being Point of (TOP-REAL 2) such that
A5: y = p and
A6: p `2 <= (2 * (p `1)) - 1 by A1, A3;
set b = (AffineMap (1,0,(1 / 2),(1 / 2))) . p;
(AffineMap (1,0,(1 / 2),(1 / 2))) . p = |[((1 * (p `1)) + 0),(((1 / 2) * (p `2)) + (1 / 2))]| by JGRAPH_2:def_2;
then A7: ( ((AffineMap (1,0,(1 / 2),(1 / 2))) . p) `1 = (1 * (p `1)) + 0 & ((AffineMap (1,0,(1 / 2),(1 / 2))) . p) `2 = ((1 / 2) * (p `2)) + (1 / 2) ) by EUCLID:52;
(1 / 2) * (p `2) <= (1 / 2) * ((2 * (p `1)) - 1) by A6, XREAL_1:64;
then ((1 / 2) * (p `2)) + (1 / 2) <= ((p `1) - (1 / 2)) + (1 / 2) by XREAL_1:6;
hence x in T by A2, A4, A5, A7; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in T or x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S )
assume A8: x in T ; ::_thesis: x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S
then A9: ex p being Point of (TOP-REAL 2) st
( x = p & p `2 <= p `1 ) by A2;
AffineMap (1,0,(1 / 2),(1 / 2)) is onto by JORDAN1K:36;
then rng (AffineMap (1,0,(1 / 2),(1 / 2))) = the carrier of (TOP-REAL 2) by FUNCT_2:def_3;
then consider y being set such that
A10: y in dom (AffineMap (1,0,(1 / 2),(1 / 2))) and
A11: x = (AffineMap (1,0,(1 / 2),(1 / 2))) . y by A8, FUNCT_1:def_3;
reconsider y = y as Point of (TOP-REAL 2) by A10;
set b = (AffineMap (1,0,(1 / 2),(1 / 2))) . y;
A12: (AffineMap (1,0,(1 / 2),(1 / 2))) . y = |[((1 * (y `1)) + 0),(((1 / 2) * (y `2)) + (1 / 2))]| by JGRAPH_2:def_2;
then ((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `1 = y `1 by EUCLID:52;
then 2 * (((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2) <= 2 * (y `1) by A9, A11, XREAL_1:64;
then A13: (2 * (((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2)) - 1 <= (2 * (y `1)) - 1 by XREAL_1:9;
((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2 = ((1 / 2) * (y `2)) + (1 / 2) by A12, EUCLID:52;
then y in S by A1, A13;
hence x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S by A10, A11, FUNCT_1:def_6; ::_thesis: verum
end;
hence (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T ; ::_thesis: verum
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
set f = AffineMap (1,0,(1 / 2),(1 / 2));
set A = 1;
set B = 0 ;
set C = 1 / 2;
set D = 1 / 2;
let S, T be Subset of (TOP-REAL 2); ::_thesis: ( 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 } implies (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T )
assume that
A1: S = { p where p is Point of (TOP-REAL 2) : p `2 >= (2 * (p `1)) - 1 } and
A2: T = { p where p is Point of (TOP-REAL 2) : p `2 >= p `1 } ; ::_thesis: (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
(AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
proof
thus (AffineMap (1,0,(1 / 2),(1 / 2))) .: S c= T :: according to XBOOLE_0:def_10 ::_thesis: T c= (AffineMap (1,0,(1 / 2),(1 / 2))) .: S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S or x in T )
assume x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S ; ::_thesis: x in T
then consider y being set such that
y in dom (AffineMap (1,0,(1 / 2),(1 / 2))) and
A3: y in S and
A4: x = (AffineMap (1,0,(1 / 2),(1 / 2))) . y by FUNCT_1:def_6;
consider p being Point of (TOP-REAL 2) such that
A5: y = p and
A6: p `2 >= (2 * (p `1)) - 1 by A1, A3;
A7: (1 / 2) * (p `2) >= (1 / 2) * ((2 * (p `1)) - 1) by A6, XREAL_1:64;
set b = (AffineMap (1,0,(1 / 2),(1 / 2))) . p;
A8: (AffineMap (1,0,(1 / 2),(1 / 2))) . p = |[((1 * (p `1)) + 0),(((1 / 2) * (p `2)) + (1 / 2))]| by JGRAPH_2:def_2;
then A9: ((AffineMap (1,0,(1 / 2),(1 / 2))) . p) `1 = (1 * (p `1)) + 0 by EUCLID:52;
((AffineMap (1,0,(1 / 2),(1 / 2))) . p) `2 = ((1 / 2) * (p `2)) + (1 / 2) by A8, EUCLID:52;
then ((AffineMap (1,0,(1 / 2),(1 / 2))) . p) `2 >= ((p `1) - (1 / 2)) + (1 / 2) by A7, XREAL_1:6;
hence x in T by A2, A4, A5, A9; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in T or x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S )
assume A10: x in T ; ::_thesis: x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S
then A11: ex p being Point of (TOP-REAL 2) st
( x = p & p `2 >= p `1 ) by A2;
AffineMap (1,0,(1 / 2),(1 / 2)) is onto by JORDAN1K:36;
then rng (AffineMap (1,0,(1 / 2),(1 / 2))) = the carrier of (TOP-REAL 2) by FUNCT_2:def_3;
then consider y being set such that
A12: y in dom (AffineMap (1,0,(1 / 2),(1 / 2))) and
A13: x = (AffineMap (1,0,(1 / 2),(1 / 2))) . y by A10, FUNCT_1:def_3;
reconsider y = y as Point of (TOP-REAL 2) by A12;
set b = (AffineMap (1,0,(1 / 2),(1 / 2))) . y;
A14: (AffineMap (1,0,(1 / 2),(1 / 2))) . y = |[((1 * (y `1)) + 0),(((1 / 2) * (y `2)) + (1 / 2))]| by JGRAPH_2:def_2;
then ((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `1 = y `1 by EUCLID:52;
then 2 * (((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2) >= 2 * (y `1) by A11, A13, XREAL_1:64;
then A15: (2 * (((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2)) - 1 >= (2 * (y `1)) - 1 by XREAL_1:9;
((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2 = ((1 / 2) * (y `2)) + (1 / 2) by A14, EUCLID:52;
then y in S by A1, A15;
hence x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S by A12, A13, FUNCT_1:def_6; ::_thesis: verum
end;
hence (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T ; ::_thesis: verum
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
set f = AffineMap (1,0,(1 / 2),(- (1 / 2)));
set A = 1;
set B = 0 ;
set C = 1 / 2;
set D = - (1 / 2);
let S, T be Subset of (TOP-REAL 2); ::_thesis: ( 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) } implies (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T )
assume that
A1: S = { p where p is Point of (TOP-REAL 2) : p `2 >= 1 - (2 * (p `1)) } and
A2: T = { p where p is Point of (TOP-REAL 2) : p `2 >= - (p `1) } ; ::_thesis: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T
(AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T
proof
thus (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S c= T :: according to XBOOLE_0:def_10 ::_thesis: T c= (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S or x in T )
assume x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S ; ::_thesis: x in T
then consider y being set such that
y in dom (AffineMap (1,0,(1 / 2),(- (1 / 2)))) and
A3: y in S and
A4: x = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y by FUNCT_1:def_6;
consider p being Point of (TOP-REAL 2) such that
A5: y = p and
A6: p `2 >= 1 - (2 * (p `1)) by A1, A3;
set b = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p;
(1 / 2) * (p `2) >= (1 / 2) * (1 - (2 * (p `1))) by A6, XREAL_1:64;
then A7: ((1 / 2) * (p `2)) + (- (1 / 2)) >= ((1 / 2) - (p `1)) + (- (1 / 2)) by XREAL_1:6;
A8: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p = |[((1 * (p `1)) + 0),(((1 / 2) * (p `2)) + (- (1 / 2)))]| by JGRAPH_2:def_2;
then ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p) `1 = (1 * (p `1)) + 0 by EUCLID:52;
then ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p) `2 >= - (((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p) `1) by A8, A7, EUCLID:52;
hence x in T by A2, A4, A5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in T or x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S )
assume A9: x in T ; ::_thesis: x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S
then A10: ex p being Point of (TOP-REAL 2) st
( x = p & p `2 >= - (p `1) ) by A2;
AffineMap (1,0,(1 / 2),(- (1 / 2))) is onto by JORDAN1K:36;
then rng (AffineMap (1,0,(1 / 2),(- (1 / 2)))) = the carrier of (TOP-REAL 2) by FUNCT_2:def_3;
then consider y being set such that
A11: y in dom (AffineMap (1,0,(1 / 2),(- (1 / 2)))) and
A12: x = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y by A9, FUNCT_1:def_3;
reconsider y = y as Point of (TOP-REAL 2) by A11;
set b = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y;
A13: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y = |[((1 * (y `1)) + 0),(((1 / 2) * (y `2)) + (- (1 / 2)))]| by JGRAPH_2:def_2;
then ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `1 = y `1 by EUCLID:52;
then 2 * (((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `2) >= 2 * (- (y `1)) by A10, A12, XREAL_1:64;
then A14: (2 * (((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `2)) + 1 >= (2 * (- (y `1))) + 1 by XREAL_1:6;
((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `2 = ((1 / 2) * (y `2)) + (- (1 / 2)) by A13, EUCLID:52;
then y `2 >= 1 - (2 * (y `1)) by A14;
then y in S by A1;
hence x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S by A11, A12, FUNCT_1:def_6; ::_thesis: verum
end;
hence (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T ; ::_thesis: verum
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
set f = AffineMap (1,0,(1 / 2),(- (1 / 2)));
set A = 1;
set B = 0 ;
set C = 1 / 2;
set D = - (1 / 2);
let S, T be Subset of (TOP-REAL 2); ::_thesis: ( 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) } implies (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T )
assume that
A1: S = { p where p is Point of (TOP-REAL 2) : p `2 <= 1 - (2 * (p `1)) } and
A2: T = { p where p is Point of (TOP-REAL 2) : p `2 <= - (p `1) } ; ::_thesis: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T
(AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T
proof
thus (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S c= T :: according to XBOOLE_0:def_10 ::_thesis: T c= (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S or x in T )
assume x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S ; ::_thesis: x in T
then consider y being set such that
y in dom (AffineMap (1,0,(1 / 2),(- (1 / 2)))) and
A3: y in S and
A4: x = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y by FUNCT_1:def_6;
consider p being Point of (TOP-REAL 2) such that
A5: y = p and
A6: p `2 <= 1 - (2 * (p `1)) by A1, A3;
set b = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p;
(1 / 2) * (p `2) <= (1 / 2) * (1 - (2 * (p `1))) by A6, XREAL_1:64;
then A7: ((1 / 2) * (p `2)) + (- (1 / 2)) <= ((1 / 2) - (p `1)) + (- (1 / 2)) by XREAL_1:6;
A8: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p = |[((1 * (p `1)) + 0),(((1 / 2) * (p `2)) + (- (1 / 2)))]| by JGRAPH_2:def_2;
then ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p) `1 = (1 * (p `1)) + 0 by EUCLID:52;
then ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p) `2 <= - (((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . p) `1) by A8, A7, EUCLID:52;
hence x in T by A2, A4, A5; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in T or x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S )
assume A9: x in T ; ::_thesis: x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S
then A10: ex p being Point of (TOP-REAL 2) st
( x = p & p `2 <= - (p `1) ) by A2;
AffineMap (1,0,(1 / 2),(- (1 / 2))) is onto by JORDAN1K:36;
then rng (AffineMap (1,0,(1 / 2),(- (1 / 2)))) = the carrier of (TOP-REAL 2) by FUNCT_2:def_3;
then consider y being set such that
A11: y in dom (AffineMap (1,0,(1 / 2),(- (1 / 2)))) and
A12: x = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y by A9, FUNCT_1:def_3;
reconsider y = y as Point of (TOP-REAL 2) by A11;
set b = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y;
A13: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y = |[((1 * (y `1)) + 0),(((1 / 2) * (y `2)) + (- (1 / 2)))]| by JGRAPH_2:def_2;
then ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `1 = y `1 by EUCLID:52;
then 2 * (((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `2) <= 2 * (- (y `1)) by A10, A12, XREAL_1:64;
then A14: (2 * (((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `2)) + 1 <= (2 * (- (y `1))) + 1 by XREAL_1:6;
((AffineMap (1,0,(1 / 2),(- (1 / 2)))) . y) `2 = ((1 / 2) * (y `2)) + (- (1 / 2)) by A13, EUCLID:52;
then y `2 <= 1 - (2 * (y `1)) by A14;
then y in S by A1;
hence x in (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S by A11, A12, FUNCT_1:def_6; ::_thesis: verum
end;
hence (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: S = T ; ::_thesis: verum
end;
begin
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
let T be non empty 1-sorted ; ::_thesis: ( T is real-membered iff for x being Element of T holds x is real )
thus ( T is real-membered implies for x being Element of T holds x is real ) ; ::_thesis: ( ( for x being Element of T holds x is real ) implies T is real-membered )
assume for x being Element of T holds x is real ; ::_thesis: T is real-membered
then for x being set st x in the carrier of T holds
x is real ;
then the carrier of T is real-membered by MEMBERED:def_3;
hence T is real-membered by TOPMETR:def_8; ::_thesis: verum
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
take I[01] ; ::_thesis: ( not I[01] is empty & I[01] is real-membered )
thus ( not I[01] is empty & I[01] is real-membered ) ; ::_thesis: verum
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
take I[01] ; ::_thesis: ( not I[01] is empty & I[01] is real-membered )
thus ( not I[01] is empty & I[01] is real-membered ) ; ::_thesis: verum
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:];
clusterp `1 -> real ;
coherence
p `1 is real
proof
p in the carrier of [:S,T:] ;
then p in [: the carrier of S, the carrier of T:] by BORSUK_1:def_2;
then p `1 in the carrier of S by MCART_1:10;
hence p `1 is real ; ::_thesis: verum
end;
clusterp `2 -> real ;
coherence
p `2 is real
proof
p in the carrier of [:S,T:] ;
then p in [: the carrier of S, the carrier of T:] by BORSUK_1:def_2;
then p `2 in the carrier of T by MCART_1:10;
hence p `2 is real ; ::_thesis: verum
end;
end;
registration
let T be non empty SubSpace of [:I[01],I[01]:];
let x be Point of T;
clusterx `1 -> real ;
coherence
x `1 is real
proof
( the carrier of T c= the carrier of [:I[01],I[01]:] & x in the carrier of T ) by BORSUK_1:1;
then reconsider x9 = x as Point of [:I[01],I[01]:] ;
x9 `1 is real ;
hence x `1 is real ; ::_thesis: verum
end;
clusterx `2 -> real ;
coherence
x `2 is real
proof
( the carrier of T c= the carrier of [:I[01],I[01]:] & x in the carrier of T ) by BORSUK_1:1;
then reconsider x9 = x as Point of [:I[01],I[01]:] ;
x9 `2 is real ;
hence x `2 is real ; ::_thesis: verum
end;
end;
begin
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
reconsider L = { p where p is Point of (TOP-REAL 2) : p `2 <= p `1 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
set f = AffineMap (1,0,(1 / 2),(1 / 2));
defpred S1[ Point of (TOP-REAL 2)] means $1 `2 <= (2 * ($1 `1)) - 1;
{ p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider K = { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1)) - 1 } as Subset of (TOP-REAL 2) ;
K c= the carrier of (TOP-REAL 2) ;
then A1: K c= dom (AffineMap (1,0,(1 / 2),(1 / 2))) by FUNCT_2:def_1;
A2: (AffineMap (1,0,(1 / 2),(1 / 2))) .: K = L by Th10;
AffineMap (1,0,(1 / 2),(1 / 2)) is one-to-one by JGRAPH_2:44;
then K = (AffineMap (1,0,(1 / 2),(1 / 2))) " ((AffineMap (1,0,(1 / 2),(1 / 2))) .: K) by A1, FUNCT_1:94;
hence { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1)) - 1 } is closed Subset of (TOP-REAL 2) by A2, PRE_TOPC:def_6; ::_thesis: verum
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
reconsider L = { p where p is Point of (TOP-REAL 2) : p `2 >= p `1 } as closed Subset of (TOP-REAL 2) by JGRAPH_2:46;
set f = AffineMap (1,0,(1 / 2),(1 / 2));
defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= (2 * ($1 `1)) - 1;
{ p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider K = { p where p is Point of (TOP-REAL 2) : p `2 >= (2 * (p `1)) - 1 } as Subset of (TOP-REAL 2) ;
K c= the carrier of (TOP-REAL 2) ;
then A1: K c= dom (AffineMap (1,0,(1 / 2),(1 / 2))) by FUNCT_2:def_1;
A2: (AffineMap (1,0,(1 / 2),(1 / 2))) .: K = L by Th11;
AffineMap (1,0,(1 / 2),(1 / 2)) is one-to-one by JGRAPH_2:44;
then K = (AffineMap (1,0,(1 / 2),(1 / 2))) " ((AffineMap (1,0,(1 / 2),(1 / 2))) .: K) by A1, FUNCT_1:94;
hence { p where p is Point of (TOP-REAL 2) : p `2 >= (2 * (p `1)) - 1 } is closed Subset of (TOP-REAL 2) by A2, PRE_TOPC:def_6; ::_thesis: verum
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
reconsider L = { p where p is Point of (TOP-REAL 2) : p `2 <= - (p `1) } as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
set f = AffineMap (1,0,(1 / 2),(- (1 / 2)));
defpred S1[ Point of (TOP-REAL 2)] means $1 `2 <= 1 - (2 * ($1 `1));
{ p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider K = { p where p is Point of (TOP-REAL 2) : p `2 <= 1 - (2 * (p `1)) } as Subset of (TOP-REAL 2) ;
K c= the carrier of (TOP-REAL 2) ;
then A1: K c= dom (AffineMap (1,0,(1 / 2),(- (1 / 2)))) by FUNCT_2:def_1;
A2: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: K = L by Th13;
AffineMap (1,0,(1 / 2),(- (1 / 2))) is one-to-one by JGRAPH_2:44;
then K = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) " ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: K) by A1, FUNCT_1:94;
hence { p where p is Point of (TOP-REAL 2) : p `2 <= 1 - (2 * (p `1)) } is closed Subset of (TOP-REAL 2) by A2, PRE_TOPC:def_6; ::_thesis: verum
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
reconsider L = { p where p is Point of (TOP-REAL 2) : p `2 >= - (p `1) } as closed Subset of (TOP-REAL 2) by JGRAPH_2:47;
set f = AffineMap (1,0,(1 / 2),(- (1 / 2)));
defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= 1 - (2 * ($1 `1));
{ p where p is Point of (TOP-REAL 2) : S1[p] } is Subset of (TOP-REAL 2) from JGRAPH_2:sch_1();
then reconsider K = { p where p is Point of (TOP-REAL 2) : p `2 >= 1 - (2 * (p `1)) } as Subset of (TOP-REAL 2) ;
K c= the carrier of (TOP-REAL 2) ;
then A1: K c= dom (AffineMap (1,0,(1 / 2),(- (1 / 2)))) by FUNCT_2:def_1;
A2: (AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: K = L by Th12;
AffineMap (1,0,(1 / 2),(- (1 / 2))) is one-to-one by JGRAPH_2:44;
then K = (AffineMap (1,0,(1 / 2),(- (1 / 2)))) " ((AffineMap (1,0,(1 / 2),(- (1 / 2)))) .: K) by A1, FUNCT_1:94;
hence { p where p is Point of (TOP-REAL 2) : p `2 >= 1 - (2 * (p `1)) } is closed Subset of (TOP-REAL 2) by A2, PRE_TOPC:def_6; ::_thesis: verum
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
defpred S1[ Point of (TOP-REAL 2)] means $1 `2 >= (2 * ($1 `1)) - 1;
reconsider L = { p where p is Point of (TOP-REAL 2) : S1[p] } as closed Subset of (TOP-REAL 2) by Th16;
defpred S2[ Point of (TOP-REAL 2)] means $1 `2 >= 1 - (2 * ($1 `1));
reconsider K = { p where p is Point of (TOP-REAL 2) : S2[p] } as closed Subset of (TOP-REAL 2) by Th18;
set T = { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } ;
{ p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = { p where p is Point of (TOP-REAL 2) : S2[p] } /\ { p where p is Point of (TOP-REAL 2) : S1[p] } from DOMAIN_1:sch_10();
then { p where p is Point of (TOP-REAL 2) : ( S2[p] & S1[p] ) } = K /\ L ;
hence { 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) ; ::_thesis: verum
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
defpred S1[ Element of REAL , Element of REAL , set ] means ex c being Element of REAL 2 st
( c = $3 & $3 = <*$1,$2*> );
A1: for x, y being Element of REAL ex u being Element of REAL 2 st S1[x,y,u]
proof
let x, y be Element of REAL ; ::_thesis: ex u being Element of REAL 2 st S1[x,y,u]
take <*x,y*> ; ::_thesis: ( <*x,y*> is Element of bool [:NAT,REAL:] & <*x,y*> is FinSequence of REAL & <*x,y*> is Element of REAL 2 & S1[x,y,<*x,y*>] )
<*x,y*> is Element of REAL 2 by FINSEQ_2:137;
hence ( <*x,y*> is Element of bool [:NAT,REAL:] & <*x,y*> is FinSequence of REAL & <*x,y*> is Element of REAL 2 & S1[x,y,<*x,y*>] ) ; ::_thesis: verum
end;
consider f being Function of [:REAL,REAL:],(REAL 2) such that
A2: for x, y being Element of REAL holds S1[x,y,f . (x,y)] from BINOP_1:sch_3(A1);
the carrier of [:R^1,R^1:] = [: the carrier of R^1, the carrier of R^1:] by BORSUK_1:def_2;
then reconsider f = f as Function of [:R^1,R^1:],(TOP-REAL 2) by EUCLID:22, TOPMETR:17;
take f ; ::_thesis: for x, y being Real holds f . [x,y] = <*x,y*>
for x, y being Real holds f . [x,y] = <*x,y*>
proof
let x, y be Real; ::_thesis: f . [x,y] = <*x,y*>
S1[x,y,f . (x,y)] by A2;
hence f . [x,y] = <*x,y*> ; ::_thesis: verum
end;
hence for x, y being Real holds f . [x,y] = <*x,y*> ; ::_thesis: verum
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
set GG = [:R^1,R^1:];
set SS = TOP-REAL 2;
defpred S1[ Point of [:R^1,R^1:]] means $1 `2 <= 1 - (2 * ($1 `1));
defpred S2[ Point of (TOP-REAL 2)] means $1 `2 <= 1 - (2 * ($1 `1));
reconsider K = { p where p is Point of [:R^1,R^1:] : S1[p] } as Subset of [:R^1,R^1:] from DOMAIN_1:sch_7();
reconsider L = { p where p is Point of (TOP-REAL 2) : S2[p] } as Subset of (TOP-REAL 2) from DOMAIN_1:sch_7();
consider f being Function of [:R^1,R^1:],(TOP-REAL 2) such that
A1: for x, y being Real holds f . [x,y] = <*x,y*> by Th20;
A2: L c= f .: K
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L or x in f .: K )
assume x in L ; ::_thesis: x in f .: K
then consider z being Point of (TOP-REAL 2) such that
A3: z = x and
A4: S2[z] ;
the carrier of (TOP-REAL 2) = REAL 2 by EUCLID:22;
then z is Tuple of 2, REAL by FINSEQ_2:131;
then consider x1, y1 being Real such that
A5: z = <*x1,y1*> by FINSEQ_2:100;
z `1 = x1 by A5, EUCLID:52;
then A6: y1 <= 1 - (2 * x1) by A4, A5, EUCLID:52;
set Y = [x1,y1];
A7: [x1,y1] in [:REAL,REAL:] by ZFMISC_1:87;
then A8: [x1,y1] in the carrier of [:R^1,R^1:] by BORSUK_1:def_2, TOPMETR:17;
reconsider Y = [x1,y1] as Point of [:R^1,R^1:] by A7, BORSUK_1:def_2, TOPMETR:17;
A9: Y in dom f by A8, FUNCT_2:def_1;
( Y `1 = x1 & Y `2 = y1 ) by MCART_1:7;
then A10: Y in K by A6;
x = f . Y by A1, A3, A5;
hence x in f .: K by A10, A9, FUNCT_1:def_6; ::_thesis: verum
end;
A11: f is being_homeomorphism by A1, TOPREAL6:76;
f .: K c= L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f .: K or x in L )
assume x in f .: K ; ::_thesis: x in L
then consider y being set such that
y in dom f and
A12: y in K and
A13: x = f . y by FUNCT_1:def_6;
consider z being Point of [:R^1,R^1:] such that
A14: z = y and
A15: S1[z] by A12;
z in the carrier of [:R^1,R^1:] ;
then z in [: the carrier of R^1, the carrier of R^1:] by BORSUK_1:def_2;
then consider x1, y1 being set such that
A16: ( x1 in the carrier of R^1 & y1 in the carrier of R^1 ) and
A17: z = [x1,y1] by ZFMISC_1:def_2;
reconsider x1 = x1, y1 = y1 as Real by A16, TOPMETR:17;
A18: x = |[x1,y1]| by A1, A13, A14, A17;
then reconsider x9 = x as Point of (TOP-REAL 2) ;
A19: ( z `1 = x1 & z `2 = y1 ) by A17, MCART_1:7;
( x9 `1 = x1 & x9 `2 = y1 ) by A18, FINSEQ_1:44;
hence x in L by A15, A19; ::_thesis: verum
end;
then f .: K = L by A2, XBOOLE_0:def_10;
hence { p where p is Point of [:R^1,R^1:] : p `2 <= 1 - (2 * (p `1)) } is closed Subset of [:R^1,R^1:] by A11, Th17, TOPS_2:58; ::_thesis: verum
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
set GG = [:R^1,R^1:];
set SS = TOP-REAL 2;
defpred S1[ Point of [:R^1,R^1:]] means $1 `2 <= (2 * ($1 `1)) - 1;
defpred S2[ Point of (TOP-REAL 2)] means $1 `2 <= (2 * ($1 `1)) - 1;
reconsider K = { p where p is Point of [:R^1,R^1:] : S1[p] } as Subset of [:R^1,R^1:] from DOMAIN_1:sch_7();
reconsider L = { p where p is Point of (TOP-REAL 2) : S2[p] } as Subset of (TOP-REAL 2) from DOMAIN_1:sch_7();
consider f being Function of [:R^1,R^1:],(TOP-REAL 2) such that
A1: for x, y being Real holds f . [x,y] = <*x,y*> by Th20;
A2: L c= f .: K
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L or x in f .: K )
assume x in L ; ::_thesis: x in f .: K
then consider z being Point of (TOP-REAL 2) such that
A3: z = x and
A4: S2[z] ;
the carrier of (TOP-REAL 2) = REAL 2 by EUCLID:22;
then z is Tuple of 2, REAL by FINSEQ_2:131;
then consider x1, y1 being Real such that
A5: z = <*x1,y1*> by FINSEQ_2:100;
z `1 = x1 by A5, EUCLID:52;
then A6: y1 <= (2 * x1) - 1 by A4, A5, EUCLID:52;
set Y = [x1,y1];
A7: [x1,y1] in [: the carrier of R^1, the carrier of R^1:] by TOPMETR:17, ZFMISC_1:87;
then A8: [x1,y1] in the carrier of [:R^1,R^1:] by BORSUK_1:def_2;
reconsider Y = [x1,y1] as Point of [:R^1,R^1:] by A7, BORSUK_1:def_2;
A9: Y in dom f by A8, FUNCT_2:def_1;
( Y `1 = x1 & Y `2 = y1 ) by MCART_1:7;
then A10: Y in K by A6;
x = f . Y by A1, A3, A5;
hence x in f .: K by A10, A9, FUNCT_1:def_6; ::_thesis: verum
end;
A11: f is being_homeomorphism by A1, TOPREAL6:76;
f .: K c= L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f .: K or x in L )
assume x in f .: K ; ::_thesis: x in L
then consider y being set such that
y in dom f and
A12: y in K and
A13: x = f . y by FUNCT_1:def_6;
consider z being Point of [:R^1,R^1:] such that
A14: z = y and
A15: S1[z] by A12;
z in the carrier of [:R^1,R^1:] ;
then z in [: the carrier of R^1, the carrier of R^1:] by BORSUK_1:def_2;
then consider x1, y1 being set such that
A16: ( x1 in the carrier of R^1 & y1 in the carrier of R^1 ) and
A17: z = [x1,y1] by ZFMISC_1:def_2;
reconsider x1 = x1, y1 = y1 as Real by A16, TOPMETR:17;
A18: x = |[x1,y1]| by A1, A13, A14, A17;
then reconsider x9 = x as Point of (TOP-REAL 2) ;
A19: ( z `1 = x1 & z `2 = y1 ) by A17, MCART_1:7;
( x9 `1 = x1 & x9 `2 = y1 ) by A18, FINSEQ_1:44;
hence x in L by A15, A19; ::_thesis: verum
end;
then f .: K = L by A2, XBOOLE_0:def_10;
hence { p where p is Point of [:R^1,R^1:] : p `2 <= (2 * (p `1)) - 1 } is closed Subset of [:R^1,R^1:] by A11, Th15, TOPS_2:58; ::_thesis: verum
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
set GG = [:R^1,R^1:];
set SS = TOP-REAL 2;
defpred S1[ Point of [:R^1,R^1:]] means ( $1 `2 >= 1 - (2 * ($1 `1)) & $1 `2 >= (2 * ($1 `1)) - 1 );
defpred S2[ Point of (TOP-REAL 2)] means ( $1 `2 >= 1 - (2 * ($1 `1)) & $1 `2 >= (2 * ($1 `1)) - 1 );
reconsider K = { p where p is Point of [:R^1,R^1:] : S1[p] } as Subset of [:R^1,R^1:] from DOMAIN_1:sch_7();
reconsider L = { p where p is Point of (TOP-REAL 2) : S2[p] } as Subset of (TOP-REAL 2) from DOMAIN_1:sch_7();
consider f being Function of [:R^1,R^1:],(TOP-REAL 2) such that
A1: for x, y being Real holds f . [x,y] = <*x,y*> by Th20;
A2: L c= f .: K
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in L or x in f .: K )
assume x in L ; ::_thesis: x in f .: K
then consider z being Point of (TOP-REAL 2) such that
A3: z = x and
A4: S2[z] ;
the carrier of (TOP-REAL 2) = REAL 2 by EUCLID:22;
then z is Tuple of 2, REAL by FINSEQ_2:131;
then consider x1, y1 being Real such that
A5: z = <*x1,y1*> by FINSEQ_2:100;
z `1 = x1 by A5, EUCLID:52;
then A6: ( y1 >= 1 - (2 * x1) & y1 >= (2 * x1) - 1 ) by A4, A5, EUCLID:52;
set Y = [x1,y1];
A7: [x1,y1] in [: the carrier of R^1, the carrier of R^1:] by TOPMETR:17, ZFMISC_1:87;
then A8: [x1,y1] in the carrier of [:R^1,R^1:] by BORSUK_1:def_2;
reconsider Y = [x1,y1] as Point of [:R^1,R^1:] by A7, BORSUK_1:def_2;
A9: Y in dom f by A8, FUNCT_2:def_1;
( Y `1 = x1 & Y `2 = y1 ) by MCART_1:7;
then A10: Y in K by A6;
x = f . Y by A1, A3, A5;
hence x in f .: K by A10, A9, FUNCT_1:def_6; ::_thesis: verum
end;
A11: f is being_homeomorphism by A1, TOPREAL6:76;
f .: K c= L
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in f .: K or x in L )
assume x in f .: K ; ::_thesis: x in L
then consider y being set such that
y in dom f and
A12: y in K and
A13: x = f . y by FUNCT_1:def_6;
consider z being Point of [:R^1,R^1:] such that
A14: z = y and
A15: S1[z] by A12;
z in the carrier of [:R^1,R^1:] ;
then z in [: the carrier of R^1, the carrier of R^1:] by BORSUK_1:def_2;
then consider x1, y1 being set such that
A16: ( x1 in the carrier of R^1 & y1 in the carrier of R^1 ) and
A17: z = [x1,y1] by ZFMISC_1:def_2;
reconsider x1 = x1, y1 = y1 as Real by A16, TOPMETR:17;
A18: x = |[x1,y1]| by A1, A13, A14, A17;
then reconsider x9 = x as Point of (TOP-REAL 2) ;
A19: ( z `1 = x1 & z `2 = y1 ) by A17, MCART_1:7;
( x9 `1 = x1 & x9 `2 = y1 ) by A18, FINSEQ_1:44;
hence x in L by A15, A19; ::_thesis: verum
end;
then f .: K = L by A2, XBOOLE_0:def_10;
hence { 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:] by A11, Th19, TOPS_2:58; ::_thesis: verum
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
set GG = [:I[01],I[01]:];
set SS = [:R^1,R^1:];
0 in the carrier of I[01] by BORSUK_1:43;
then [0,0] in [: the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
then reconsider x = [0,0] as Point of [:I[01],I[01]:] by BORSUK_1:def_2;
reconsider PA = { p where p is Point of [:R^1,R^1:] : p `2 <= 1 - (2 * (p `1)) } as closed Subset of [:R^1,R^1:] by Th21;
set P0 = { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ;
A1: [:I[01],I[01]:] is SubSpace of [:R^1,R^1:] by BORSUK_3:21;
A2: { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } = PA /\ ([#] [:I[01],I[01]:])
proof
thus { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } c= PA /\ ([#] [:I[01],I[01]:]) :: according to XBOOLE_0:def_10 ::_thesis: PA /\ ([#] [:I[01],I[01]:]) c= { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } or x in PA /\ ([#] [:I[01],I[01]:]) )
A3: the carrier of [:I[01],I[01]:] c= the carrier of [:R^1,R^1:] by A1, BORSUK_1:1;
assume x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ; ::_thesis: x in PA /\ ([#] [:I[01],I[01]:])
then A4: ex p being Point of [:I[01],I[01]:] st
( x = p & p `2 <= 1 - (2 * (p `1)) ) ;
then x in the carrier of [:I[01],I[01]:] ;
then reconsider a = x as Point of [:R^1,R^1:] by A3;
a `2 <= 1 - (2 * (a `1)) by A4;
then x in PA ;
hence x in PA /\ ([#] [:I[01],I[01]:]) by A4, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in PA /\ ([#] [:I[01],I[01]:]) or x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } )
assume A5: x in PA /\ ([#] [:I[01],I[01]:]) ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) }
then x in PA by XBOOLE_0:def_4;
then ex p being Point of [:R^1,R^1:] st
( x = p & p `2 <= 1 - (2 * (p `1)) ) ;
hence x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } by A5; ::_thesis: verum
end;
x `1 = 0 by MCART_1:7;
then x `2 <= 1 - (2 * (x `1)) by MCART_1:7;
then x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ;
hence { 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]:] by A1, A2, PRE_TOPC:13; ::_thesis: verum
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
set GG = [:I[01],I[01]:];
set SS = [:R^1,R^1:];
1 in the carrier of I[01] by BORSUK_1:43;
then [1,1] in [: the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
then reconsider x = [1,1] as Point of [:I[01],I[01]:] by BORSUK_1:def_2;
reconsider PA = { p where p is Point of [:R^1,R^1:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } as closed Subset of [:R^1,R^1:] by Th23;
set P0 = { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } ;
A1: x `1 = 1 by MCART_1:7;
then A2: x `2 >= (2 * (x `1)) - 1 by MCART_1:7;
A3: [:I[01],I[01]:] is SubSpace of [:R^1,R^1:] by BORSUK_3:21;
A4: { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } = PA /\ ([#] [:I[01],I[01]:])
proof
thus { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } c= PA /\ ([#] [:I[01],I[01]:]) :: according to XBOOLE_0:def_10 ::_thesis: PA /\ ([#] [:I[01],I[01]:]) c= { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } or x in PA /\ ([#] [:I[01],I[01]:]) )
A5: the carrier of [:I[01],I[01]:] c= the carrier of [:R^1,R^1:] by A3, BORSUK_1:1;
assume x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } ; ::_thesis: x in PA /\ ([#] [:I[01],I[01]:])
then A6: ex p being Point of [:I[01],I[01]:] st
( x = p & p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) ;
then x in the carrier of [:I[01],I[01]:] ;
then reconsider a = x as Point of [:R^1,R^1:] by A5;
a `2 >= 1 - (2 * (a `1)) by A6;
then x in PA by A6;
hence x in PA /\ ([#] [:I[01],I[01]:]) by A6, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in PA /\ ([#] [:I[01],I[01]:]) or x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } )
assume A7: x in PA /\ ([#] [:I[01],I[01]:]) ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) }
then x in PA by XBOOLE_0:def_4;
then ex p being Point of [:R^1,R^1:] st
( x = p & p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) ;
hence x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } by A7; ::_thesis: verum
end;
x `2 = 1 by MCART_1:7;
then x `2 >= 1 - (2 * (x `1)) by A1;
then x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } by A2;
hence { 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]:] by A3, A4, PRE_TOPC:13; ::_thesis: verum
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
set GG = [:I[01],I[01]:];
set SS = [:R^1,R^1:];
( 0 in the carrier of I[01] & 1 in the carrier of I[01] ) by BORSUK_1:43;
then [1,0] in [: the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
then reconsider x = [1,0] as Point of [:I[01],I[01]:] by BORSUK_1:def_2;
reconsider PA = { p where p is Point of [:R^1,R^1:] : p `2 <= (2 * (p `1)) - 1 } as closed Subset of [:R^1,R^1:] by Th22;
set P0 = { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } ;
A1: [:I[01],I[01]:] is SubSpace of [:R^1,R^1:] by BORSUK_3:21;
A2: { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } = PA /\ ([#] [:I[01],I[01]:])
proof
thus { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } c= PA /\ ([#] [:I[01],I[01]:]) :: according to XBOOLE_0:def_10 ::_thesis: PA /\ ([#] [:I[01],I[01]:]) c= { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } or x in PA /\ ([#] [:I[01],I[01]:]) )
A3: the carrier of [:I[01],I[01]:] c= the carrier of [:R^1,R^1:] by A1, BORSUK_1:1;
assume x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } ; ::_thesis: x in PA /\ ([#] [:I[01],I[01]:])
then A4: ex p being Point of [:I[01],I[01]:] st
( x = p & p `2 <= (2 * (p `1)) - 1 ) ;
then x in the carrier of [:I[01],I[01]:] ;
then reconsider a = x as Point of [:R^1,R^1:] by A3;
a `2 <= (2 * (a `1)) - 1 by A4;
then x in PA ;
hence x in PA /\ ([#] [:I[01],I[01]:]) by A4, XBOOLE_0:def_4; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in PA /\ ([#] [:I[01],I[01]:]) or x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } )
assume A5: x in PA /\ ([#] [:I[01],I[01]:]) ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 }
then x in PA by XBOOLE_0:def_4;
then ex p being Point of [:R^1,R^1:] st
( x = p & p `2 <= (2 * (p `1)) - 1 ) ;
hence x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } by A5; ::_thesis: verum
end;
x `1 = 1 by MCART_1:7;
then x `2 <= (2 * (x `1)) - 1 by MCART_1:7;
then x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } ;
hence { 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]:] by A1, A2, PRE_TOPC:13; ::_thesis: verum
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
let S, T be non empty TopSpace; ::_thesis: for p being Point of [:S,T:] holds
( p `1 is Point of S & p `2 is Point of T )
let p be Point of [:S,T:]; ::_thesis: ( p `1 is Point of S & p `2 is Point of T )
p in the carrier of [:S,T:] ;
then p in [: the carrier of S, the carrier of T:] by BORSUK_1:def_2;
hence ( p `1 is Point of S & p `2 is Point of T ) by MCART_1:10; ::_thesis: verum
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
let A, B be Subset of [:I[01],I[01]:]; ::_thesis: ( A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] implies ([#] ([:I[01],I[01]:] | A)) \/ ([#] ([:I[01],I[01]:] | B)) = [#] [:I[01],I[01]:] )
assume A1: ( A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] ) ; ::_thesis: ([#] ([:I[01],I[01]:] | A)) \/ ([#] ([:I[01],I[01]:] | B)) = [#] [:I[01],I[01]:]
([#] ([:I[01],I[01]:] | A)) \/ ([#] ([:I[01],I[01]:] | B)) = A \/ ([#] ([:I[01],I[01]:] | B)) by PRE_TOPC:def_5
.= A \/ B by PRE_TOPC:def_5
.= [:([.0,(1 / 2).] \/ [.(1 / 2),1.]),[.0,1.]:] by A1, ZFMISC_1:97
.= [:[.0,1.],[.0,1.]:] by XXREAL_1:174
.= [#] [:I[01],I[01]:] by BORSUK_1:40, BORSUK_1:def_2 ;
hence ([#] ([:I[01],I[01]:] | A)) \/ ([#] ([:I[01],I[01]:] | B)) = [#] [:I[01],I[01]:] ; ::_thesis: verum
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
let A, B be Subset of [:I[01],I[01]:]; ::_thesis: ( A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] implies ([#] ([:I[01],I[01]:] | A)) /\ ([#] ([:I[01],I[01]:] | B)) = [:{(1 / 2)},[.0,1.]:] )
assume A1: ( A = [:[.0,(1 / 2).],[.0,1.]:] & B = [:[.(1 / 2),1.],[.0,1.]:] ) ; ::_thesis: ([#] ([:I[01],I[01]:] | A)) /\ ([#] ([:I[01],I[01]:] | B)) = [:{(1 / 2)},[.0,1.]:]
([#] ([:I[01],I[01]:] | A)) /\ ([#] ([:I[01],I[01]:] | B)) = A /\ ([#] ([:I[01],I[01]:] | B)) by PRE_TOPC:def_5
.= A /\ B by PRE_TOPC:def_5
.= [:([.0,(1 / 2).] /\ [.(1 / 2),1.]),[.0,1.]:] by A1, ZFMISC_1:99
.= [:{(1 / 2)},[.0,1.]:] by XXREAL_1:418 ;
hence ([#] ([:I[01],I[01]:] | A)) /\ ([#] ([:I[01],I[01]:] | B)) = [:{(1 / 2)},[.0,1.]:] ; ::_thesis: verum
end;
begin
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
take {} T ; ::_thesis: ( {} T is empty & {} T is compact )
thus ( {} T is empty & {} T is compact ) ; ::_thesis: verum
end;
end;
theorem Th30: :: BORSUK_6:30
for T being TopStruct holds {} is empty compact Subset of T
proof
let T be TopStruct ; ::_thesis: {} is empty compact Subset of T
{} T c= the carrier of T ;
hence {} is empty compact Subset of T ; ::_thesis: verum
end;
theorem Th31: :: BORSUK_6:31
for T being TopStruct
for a, b being real number st a > b holds
[.a,b.] is empty compact Subset of T
proof
let T be TopStruct ; ::_thesis: for a, b being real number st a > b holds
[.a,b.] is empty compact Subset of T
let a, b be real number ; ::_thesis: ( a > b implies [.a,b.] is empty compact Subset of T )
assume a > b ; ::_thesis: [.a,b.] is empty compact Subset of T
then [.a,b.] = {} T by XXREAL_1:29;
hence [.a,b.] is empty compact Subset of T ; ::_thesis: verum
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
let a, b, c, d be Point of I[01]; ::_thesis: [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:]
percases ( ( a <= b & c <= d ) or ( a > b & c <= d ) or ( a > b & c > d ) or ( a <= b & c > d ) ) ;
suppose ( a <= b & c <= d ) ; ::_thesis: [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:]
hence [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:] by Th9; ::_thesis: verum
end;
suppose ( a > b & c <= d ) ; ::_thesis: [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:]
then reconsider A = [.a,b.] as empty Subset of I[01] by Th31;
[:A,[.c,d.]:] = {} ;
hence [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:] by Th30; ::_thesis: verum
end;
suppose ( a > b & c > d ) ; ::_thesis: [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:]
then reconsider A = [.c,d.] as empty Subset of I[01] by Th31;
[:[.a,b.],A:] = {} ;
hence [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:] by Th30; ::_thesis: verum
end;
suppose ( a <= b & c > d ) ; ::_thesis: [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:]
then reconsider A = [.c,d.] as empty Subset of I[01] by Th31;
[:[.a,b.],A:] = {} ;
hence [:[.a,b.],[.c,d.]:] is compact Subset of [:I[01],I[01]:] by Th30; ::_thesis: verum
end;
end;
end;
begin
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 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 number 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 number st a < b & c < d holds
( (L[01] (a,b,c,d)) . a = c & (L[01] (a,b,c,d)) . b = d )
proof
let a, b, c, d be real number ; ::_thesis: ( a < b & c < d implies ( (L[01] (a,b,c,d)) . a = c & (L[01] (a,b,c,d)) . b = d ) )
assume that
A1: a < b and
A2: c < d ; ::_thesis: ( (L[01] (a,b,c,d)) . a = c & (L[01] (a,b,c,d)) . b = d )
a in [.a,b.] by A1, XXREAL_1:1;
then a in the carrier of (Closed-Interval-TSpace (a,b)) by A1, TOPMETR:18;
then a in dom (P[01] (a,b,((#) (0,1)),((0,1) (#)))) by FUNCT_2:def_1;
hence (L[01] (a,b,c,d)) . a = (L[01] (((#) (c,d)),((c,d) (#)))) . ((P[01] (a,b,((#) (0,1)),((0,1) (#)))) . a) by FUNCT_1:13
.= (L[01] (((#) (c,d)),((c,d) (#)))) . ((P[01] (a,b,((#) (0,1)),((0,1) (#)))) . ((#) (a,b))) by A1, TREAL_1:def_1
.= (L[01] (((#) (c,d)),((c,d) (#)))) . ((#) (0,1)) by A1, TREAL_1:13
.= (#) (c,d) by A2, TREAL_1:9
.= c by A2, TREAL_1:def_1 ;
::_thesis: (L[01] (a,b,c,d)) . b = d
b in [.a,b.] by A1, XXREAL_1:1;
then b in the carrier of (Closed-Interval-TSpace (a,b)) by A1, TOPMETR:18;
then b in dom (P[01] (a,b,((#) (0,1)),((0,1) (#)))) by FUNCT_2:def_1;
hence (L[01] (a,b,c,d)) . b = (L[01] (((#) (c,d)),((c,d) (#)))) . ((P[01] (a,b,((#) (0,1)),((0,1) (#)))) . b) by FUNCT_1:13
.= (L[01] (((#) (c,d)),((c,d) (#)))) . ((P[01] (a,b,((#) (0,1)),((0,1) (#)))) . ((a,b) (#))) by A1, TREAL_1:def_2
.= (L[01] (((#) (c,d)),((c,d) (#)))) . ((0,1) (#)) by A1, TREAL_1:13
.= (c,d) (#) by A2, TREAL_1:9
.= d by A2, TREAL_1:def_2 ;
::_thesis: verum
end;
theorem Th34: :: BORSUK_6:34
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
let a, b, c, d be real number ; ::_thesis: ( a < b & c <= d implies L[01] (a,b,c,d) is continuous Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (c,d)) )
assume ( a < b & c <= d ) ; ::_thesis: L[01] (a,b,c,d) is continuous Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (c,d))
then ( L[01] (((#) (c,d)),((c,d) (#))) is continuous Function of (Closed-Interval-TSpace (0,1)),(Closed-Interval-TSpace (c,d)) & P[01] (a,b,((#) (0,1)),((0,1) (#))) is continuous Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (0,1)) ) by TREAL_1:8, TREAL_1:12;
hence L[01] (a,b,c,d) is continuous Function of (Closed-Interval-TSpace (a,b)),(Closed-Interval-TSpace (c,d)) ; ::_thesis: verum
end;
theorem Th35: :: BORSUK_6:35
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
A1: ( 0 = (#) (0,1) & 1 = (0,1) (#) ) by TREAL_1:def_1, TREAL_1:def_2;
let a, b, c, d be real number ; ::_thesis: ( a < b & c <= d implies 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 )
assume A2: a < b ; ::_thesis: ( not c <= d or 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 )
set G = P[01] (a,b,((#) (0,1)),((0,1) (#)));
set F = L[01] (((#) (c,d)),((c,d) (#)));
set f = L[01] (a,b,c,d);
assume A3: c <= d ; ::_thesis: 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
then A4: ( (#) (c,d) = c & (c,d) (#) = d ) by TREAL_1:def_1, TREAL_1:def_2;
let x be real number ; ::_thesis: ( a <= x & x <= b implies (L[01] (a,b,c,d)) . x = (((d - c) / (b - a)) * (x - a)) + c )
assume A5: a <= x ; ::_thesis: ( not x <= b or (L[01] (a,b,c,d)) . x = (((d - c) / (b - a)) * (x - a)) + c )
set X = (x - a) / (b - a);
assume A6: x <= b ; ::_thesis: (L[01] (a,b,c,d)) . x = (((d - c) / (b - a)) * (x - a)) + c
then A7: (x - a) / (b - a) in the carrier of (Closed-Interval-TSpace (0,1)) by A5, Th2;
x in [.a,b.] by A5, A6, XXREAL_1:1;
then A8: x in the carrier of (Closed-Interval-TSpace (a,b)) by A2, TOPMETR:18;
then x in dom (P[01] (a,b,((#) (0,1)),((0,1) (#)))) by FUNCT_2:def_1;
then (L[01] (a,b,c,d)) . x = (L[01] (((#) (c,d)),((c,d) (#)))) . ((P[01] (a,b,((#) (0,1)),((0,1) (#)))) . x) by FUNCT_1:13
.= (L[01] (((#) (c,d)),((c,d) (#)))) . ((((b - x) * 0) + ((x - a) * 1)) / (b - a)) by A2, A8, A1, TREAL_1:def_4
.= ((1 - ((x - a) / (b - a))) * c) + (((x - a) / (b - a)) * d) by A3, A4, A7, TREAL_1:def_3
.= (((d - c) / (b - a)) * (x - a)) + c by XCMPLX_1:234 ;
hence (L[01] (a,b,c,d)) . x = (((d - c) / (b - a)) * (x - a)) + c ; ::_thesis: verum
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 number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 * r2 ) & g is continuous )
proof
reconsider A = [.0,1.] as non empty Subset of R^1 by TOPMETR:17, XXREAL_1:1;
set X = [:I[01],I[01]:];
let f1, f2 be Function of [:I[01],I[01]:],I[01]; ::_thesis: ( 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] ) implies 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 ) )
assume that
A1: ( f1 is continuous & f2 is continuous ) and
A2: for p being Point of [:I[01],I[01]:] holds (f1 . p) * (f2 . p) is Point of I[01] ; ::_thesis: 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 )
reconsider f19 = f1, f29 = f2 as Function of [:I[01],I[01]:],R^1 by BORSUK_1:40, FUNCT_2:7, TOPMETR:17;
( f19 is continuous & f29 is continuous ) by A1, PRE_TOPC:26;
then consider g being Function of [:I[01],I[01]:],R^1 such that
A3: for p being Point of [:I[01],I[01]:]
for r1, r2 being real number st f19 . p = r1 & f29 . p = r2 holds
g . p = r1 * r2 and
A4: g is continuous by JGRAPH_2:25;
A5: rng g c= [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g or x in [.0,1.] )
assume x in rng g ; ::_thesis: x in [.0,1.]
then consider y being set such that
A6: y in dom g and
A7: x = g . y by FUNCT_1:def_3;
reconsider y = y as Point of [:I[01],I[01]:] by A6;
g . y = (f1 . y) * (f2 . y) by A3;
then g . y is Point of I[01] by A2;
hence x in [.0,1.] by A7, BORSUK_1:40; ::_thesis: verum
end;
( [.0,1.] = the carrier of (R^1 | A) & dom g = the carrier of [:I[01],I[01]:] ) by FUNCT_2:def_1, PRE_TOPC:8;
then reconsider g = g as Function of [:I[01],I[01]:],(R^1 | A) by A5, FUNCT_2:2;
R^1 | A = I[01] by BORSUK_1:def_13, TOPMETR:def_6;
then reconsider g = g as continuous Function of [:I[01],I[01]:],I[01] by A4, JGRAPH_1:45;
take g ; ::_thesis: ( ( 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 )
thus ( ( 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 ) by A3; ::_thesis: verum
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 number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 + r2 ) & g is continuous )
proof
reconsider A = [.0,1.] as non empty Subset of R^1 by TOPMETR:17, XXREAL_1:1;
set X = [:I[01],I[01]:];
let f1, f2 be Function of [:I[01],I[01]:],I[01]; ::_thesis: ( 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] ) implies 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 ) )
assume that
A1: ( f1 is continuous & f2 is continuous ) and
A2: for p being Point of [:I[01],I[01]:] holds (f1 . p) + (f2 . p) is Point of I[01] ; ::_thesis: 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 )
reconsider f19 = f1, f29 = f2 as Function of [:I[01],I[01]:],R^1 by BORSUK_1:40, FUNCT_2:7, TOPMETR:17;
( f19 is continuous & f29 is continuous ) by A1, PRE_TOPC:26;
then consider g being Function of [:I[01],I[01]:],R^1 such that
A3: for p being Point of [:I[01],I[01]:]
for r1, r2 being real number st f19 . p = r1 & f29 . p = r2 holds
g . p = r1 + r2 and
A4: g is continuous by JGRAPH_2:19;
A5: rng g c= [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g or x in [.0,1.] )
assume x in rng g ; ::_thesis: x in [.0,1.]
then consider y being set such that
A6: y in dom g and
A7: x = g . y by FUNCT_1:def_3;
reconsider y = y as Point of [:I[01],I[01]:] by A6;
g . y = (f1 . y) + (f2 . y) by A3;
then g . y is Point of I[01] by A2;
hence x in [.0,1.] by A7, BORSUK_1:40; ::_thesis: verum
end;
( [.0,1.] = the carrier of (R^1 | A) & dom g = the carrier of [:I[01],I[01]:] ) by FUNCT_2:def_1, PRE_TOPC:8;
then reconsider g = g as Function of [:I[01],I[01]:],(R^1 | A) by A5, FUNCT_2:2;
R^1 | A = I[01] by BORSUK_1:def_13, TOPMETR:def_6;
then reconsider g = g as continuous Function of [:I[01],I[01]:],I[01] by A4, JGRAPH_1:45;
take g ; ::_thesis: ( ( 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 )
thus ( ( 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 ) by A3; ::_thesis: verum
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 number st f1 . p = r1 & f2 . p = r2 holds
g . p = r1 - r2 ) & g is continuous )
proof
reconsider A = [.0,1.] as non empty Subset of R^1 by TOPMETR:17, XXREAL_1:1;
set X = [:I[01],I[01]:];
let f1, f2 be Function of [:I[01],I[01]:],I[01]; ::_thesis: ( 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] ) implies 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 ) )
assume that
A1: ( f1 is continuous & f2 is continuous ) and
A2: for p being Point of [:I[01],I[01]:] holds (f1 . p) - (f2 . p) is Point of I[01] ; ::_thesis: 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 )
reconsider f19 = f1, f29 = f2 as Function of [:I[01],I[01]:],R^1 by BORSUK_1:40, FUNCT_2:7, TOPMETR:17;
( f19 is continuous & f29 is continuous ) by A1, PRE_TOPC:26;
then consider g being Function of [:I[01],I[01]:],R^1 such that
A3: for p being Point of [:I[01],I[01]:]
for r1, r2 being real number st f19 . p = r1 & f29 . p = r2 holds
g . p = r1 - r2 and
A4: g is continuous by JGRAPH_2:21;
A5: rng g c= [.0,1.]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in rng g or x in [.0,1.] )
assume x in rng g ; ::_thesis: x in [.0,1.]
then consider y being set such that
A6: y in dom g and
A7: x = g . y by FUNCT_1:def_3;
reconsider y = y as Point of [:I[01],I[01]:] by A6;
g . y = (f1 . y) - (f2 . y) by A3;
then g . y is Point of I[01] by A2;
hence x in [.0,1.] by A7, BORSUK_1:40; ::_thesis: verum
end;
( [.0,1.] = the carrier of (R^1 | A) & dom g = the carrier of [:I[01],I[01]:] ) by FUNCT_2:def_1, PRE_TOPC:8;
then reconsider g = g as Function of [:I[01],I[01]:],(R^1 | A) by A5, FUNCT_2:2;
R^1 | A = I[01] by BORSUK_1:def_13, TOPMETR:def_6;
then reconsider g = g as continuous Function of [:I[01],I[01]:],I[01] by A4, JGRAPH_1:45;
take g ; ::_thesis: ( ( 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 )
thus ( ( 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 ) by A3; ::_thesis: verum
end;
begin
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
reconsider g = L[01] (((0,1) (#)),((#) (0,1))) as Function of I[01],I[01] by TOPMETR:20;
let P be Path of a,b; ::_thesis: ( P is continuous implies P * (L[01] (((0,1) (#)),((#) (0,1)))) is continuous Function of I[01],T )
assume A1: P is continuous ; ::_thesis: P * (L[01] (((0,1) (#)),((#) (0,1)))) is continuous Function of I[01],T
reconsider f = P * g as Function of I[01],T ;
g is continuous by TOPMETR:20, TREAL_1:8;
then f is continuous by A1;
hence P * (L[01] (((0,1) (#)),((#) (0,1)))) is continuous Function of I[01],T ; ::_thesis: verum
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
A1: 0 in [.0,1.] by XXREAL_1:1;
set e = L[01] (((0,1) (#)),((#) (0,1)));
let X be non empty TopStruct ; ::_thesis: 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 )
let a, b be Point of X; ::_thesis: 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 )
let P be Path of a,b; ::_thesis: ( P . 0 = a & P . 1 = b implies ( (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 0 = b & (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 1 = a ) )
assume that
A2: P . 0 = a and
A3: P . 1 = b ; ::_thesis: ( (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 0 = b & (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 1 = a )
A4: the carrier of (Closed-Interval-TSpace (0,1)) = [.0,1.] by TOPMETR:18;
(L[01] (((0,1) (#)),((#) (0,1)))) . 0 = (L[01] (((0,1) (#)),((#) (0,1)))) . ((#) (0,1)) by TREAL_1:def_1
.= (0,1) (#) by TREAL_1:9
.= 1 by TREAL_1:def_2 ;
hence (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 0 = b by A3, A4, A1, FUNCT_2:15; ::_thesis: (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 1 = a
A5: 1 in [.0,1.] by XXREAL_1:1;
(L[01] (((0,1) (#)),((#) (0,1)))) . 1 = (L[01] (((0,1) (#)),((#) (0,1)))) . ((0,1) (#)) by TREAL_1:def_2
.= (#) (0,1) by TREAL_1:9
.= 0 by TREAL_1:def_1 ;
hence (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 1 = a by A2, A4, A5, FUNCT_2:15; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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 )
let a, b be Point of T; ::_thesis: 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 )
let P be Path of a,b; ::_thesis: ( P is continuous & P . 0 = a & P . 1 = b implies ( - P is continuous & (- P) . 0 = b & (- P) . 1 = a ) )
assume that
A1: P is continuous and
A2: ( P . 0 = a & P . 1 = b ) ; ::_thesis: ( - P is continuous & (- P) . 0 = b & (- P) . 1 = a )
A3: (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 1 = a by A2, Th40;
( P * (L[01] (((0,1) (#)),((#) (0,1)))) is continuous Function of I[01],T & (P * (L[01] (((0,1) (#)),((#) (0,1))))) . 0 = b ) by A1, A2, Th39, Th40;
then b,a are_connected by A3, BORSUK_2:def_1;
hence ( - P is continuous & (- P) . 0 = b & (- P) . 1 = a ) by BORSUK_2:def_2; ::_thesis: verum
end;
definition
let T be non empty TopSpace;
let a, b be Point of T;
:: original: are_connected
redefine preda,b are_connected ;
reflexivity
for a being Point of T holds R61(T,b1,b1)
proof
let a be Point of T; ::_thesis: R61(T,a,a)
thus a,a are_connected ; ::_thesis: verum
end;
symmetry
for a, b being Point of T st R61(T,b1,b2) holds
R61(T,b2,b1)
proof
let a, b be Point of T; ::_thesis: ( R61(T,a,b) implies R61(T,b,a) )
set P = the Path of a,b;
assume A1: a,b are_connected ; ::_thesis: R61(T,b,a)
then A2: the Path of a,b . 1 = b by BORSUK_2:def_2;
take - the Path of a,b ; :: according to BORSUK_2:def_1 ::_thesis: ( - the Path of a,b is continuous & (- the Path of a,b) . 0 = b & (- the Path of a,b) . 1 = a )
( the Path of a,b is continuous & the Path of a,b . 0 = a ) by A1, BORSUK_2:def_2;
hence ( - the Path of a,b is continuous & (- the Path of a,b) . 0 = b & (- the Path of a,b) . 1 = a ) by A2, Th41; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: for a, b, c being Point of T st a,b are_connected & b,c are_connected holds
a,c are_connected
let a, b, c be Point of T; ::_thesis: ( a,b are_connected & b,c are_connected implies a,c are_connected )
assume that
A1: a,b are_connected and
A2: b,c are_connected ; ::_thesis: a,c are_connected
set P = the Path of a,b;
set R = the Path of b,c;
A3: ( the Path of a,b is continuous & the Path of a,b . 0 = a ) by A1, BORSUK_2:def_2;
take the Path of a,b + the Path of b,c ; :: according to BORSUK_2:def_1 ::_thesis: ( the Path of a,b + the Path of b,c is continuous & ( the Path of a,b + the Path of b,c) . 0 = a & ( the Path of a,b + the Path of b,c) . 1 = c )
A4: ( the Path of b,c . 0 = b & the Path of b,c . 1 = c ) by A2, BORSUK_2:def_2;
( the Path of a,b . 1 = b & the Path of b,c is continuous ) by A1, A2, BORSUK_2:def_2;
hence ( the Path of a,b + the Path of b,c is continuous & ( the Path of a,b + the Path of b,c) . 0 = a & ( the Path of a,b + the Path of b,c) . 1 = c ) by A3, A4, BORSUK_2:14; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: for a, b being Point of T st a,b are_connected holds
for A being Path of a,b holds A = - (- A)
let a, b be Point of T; ::_thesis: ( a,b are_connected implies for A being Path of a,b holds A = - (- A) )
set I = the carrier of I[01];
assume A1: a,b are_connected ; ::_thesis: for A being Path of a,b holds A = - (- A)
let A be Path of a,b; ::_thesis: A = - (- A)
for x being Element of the carrier of I[01] holds A . x = (- (- A)) . x
proof
let x be Element of the carrier of I[01]; ::_thesis: A . x = (- (- A)) . x
reconsider z = 1 - x as Point of I[01] by JORDAN5B:4;
thus (- (- A)) . x = (- A) . (1 - x) by A1, BORSUK_2:def_6
.= A . (1 - z) by A1, BORSUK_2:def_6
.= A . x ; ::_thesis: verum
end;
hence A = - (- A) by FUNCT_2:63; ::_thesis: verum
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)
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: for a, b being Point of T
for A being Path of a,b holds A = - (- A)
let a, b be Point of T; ::_thesis: for A being Path of a,b holds A = - (- A)
a,b are_connected by BORSUK_2:def_3;
hence for A being Path of a,b holds A = - (- A) by Th43; ::_thesis: verum
end;
begin
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
let X be Path of a,c; ::_thesis: ( X = P + Q iff for t being Point of I[01] holds
( ( t <= 1 / 2 implies X . t = P . (2 * t) ) & ( 1 / 2 <= t implies X . t = Q . ((2 * t) - 1) ) ) )
( a,b are_connected & b,c are_connected ) by BORSUK_2:def_3;
hence ( X = P + Q iff for t being Point of I[01] holds
( ( t <= 1 / 2 implies X . t = P . (2 * t) ) & ( 1 / 2 <= t implies X . t = Q . ((2 * t) - 1) ) ) ) by BORSUK_2:def_5; ::_thesis: verum
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
let X be Path of b,a; ::_thesis: ( X = - P iff for t being Point of I[01] holds X . t = P . (1 - t) )
b,a are_connected by BORSUK_2:def_3;
hence ( X = - P iff for t being Point of I[01] holds X . t = P . (1 - t) ) by BORSUK_2:def_6; ::_thesis: verum
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) );
begin
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
set PF = P * f;
0 in the carrier of I[01] by BORSUK_1:43;
then 0 in dom f by FUNCT_2:def_1;
then A4: (P * f) . 0 = P . (f . 0) by FUNCT_1:13
.= a by A1, A3, BORSUK_2:def_2 ;
1 in the carrier of I[01] by BORSUK_1:43;
then 1 in dom f by FUNCT_2:def_1;
then A5: (P * f) . 1 = P . (f . 1) by FUNCT_1:13
.= b by A2, A3, BORSUK_2:def_2 ;
P is continuous by A3, BORSUK_2:def_2;
hence P * f is Path of a,b by A3, A4, A5, BORSUK_2:def_2; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
set X = [:I[01],I[01]:];
reconsider G2 = pr2 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:40;
reconsider F2 = pr1 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:39;
reconsider f3 = pr1 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:39;
reconsider f4 = pr2 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:40;
reconsider ID = id I[01] as Path of 0[01] , 1[01] by Th8;
let P be Path of a,b; ::_thesis: 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
let f be continuous Function of I[01],I[01]; ::_thesis: ( f . 0 = 0 & f . 1 = 1 & a,b are_connected implies RePar (P,f),P are_homotopic )
assume that
A1: f . 0 = 0 and
A2: f . 1 = 1 and
A3: a,b are_connected ; ::_thesis: RePar (P,f),P are_homotopic
reconsider f2 = f * F2 as continuous Function of [:I[01],I[01]:],I[01] ;
set G1 = - ID;
reconsider f1 = (- ID) * G2 as continuous Function of [:I[01],I[01]:],I[01] ;
A4: for s, t being Point of I[01] holds f1 . [s,t] = 1 - t
proof
let s, t be Point of I[01]; ::_thesis: f1 . [s,t] = 1 - t
A5: 1 - t in the carrier of I[01] by JORDAN5B:4;
[s,t] in [: the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
then [s,t] in dom G2 by FUNCT_2:def_1;
then f1 . [s,t] = (- ID) . (G2 . (s,t)) by FUNCT_1:13
.= (- ID) . t by FUNCT_3:def_5
.= ID . (1 - t) by Def3
.= 1 - t by A5, FUNCT_1:18 ;
hence f1 . [s,t] = 1 - t ; ::_thesis: verum
end;
for p being Point of [:I[01],I[01]:] holds (f3 . p) * (f4 . p) is Point of I[01] by Th5;
then consider g2 being Function of [:I[01],I[01]:],I[01] such that
A6: for p being Point of [:I[01],I[01]:]
for r1, r2 being real number st f3 . p = r1 & f4 . p = r2 holds
g2 . p = r1 * r2 and
A7: g2 is continuous by Th36;
for p being Point of [:I[01],I[01]:] holds (f1 . p) * (f2 . p) is Point of I[01] by Th5;
then consider g1 being Function of [:I[01],I[01]:],I[01] such that
A8: for p being Point of [:I[01],I[01]:]
for r1, r2 being real number st f1 . p = r1 & f2 . p = r2 holds
g1 . p = r1 * r2 and
A9: g1 is continuous by Th36;
A10: for s, t being Point of I[01] holds f2 . (s,t) = f . s
proof
let s, t be Point of I[01]; ::_thesis: f2 . (s,t) = f . s
[s,t] in [: the carrier of I[01], the carrier of I[01]:] by ZFMISC_1:87;
then [s,t] in dom F2 by FUNCT_2:def_1;
hence f2 . (s,t) = f . (F2 . (s,t)) by FUNCT_1:13
.= f . s by FUNCT_3:def_4 ;
::_thesis: verum
end;
A11: for t, s being Point of I[01] holds g1 . [s,t] = (1 - t) * (f . s)
proof
let t, s be Point of I[01]; ::_thesis: g1 . [s,t] = (1 - t) * (f . s)
( f1 . (s,t) = 1 - t & f2 . (s,t) = f . s ) by A4, A10;
hence g1 . [s,t] = (1 - t) * (f . s) by A8; ::_thesis: verum
end;
A12: for t, s being Point of I[01] holds g2 . [s,t] = t * s
proof
let t, s be Point of I[01]; ::_thesis: g2 . [s,t] = t * s
( f3 . (s,t) = s & f4 . (s,t) = t ) by FUNCT_3:def_4, FUNCT_3:def_5;
hence g2 . [s,t] = t * s by A6; ::_thesis: verum
end;
for p being Point of [:I[01],I[01]:] holds (g1 . p) + (g2 . p) is Point of I[01]
proof
let p be Point of [:I[01],I[01]:]; ::_thesis: (g1 . p) + (g2 . p) is Point of I[01]
p in the carrier of [:I[01],I[01]:] ;
then p in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then consider s, t being set such that
A13: ( s in the carrier of I[01] & t in the carrier of I[01] ) and
A14: p = [s,t] by ZFMISC_1:def_2;
reconsider s = s, t = t as Point of I[01] by A13;
set a = f . s;
percases ( f . s <= s or f . s > s ) ;
supposeA15: f . s <= s ; ::_thesis: (g1 . p) + (g2 . p) is Point of I[01]
A16: s <= 1 by BORSUK_1:40, XXREAL_1:1;
A17: t <= 1 by BORSUK_1:40, XXREAL_1:1;
then ((1 - t) * (f . s)) + (t * s) <= s by A15, XREAL_1:172;
then A18: ( 0 <= f . s & ((1 - t) * (f . s)) + (t * s) <= 1 ) by A16, BORSUK_1:40, XXREAL_0:2, XXREAL_1:1;
0 <= t by BORSUK_1:40, XXREAL_1:1;
then A19: f . s <= ((1 - t) * (f . s)) + (t * s) by A15, A17, XREAL_1:173;
(g1 . p) + (g2 . p) = ((1 - t) * (f . s)) + (g2 . p) by A11, A14
.= ((1 - t) * (f . s)) + (t * s) by A12, A14 ;
hence (g1 . p) + (g2 . p) is Point of I[01] by A19, A18, BORSUK_1:40, XXREAL_1:1; ::_thesis: verum
end;
supposeA20: f . s > s ; ::_thesis: (g1 . p) + (g2 . p) is Point of I[01]
set j = 1 - t;
A21: f . s <= 1 by BORSUK_1:40, XXREAL_1:1;
A22: 1 - t in the carrier of I[01] by JORDAN5B:4;
then A23: 1 - t <= 1 by BORSUK_1:43;
then ((1 - (1 - t)) * s) + ((1 - t) * (f . s)) <= f . s by A20, XREAL_1:172;
then A24: ( 0 <= s & ((1 - t) * (f . s)) + (t * s) <= 1 ) by A21, BORSUK_1:40, XXREAL_0:2, XXREAL_1:1;
0 <= 1 - t by A22, BORSUK_1:43;
then A25: s <= ((1 - (1 - t)) * s) + ((1 - t) * (f . s)) by A20, A23, XREAL_1:173;
(g1 . p) + (g2 . p) = ((1 - t) * (f . s)) + (g2 . p) by A11, A14
.= ((1 - t) * (f . s)) + (t * s) by A12, A14 ;
hence (g1 . p) + (g2 . p) is Point of I[01] by A25, A24, BORSUK_1:40, XXREAL_1:1; ::_thesis: verum
end;
end;
end;
then consider h being Function of [:I[01],I[01]:],I[01] such that
A26: for p being Point of [:I[01],I[01]:]
for r1, r2 being real number st g1 . p = r1 & g2 . p = r2 holds
h . p = r1 + r2 and
A27: h is continuous by A9, A7, Th37;
A28: for t, s being Point of I[01] holds h . [s,t] = ((1 - t) * (f . s)) + (t * s)
proof
let t, s be Point of I[01]; ::_thesis: h . [s,t] = ((1 - t) * (f . s)) + (t * s)
( g1 . [s,t] = (1 - t) * (f . s) & g2 . [s,t] = t * s ) by A11, A12;
hence h . [s,t] = ((1 - t) * (f . s)) + (t * s) by A26; ::_thesis: verum
end;
A29: for t being Point of I[01] holds h . [1,t] = 1
proof
reconsider oo = 1 as Point of I[01] by BORSUK_1:43;
let t be Point of I[01]; ::_thesis: h . [1,t] = 1
thus h . [1,t] = ((1 - t) * (f . oo)) + (t * 1) by A28
.= 1 by A2 ; ::_thesis: verum
end;
set H = P * h;
A30: dom h = the carrier of [:I[01],I[01]:] by FUNCT_2:def_1
.= [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2 ;
set Q = RePar (P,f);
A31: 1 is Point of I[01] by BORSUK_1:43;
A32: for s being Point of I[01] holds h . [s,1] = s
proof
let s be Point of I[01]; ::_thesis: h . [s,1] = s
thus h . [s,1] = ((1 - 1) * (f . s)) + (1 * s) by A31, A28
.= s ; ::_thesis: verum
end;
A33: 0 is Point of I[01] by BORSUK_1:43;
A34: for s being Point of I[01] holds h . [s,0] = f . s
proof
let s be Point of I[01]; ::_thesis: h . [s,0] = f . s
thus h . [s,0] = ((1 - 0) * (f . s)) + (0 * s) by A33, A28
.= f . s ; ::_thesis: verum
end;
A35: for s being Point of I[01] holds
( (P * h) . (s,0) = (RePar (P,f)) . s & (P * h) . (s,1) = P . s )
proof
let s be Point of I[01]; ::_thesis: ( (P * h) . (s,0) = (RePar (P,f)) . s & (P * h) . (s,1) = P . s )
s in the carrier of I[01] ;
then A36: s in dom f by FUNCT_2:def_1;
0 in the carrier of I[01] by BORSUK_1:43;
then [s,0] in dom h by A30, ZFMISC_1:87;
hence (P * h) . (s,0) = P . (h . [s,0]) by FUNCT_1:13
.= P . (f . s) by A34
.= (P * f) . s by A36, FUNCT_1:13
.= (RePar (P,f)) . s by A1, A2, A3, Def4 ;
::_thesis: (P * h) . (s,1) = P . s
1 in the carrier of I[01] by BORSUK_1:43;
then [s,1] in dom h by A30, ZFMISC_1:87;
hence (P * h) . (s,1) = P . (h . [s,1]) by FUNCT_1:13
.= P . s by A32 ;
::_thesis: verum
end;
A37: for t being Point of I[01] holds h . [0,t] = 0
proof
reconsider oo = 0 as Point of I[01] by BORSUK_1:43;
let t be Point of I[01]; ::_thesis: h . [0,t] = 0
thus h . [0,t] = ((1 - t) * (f . oo)) + (t * 0) by A28
.= 0 by A1 ; ::_thesis: verum
end;
A38: for t being Point of I[01] holds
( (P * h) . (0,t) = a & (P * h) . (1,t) = b )
proof
let t be Point of I[01]; ::_thesis: ( (P * h) . (0,t) = a & (P * h) . (1,t) = b )
0 in the carrier of I[01] by BORSUK_1:43;
then [0,t] in dom h by A30, ZFMISC_1:87;
hence (P * h) . (0,t) = P . (h . [0,t]) by FUNCT_1:13
.= P . 0 by A37
.= a by A3, BORSUK_2:def_2 ;
::_thesis: (P * h) . (1,t) = b
1 in the carrier of I[01] by BORSUK_1:43;
then [1,t] in dom h by A30, ZFMISC_1:87;
hence (P * h) . (1,t) = P . (h . [1,t]) by FUNCT_1:13
.= P . 1 by A29
.= b by A3, BORSUK_2:def_2 ;
::_thesis: verum
end;
P is continuous by A3, BORSUK_2:def_2;
hence RePar (P,f),P are_homotopic by A27, A35, A38, BORSUK_2:def_7; ::_thesis: verum
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
proof
let T be non empty pathwise_connected TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
let P be Path of a,b; ::_thesis: for f being continuous Function of I[01],I[01] st f . 0 = 0 & f . 1 = 1 holds
RePar (P,f),P are_homotopic
let f be continuous Function of I[01],I[01]; ::_thesis: ( f . 0 = 0 & f . 1 = 1 implies RePar (P,f),P are_homotopic )
a,b are_connected by BORSUK_2:def_3;
hence ( f . 0 = 0 & f . 1 = 1 implies RePar (P,f),P are_homotopic ) by Th45; ::_thesis: verum
end;
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
deffunc H1( set ) -> Element of NAT = 1;
deffunc H2( real number ) -> Element of REAL = 2 * $1;
defpred S1[ real number ] means $1 <= 1 / 2;
consider f being Function such that
A1: ( dom f = the carrier of I[01] & ( for x being Element of I[01] holds
( ( S1[x] implies f . x = H2(x) ) & ( not S1[x] implies f . x = H1(x) ) ) ) ) from PARTFUN1:sch_4();
for x being set st x in the carrier of I[01] holds
f . x in the carrier of I[01]
proof
let x be set ; ::_thesis: ( x in the carrier of I[01] implies f . x in the carrier of I[01] )
assume x in the carrier of I[01] ; ::_thesis: f . x in the carrier of I[01]
then reconsider x = x as Point of I[01] ;
percases ( S1[x] or not S1[x] ) ;
supposeA2: S1[x] ; ::_thesis: f . x in the carrier of I[01]
then f . x = 2 * x by A1;
then f . x is Point of I[01] by A2, Th3;
hence f . x in the carrier of I[01] ; ::_thesis: verum
end;
suppose not S1[x] ; ::_thesis: f . x in the carrier of I[01]
then f . x = H1(x) by A1;
hence f . x in the carrier of I[01] by BORSUK_1:43; ::_thesis: verum
end;
end;
end;
then reconsider f = f as Function of I[01],I[01] by A1, FUNCT_2:3;
for t being Point of I[01] holds
( ( t <= 1 / 2 implies f . t = 2 * t ) & ( t > 1 / 2 implies f . t = 1 ) ) by A1;
hence 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 ) ) ; ::_thesis: verum
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
let f1, f2 be Function of I[01],I[01]; ::_thesis: ( ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies f1 . t = 2 * t ) & ( t > 1 / 2 implies f1 . t = 1 ) ) ) & ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies f2 . t = 2 * t ) & ( t > 1 / 2 implies f2 . t = 1 ) ) ) implies f1 = f2 )
assume that
A3: for t being Point of I[01] holds
( ( t <= 1 / 2 implies f1 . t = 2 * t ) & ( t > 1 / 2 implies f1 . t = 1 ) ) and
A4: for t being Point of I[01] holds
( ( t <= 1 / 2 implies f2 . t = 2 * t ) & ( t > 1 / 2 implies f2 . t = 1 ) ) ; ::_thesis: f1 = f2
for t being Point of I[01] holds f1 . t = f2 . t
proof
let t be Point of I[01]; ::_thesis: f1 . t = f2 . t
percases ( t <= 1 / 2 or t > 1 / 2 ) ;
supposeA5: t <= 1 / 2 ; ::_thesis: f1 . t = f2 . t
then f1 . t = 2 * t by A3
.= f2 . t by A4, A5 ;
hence f1 . t = f2 . t ; ::_thesis: verum
end;
supposeA6: t > 1 / 2 ; ::_thesis: f1 . t = f2 . t
then f1 . t = 1 by A3
.= f2 . t by A4, A6 ;
hence f1 . t = f2 . t ; ::_thesis: verum
end;
end;
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
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
A1: 1 / 2 is Point of I[01] by BORSUK_1:43;
1 is Point of I[01] by BORSUK_1:43;
then reconsider B = [.(1 / 2),1.] as non empty compact Subset of I[01] by A1, BORSUK_4:24;
0 is Point of I[01] by BORSUK_1:43;
then reconsider A = [.0,(1 / 2).] as non empty compact Subset of I[01] by A1, BORSUK_4:24;
set T1 = I[01] | A;
set T2 = I[01] | B;
set T = I[01] ;
reconsider g = (I[01] | B) --> 1[01] as continuous Function of (I[01] | B),I[01] ;
I[01] | A = Closed-Interval-TSpace (0,(1 / 2)) by TOPMETR:24;
then reconsider f = L[01] (0,(1 / 2),0,1) as continuous Function of (I[01] | A),I[01] by Th34, TOPMETR:20;
A2: for p being set st p in ([#] (I[01] | A)) /\ ([#] (I[01] | B)) holds
f . p = g . p
proof
let p be set ; ::_thesis: ( p in ([#] (I[01] | A)) /\ ([#] (I[01] | B)) implies f . p = g . p )
assume p in ([#] (I[01] | A)) /\ ([#] (I[01] | B)) ; ::_thesis: f . p = g . p
then p in [.0,(1 / 2).] /\ ([#] (I[01] | B)) by PRE_TOPC:def_5;
then p in [.0,(1 / 2).] /\ [.(1 / 2),1.] by PRE_TOPC:def_5;
then p in {(1 / 2)} by XXREAL_1:418;
then A3: p = 1 / 2 by TARSKI:def_1;
then p in [.(1 / 2),1.] by XXREAL_1:1;
then A4: p in the carrier of (I[01] | B) by PRE_TOPC:8;
f . p = (((1 - 0) / ((1 / 2) - 0)) * ((1 / 2) - 0)) + 0 by A3, Th35
.= g . p by A4, BORSUK_1:def_15, FUNCOP_1:7 ;
hence f . p = g . p ; ::_thesis: verum
end;
A5: for x being Element of I[01] holds 1RP . x = (f +* g) . x
proof
let x be Element of I[01]; ::_thesis: 1RP . x = (f +* g) . x
A6: dom g = the carrier of (I[01] | B) by FUNCT_2:def_1
.= [.(1 / 2),1.] by PRE_TOPC:8 ;
percases ( x < 1 / 2 or x = 1 / 2 or x > 1 / 2 ) by XXREAL_0:1;
supposeA7: x < 1 / 2 ; ::_thesis: 1RP . x = (f +* g) . x
0 <= x by BORSUK_1:43;
then A8: f . x = (((1 - 0) / ((1 / 2) - 0)) * (x - 0)) + 0 by A7, Th35
.= (1 / (1 / 2)) * x ;
A9: not x in dom g by A6, A7, XXREAL_1:1;
thus 1RP . x = 2 * x by A7, Def5
.= (f +* g) . x by A9, A8, FUNCT_4:11 ; ::_thesis: verum
end;
supposeA10: x = 1 / 2 ; ::_thesis: 1RP . x = (f +* g) . x
then A11: x in dom g by A6, XXREAL_1:1;
thus 1RP . x = 2 * (1 / 2) by A10, Def5
.= g . x by A11, BORSUK_1:def_15, FUNCOP_1:7
.= (f +* g) . x by A11, FUNCT_4:13 ; ::_thesis: verum
end;
supposeA12: x > 1 / 2 ; ::_thesis: 1RP . x = (f +* g) . x
x <= 1 by BORSUK_1:43;
then A13: x in dom g by A6, A12, XXREAL_1:1;
thus 1RP . x = 1 by A12, Def5
.= g . x by A13, BORSUK_1:def_15, FUNCOP_1:7
.= (f +* g) . x by A13, FUNCT_4:13 ; ::_thesis: verum
end;
end;
end;
([#] (I[01] | A)) \/ ([#] (I[01] | B)) = [.0,(1 / 2).] \/ ([#] (I[01] | B)) by PRE_TOPC:def_5
.= [.0,(1 / 2).] \/ [.(1 / 2),1.] by PRE_TOPC:def_5
.= [#] I[01] by BORSUK_1:40, XXREAL_1:174 ;
then ex h being Function of I[01],I[01] st
( h = f +* g & h is continuous ) by A2, BORSUK_2:1;
hence 1RP is continuous by A5, FUNCT_2:63; ::_thesis: verum
end;
end;
theorem Th47: :: BORSUK_6:47
( 1RP . 0 = 0 & 1RP . 1 = 1 )
proof
reconsider x = 0 , y = 1 as Point of I[01] by BORSUK_1:43;
thus 1RP . 0 = 2 * x by Def5
.= 0 ; ::_thesis: 1RP . 1 = 1
thus 1RP . 1 = 1RP . y
.= 1 by Def5 ; ::_thesis: verum
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
deffunc H1( set ) -> Element of NAT = 0 ;
deffunc H2( real number ) -> Element of REAL = (2 * $1) - 1;
defpred S1[ real number ] means $1 <= 1 / 2;
consider f being Function such that
A1: ( dom f = the carrier of I[01] & ( for x being Element of I[01] holds
( ( S1[x] implies f . x = H1(x) ) & ( not S1[x] implies f . x = H2(x) ) ) ) ) from PARTFUN1:sch_4();
for x being set st x in the carrier of I[01] holds
f . x in the carrier of I[01]
proof
let x be set ; ::_thesis: ( x in the carrier of I[01] implies f . x in the carrier of I[01] )
assume x in the carrier of I[01] ; ::_thesis: f . x in the carrier of I[01]
then reconsider x = x as Point of I[01] ;
percases ( S1[x] or not S1[x] ) ;
suppose S1[x] ; ::_thesis: f . x in the carrier of I[01]
then f . x = 0 by A1;
hence f . x in the carrier of I[01] by BORSUK_1:43; ::_thesis: verum
end;
supposeA2: not S1[x] ; ::_thesis: f . x in the carrier of I[01]
then f . x = H2(x) by A1;
then f . x is Point of I[01] by A2, Th4;
hence f . x in the carrier of I[01] ; ::_thesis: verum
end;
end;
end;
then reconsider f = f as Function of I[01],I[01] by A1, FUNCT_2:3;
for t being Point of I[01] holds
( ( t <= 1 / 2 implies f . t = 0 ) & ( t > 1 / 2 implies f . t = (2 * t) - 1 ) ) by A1;
hence 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 ) ) ; ::_thesis: verum
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
let f1, f2 be Function of I[01],I[01]; ::_thesis: ( ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies f1 . t = 0 ) & ( t > 1 / 2 implies f1 . t = (2 * t) - 1 ) ) ) & ( for t being Point of I[01] holds
( ( t <= 1 / 2 implies f2 . t = 0 ) & ( t > 1 / 2 implies f2 . t = (2 * t) - 1 ) ) ) implies f1 = f2 )
assume that
A3: for t being Point of I[01] holds
( ( t <= 1 / 2 implies f1 . t = 0 ) & ( t > 1 / 2 implies f1 . t = (2 * t) - 1 ) ) and
A4: for t being Point of I[01] holds
( ( t <= 1 / 2 implies f2 . t = 0 ) & ( t > 1 / 2 implies f2 . t = (2 * t) - 1 ) ) ; ::_thesis: f1 = f2
for t being Point of I[01] holds f1 . t = f2 . t
proof
let t be Point of I[01]; ::_thesis: f1 . t = f2 . t
percases ( t <= 1 / 2 or t > 1 / 2 ) ;
supposeA5: t <= 1 / 2 ; ::_thesis: f1 . t = f2 . t
then f1 . t = 0 by A3
.= f2 . t by A4, A5 ;
hence f1 . t = f2 . t ; ::_thesis: verum
end;
supposeA6: t > 1 / 2 ; ::_thesis: f1 . t = f2 . t
then f1 . t = (2 * t) - 1 by A3
.= f2 . t by A4, A6 ;
hence f1 . t = f2 . t ; ::_thesis: verum
end;
end;
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
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
A1: 1 / 2 is Point of I[01] by BORSUK_1:43;
1 is Point of I[01] by BORSUK_1:43;
then reconsider B = [.(1 / 2),1.] as non empty compact Subset of I[01] by A1, BORSUK_4:24;
0 is Point of I[01] by BORSUK_1:43;
then reconsider A = [.0,(1 / 2).] as non empty compact Subset of I[01] by A1, BORSUK_4:24;
set T1 = I[01] | A;
set T2 = I[01] | B;
set T = I[01] ;
reconsider g = (I[01] | A) --> 0[01] as continuous Function of (I[01] | A),I[01] ;
I[01] | B = Closed-Interval-TSpace ((1 / 2),1) by TOPMETR:24;
then reconsider f = L[01] ((1 / 2),1,0,1) as continuous Function of (I[01] | B),I[01] by Th34, TOPMETR:20;
A2: for p being set st p in ([#] (I[01] | B)) /\ ([#] (I[01] | A)) holds
f . p = g . p
proof
let p be set ; ::_thesis: ( p in ([#] (I[01] | B)) /\ ([#] (I[01] | A)) implies f . p = g . p )
assume p in ([#] (I[01] | B)) /\ ([#] (I[01] | A)) ; ::_thesis: f . p = g . p
then p in [.0,(1 / 2).] /\ ([#] (I[01] | B)) by PRE_TOPC:def_5;
then p in [.0,(1 / 2).] /\ [.(1 / 2),1.] by PRE_TOPC:def_5;
then p in {(1 / 2)} by XXREAL_1:418;
then A3: p = 1 / 2 by TARSKI:def_1;
then p in [.0,(1 / 2).] by XXREAL_1:1;
then A4: p in the carrier of (I[01] | A) by PRE_TOPC:8;
f . p = (((1 - 0) / (1 - (1 / 2))) * ((1 / 2) - (1 / 2))) + 0 by A3, Th35
.= g . p by A4, BORSUK_1:def_14, FUNCOP_1:7 ;
hence f . p = g . p ; ::_thesis: verum
end;
A5: for x being Element of I[01] holds 2RP . x = (g +* f) . x
proof
let x be Element of I[01]; ::_thesis: 2RP . x = (g +* f) . x
A6: dom f = the carrier of (I[01] | B) by FUNCT_2:def_1
.= [.(1 / 2),1.] by PRE_TOPC:8 ;
percases ( x > 1 / 2 or x = 1 / 2 or x < 1 / 2 ) by XXREAL_0:1;
supposeA7: x > 1 / 2 ; ::_thesis: 2RP . x = (g +* f) . x
1 >= x by BORSUK_1:43;
then A8: f . x = (((1 - 0) / (1 - (1 / 2))) * (x - (1 / 2))) + 0 by A7, Th35
.= (2 * x) - 1 ;
x <= 1 by BORSUK_1:43;
then A9: x in dom f by A6, A7, XXREAL_1:1;
thus 2RP . x = (2 * x) - 1 by A7, Def6
.= (g +* f) . x by A9, A8, FUNCT_4:13 ; ::_thesis: verum
end;
supposeA10: x = 1 / 2 ; ::_thesis: 2RP . x = (g +* f) . x
then A11: x in dom f by A6, XXREAL_1:1;
thus 2RP . x = (((1 - 0) / (1 - (1 / 2))) * ((1 / 2) - (1 / 2))) + 0 by A10, Def6
.= f . x by A10, Th35
.= (g +* f) . x by A11, FUNCT_4:13 ; ::_thesis: verum
end;
supposeA12: x < 1 / 2 ; ::_thesis: 2RP . x = (g +* f) . x
x >= 0 by BORSUK_1:43;
then x in [.0,(1 / 2).] by A12, XXREAL_1:1;
then A13: x in the carrier of (I[01] | A) by PRE_TOPC:8;
A14: not x in dom f by A6, A12, XXREAL_1:1;
thus 2RP . x = 0 by A12, Def6
.= g . x by A13, BORSUK_1:def_14, FUNCOP_1:7
.= (g +* f) . x by A14, FUNCT_4:11 ; ::_thesis: verum
end;
end;
end;
([#] (I[01] | B)) \/ ([#] (I[01] | A)) = [.0,(1 / 2).] \/ ([#] (I[01] | B)) by PRE_TOPC:def_5
.= [.0,(1 / 2).] \/ [.(1 / 2),1.] by PRE_TOPC:def_5
.= [#] I[01] by BORSUK_1:40, XXREAL_1:174 ;
then ex h being Function of I[01],I[01] st
( h = g +* f & h is continuous ) by A2, BORSUK_2:1;
hence 2RP is continuous by A5, FUNCT_2:63; ::_thesis: verum
end;
end;
theorem Th48: :: BORSUK_6:48
( 2RP . 0 = 0 & 2RP . 1 = 1 )
proof
reconsider x = 0 , y = 1 as Point of I[01] by BORSUK_1:43;
thus 2RP . 0 = 2RP . x
.= 0 by Def6 ; ::_thesis: 2RP . 1 = 1
thus 2RP . 1 = (2 * y) - 1 by Def6
.= 1 ; ::_thesis: verum
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
deffunc H1( real number ) -> Element of REAL = (2 * $1) - 1;
deffunc H2( real number ) -> Element of REAL = $1 - (1 / 4);
deffunc H3( real number ) -> Element of REAL = (1 / 2) * $1;
defpred S1[ real number ] means $1 > 3 / 4;
defpred S2[ real number ] means ( $1 > 1 / 2 & $1 <= 3 / 4 );
defpred S3[ real number ] means $1 <= 1 / 2;
A1: for c being Element of I[01] holds
( S3[c] or S2[c] or S1[c] ) ;
A2: for c being Element of I[01] holds
( ( S3[c] implies not S2[c] ) & ( S3[c] implies not S1[c] ) & ( S2[c] implies not S1[c] ) ) by XXREAL_0:2;
consider f being Function such that
A3: ( dom f = the carrier of I[01] & ( for c being Element of I[01] holds
( ( S3[c] implies f . c = H3(c) ) & ( S2[c] implies f . c = H2(c) ) & ( S1[c] implies f . c = H1(c) ) ) ) ) from BORSUK_6:sch_1(A2, A1);
for x being set st x in the carrier of I[01] holds
f . x in the carrier of I[01]
proof
let x be set ; ::_thesis: ( x in the carrier of I[01] implies f . x in the carrier of I[01] )
assume x in the carrier of I[01] ; ::_thesis: f . x in the carrier of I[01]
then reconsider x = x as Point of I[01] ;
percases ( S3[x] or S2[x] or S1[x] ) ;
suppose S3[x] ; ::_thesis: f . x in the carrier of I[01]
then f . x = (1 / 2) * x by A3;
then f . x is Point of I[01] by Th6;
hence f . x in the carrier of I[01] ; ::_thesis: verum
end;
supposeA4: S2[x] ; ::_thesis: f . x in the carrier of I[01]
then f . x = H2(x) by A3;
then f . x is Point of I[01] by A4, Th7;
hence f . x in the carrier of I[01] ; ::_thesis: verum
end;
supposeA5: S1[x] ; ::_thesis: f . x in the carrier of I[01]
then f . x = (2 * x) - 1 by A3;
then f . x is Point of I[01] by A5, Th4, XXREAL_0:2;
hence f . x in the carrier of I[01] ; ::_thesis: verum
end;
end;
end;
then reconsider f = f as Function of I[01],I[01] by A3, FUNCT_2:3;
for x being Point of I[01] holds
( ( x <= 1 / 2 implies f . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies f . x = x - (1 / 4) ) & ( x > 3 / 4 implies f . x = (2 * x) - 1 ) ) by A3;
hence 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 ) ) ; ::_thesis: verum
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
let f1, f2 be Function of I[01],I[01]; ::_thesis: ( ( for x being Point of I[01] holds
( ( x <= 1 / 2 implies f1 . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies f1 . x = x - (1 / 4) ) & ( x > 3 / 4 implies f1 . x = (2 * x) - 1 ) ) ) & ( for x being Point of I[01] holds
( ( x <= 1 / 2 implies f2 . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies f2 . x = x - (1 / 4) ) & ( x > 3 / 4 implies f2 . x = (2 * x) - 1 ) ) ) implies f1 = f2 )
assume that
A6: for x being Point of I[01] holds
( ( x <= 1 / 2 implies f1 . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies f1 . x = x - (1 / 4) ) & ( x > 3 / 4 implies f1 . x = (2 * x) - 1 ) ) and
A7: for x being Point of I[01] holds
( ( x <= 1 / 2 implies f2 . x = (1 / 2) * x ) & ( x > 1 / 2 & x <= 3 / 4 implies f2 . x = x - (1 / 4) ) & ( x > 3 / 4 implies f2 . x = (2 * x) - 1 ) ) ; ::_thesis: f1 = f2
for x being Point of I[01] holds f1 . x = f2 . x
proof
let x be Point of I[01]; ::_thesis: f1 . x = f2 . x
percases ( x <= 1 / 2 or ( x > 1 / 2 & x <= 3 / 4 ) or x > 3 / 4 ) ;
supposeA8: x <= 1 / 2 ; ::_thesis: f1 . x = f2 . x
then f1 . x = (1 / 2) * x by A6
.= f2 . x by A7, A8 ;
hence f1 . x = f2 . x ; ::_thesis: verum
end;
supposeA9: ( x > 1 / 2 & x <= 3 / 4 ) ; ::_thesis: f1 . x = f2 . x
then f1 . x = x - (1 / 4) by A6
.= f2 . x by A7, A9 ;
hence f1 . x = f2 . x ; ::_thesis: verum
end;
supposeA10: x > 3 / 4 ; ::_thesis: f1 . x = f2 . x
then f1 . x = (2 * x) - 1 by A6
.= f2 . x by A7, A10 ;
hence f1 . x = f2 . x ; ::_thesis: verum
end;
end;
end;
hence f1 = f2 by FUNCT_2:63; ::_thesis: verum
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
A1: 1 / 2 is Point of I[01] by BORSUK_1:43;
A2: 3 / 4 is Point of I[01] by BORSUK_1:43;
then reconsider B = [.(1 / 2),(3 / 4).] as non empty compact Subset of I[01] by A1, BORSUK_4:24;
A3: 0 is Point of I[01] by BORSUK_1:43;
then reconsider A = [.0,(1 / 2).] as non empty compact Subset of I[01] by A1, BORSUK_4:24;
reconsider G = [.0,(3 / 4).] as non empty compact Subset of I[01] by A3, A2, BORSUK_4:24;
A4: 1 / 4 is Point of I[01] by BORSUK_1:43;
then reconsider E = [.(1 / 4),(1 / 2).] as non empty compact Subset of I[01] by A1, BORSUK_4:24;
reconsider F = [.0,(1 / 4).] as non empty compact Subset of I[01] by A3, A4, BORSUK_4:24;
A5: 1 is Point of I[01] by BORSUK_1:43;
then reconsider C = [.(3 / 4),1.] as non empty compact Subset of I[01] by A2, BORSUK_4:24;
reconsider D = [.(1 / 2),1.] as non empty compact Subset of I[01] by A1, A5, BORSUK_4:24;
set T = I[01] ;
set T1 = I[01] | A;
set T2 = I[01] | B;
set T3 = I[01] | C;
set T4 = I[01] | D;
set T5 = I[01] | E;
set T6 = I[01] | F;
set f = L[01] (0,(1 / 2),0,(1 / 4));
set g = L[01] ((1 / 2),(3 / 4),(1 / 4),(1 / 2));
set h = L[01] ((3 / 4),1,(1 / 2),1);
reconsider TT1 = I[01] | A, TT2 = I[01] | B as SubSpace of I[01] | G by TOPMETR:22, XXREAL_1:34;
( Closed-Interval-TSpace ((3 / 4),1) = I[01] | C & Closed-Interval-TSpace ((1 / 2),1) = I[01] | D ) by TOPMETR:24;
then reconsider h = L[01] ((3 / 4),1,(1 / 2),1) as continuous Function of (I[01] | C),(I[01] | D) by Th34;
reconsider h = h as continuous Function of (I[01] | C),I[01] by JORDAN6:3;
A6: for x being Point of (I[01] | C) holds h . x = (2 * x) - 1
proof
let x be Point of (I[01] | C); ::_thesis: h . x = (2 * x) - 1
x in the carrier of (I[01] | C) ;
then x in C by PRE_TOPC:8;
then ( 3 / 4 <= x & x <= 1 ) by XXREAL_1:1;
then h . x = (((1 - (1 / 2)) / (1 - (3 / 4))) * (x - (3 / 4))) + (1 / 2) by Th35
.= (2 * x) - 1 ;
hence h . x = (2 * x) - 1 ; ::_thesis: verum
end;
( Closed-Interval-TSpace (0,(1 / 4)) = I[01] | F & Closed-Interval-TSpace (0,(1 / 2)) = I[01] | A ) by TOPMETR:24;
then reconsider f = L[01] (0,(1 / 2),0,(1 / 4)) as continuous Function of (I[01] | A),(I[01] | F) by Th34;
( Closed-Interval-TSpace ((1 / 4),(1 / 2)) = I[01] | E & Closed-Interval-TSpace ((1 / 2),(3 / 4)) = I[01] | B ) by TOPMETR:24;
then reconsider g = L[01] ((1 / 2),(3 / 4),(1 / 4),(1 / 2)) as continuous Function of (I[01] | B),(I[01] | E) by Th34;
reconsider g = g as continuous Function of (I[01] | B),I[01] by JORDAN6:3;
reconsider f = f as continuous Function of (I[01] | A),I[01] by JORDAN6:3;
set f1 = f;
set g1 = g;
A7: for x being Point of (I[01] | B) holds g . x = x - (1 / 4)
proof
let x be Point of (I[01] | B); ::_thesis: g . x = x - (1 / 4)
x in the carrier of (I[01] | B) ;
then x in B by PRE_TOPC:8;
then ( 1 / 2 <= x & x <= 3 / 4 ) by XXREAL_1:1;
then g . x = ((((1 / 2) - (1 / 4)) / ((3 / 4) - (1 / 2))) * (x - (1 / 2))) + (1 / 4) by Th35
.= x - (1 / 4) ;
hence g . x = x - (1 / 4) ; ::_thesis: verum
end;
A8: ([#] TT1) /\ ([#] TT2) = A /\ ([#] TT2) by PRE_TOPC:def_5
.= A /\ B by PRE_TOPC:def_5
.= {(1 / 2)} by XXREAL_1:418 ;
A9: for p being set st p in ([#] TT1) /\ ([#] TT2) holds
f . p = g . p
proof
let p be set ; ::_thesis: ( p in ([#] TT1) /\ ([#] TT2) implies f . p = g . p )
assume p in ([#] TT1) /\ ([#] TT2) ; ::_thesis: f . p = g . p
then A10: p = 1 / 2 by A8, TARSKI:def_1;
then reconsider p = p as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
f . p = ((((1 / 4) - 0) / ((1 / 2) - 0)) * (p - 0)) + 0 by A10, Th35
.= ((((1 / 2) - (1 / 4)) / ((3 / 4) - (1 / 2))) * (p - (1 / 2))) + (1 / 4) by A10
.= g . p by A10, Th35 ;
hence f . p = g . p ; ::_thesis: verum
end;
([#] TT1) \/ ([#] TT2) = A \/ ([#] TT2) by PRE_TOPC:def_5
.= A \/ B by PRE_TOPC:def_5
.= [.0,(3 / 4).] by XXREAL_1:174
.= [#] (I[01] | G) by PRE_TOPC:def_5 ;
then consider FF being Function of (I[01] | G),I[01] such that
A11: FF = f +* g and
A12: FF is continuous by A9, BORSUK_2:1;
A13: ([#] (I[01] | G)) /\ ([#] (I[01] | C)) = G /\ ([#] (I[01] | C)) by PRE_TOPC:def_5
.= G /\ C by PRE_TOPC:def_5
.= {(3 / 4)} by XXREAL_1:418 ;
A14: for p being set st p in ([#] (I[01] | G)) /\ ([#] (I[01] | C)) holds
FF . p = h . p
proof
let p be set ; ::_thesis: ( p in ([#] (I[01] | G)) /\ ([#] (I[01] | C)) implies FF . p = h . p )
assume p in ([#] (I[01] | G)) /\ ([#] (I[01] | C)) ; ::_thesis: FF . p = h . p
then A15: p = 3 / 4 by A13, TARSKI:def_1;
then reconsider p = p as Point of I[01] by BORSUK_1:43;
p in [.(1 / 2),(3 / 4).] by A15, XXREAL_1:1;
then p in the carrier of (I[01] | B) by PRE_TOPC:8;
then p in dom g by FUNCT_2:def_1;
then FF . p = g . p by A11, FUNCT_4:13
.= 1 / 2 by A15, Th33
.= h . p by A15, Th33 ;
hence FF . p = h . p ; ::_thesis: verum
end;
([#] (I[01] | G)) \/ ([#] (I[01] | C)) = G \/ ([#] (I[01] | C)) by PRE_TOPC:def_5
.= G \/ C by PRE_TOPC:def_5
.= [#] I[01] by BORSUK_1:40, XXREAL_1:174 ;
then consider HH being Function of I[01],I[01] such that
A16: HH = FF +* h and
A17: HH is continuous by A12, A14, BORSUK_2:1;
A18: for x being Point of (I[01] | A) holds f . x = (1 / 2) * x
proof
let x be Point of (I[01] | A); ::_thesis: f . x = (1 / 2) * x
x in the carrier of (I[01] | A) ;
then x in A by PRE_TOPC:8;
then ( 0 <= x & x <= 1 / 2 ) by XXREAL_1:1;
then f . x = ((((1 / 4) - 0) / ((1 / 2) - 0)) * (x - 0)) + 0 by Th35
.= (1 / 2) * x ;
hence f . x = (1 / 2) * x ; ::_thesis: verum
end;
for x being Element of I[01] holds HH . x = 3RP . x
proof
let x be Element of I[01]; ::_thesis: HH . x = 3RP . x
A19: 0 <= x by BORSUK_1:43;
A20: x <= 1 by BORSUK_1:43;
percases ( x < 1 / 2 or x = 1 / 2 or ( x > 1 / 2 & x < 3 / 4 ) or x = 3 / 4 or x > 3 / 4 ) by XXREAL_0:1;
supposeA21: x < 1 / 2 ; ::_thesis: HH . x = 3RP . x
then not x in [.(1 / 2),(3 / 4).] by XXREAL_1:1;
then not x in the carrier of (I[01] | B) by PRE_TOPC:8;
then A22: not x in dom g ;
x in [.0,(1 / 2).] by A19, A21, XXREAL_1:1;
then A23: x is Point of (I[01] | A) by PRE_TOPC:8;
x < 3 / 4 by A21, XXREAL_0:2;
then not x in [.(3 / 4),1.] by XXREAL_1:1;
then not x in the carrier of (I[01] | C) by PRE_TOPC:8;
then not x in dom h ;
then HH . x = FF . x by A16, FUNCT_4:11
.= f . x by A11, A22, FUNCT_4:11
.= (1 / 2) * x by A18, A23
.= 3RP . x by A21, Def7 ;
hence HH . x = 3RP . x ; ::_thesis: verum
end;
supposeA24: x = 1 / 2 ; ::_thesis: HH . x = 3RP . x
then x in [.(1 / 2),(3 / 4).] by XXREAL_1:1;
then x in the carrier of (I[01] | B) by PRE_TOPC:8;
then A25: x in dom g by FUNCT_2:def_1;
not x in [.(3 / 4),1.] by A24, XXREAL_1:1;
then not x in the carrier of (I[01] | C) by PRE_TOPC:8;
then not x in dom h ;
then HH . x = FF . x by A16, FUNCT_4:11
.= g . x by A11, A25, FUNCT_4:13
.= (1 / 2) * x by A24, Th33
.= 3RP . x by A24, Def7 ;
hence HH . x = 3RP . x ; ::_thesis: verum
end;
supposeA26: ( x > 1 / 2 & x < 3 / 4 ) ; ::_thesis: HH . x = 3RP . x
then x in [.(1 / 2),(3 / 4).] by XXREAL_1:1;
then A27: x in the carrier of (I[01] | B) by PRE_TOPC:8;
then A28: x in dom g by FUNCT_2:def_1;
not x in [.(3 / 4),1.] by A26, XXREAL_1:1;
then not x in the carrier of (I[01] | C) by PRE_TOPC:8;
then not x in dom h ;
then HH . x = FF . x by A16, FUNCT_4:11
.= g . x by A11, A28, FUNCT_4:13
.= x - (1 / 4) by A7, A27
.= 3RP . x by A26, Def7 ;
hence HH . x = 3RP . x ; ::_thesis: verum
end;
supposeA29: x = 3 / 4 ; ::_thesis: HH . x = 3RP . x
then x in [.(3 / 4),1.] by XXREAL_1:1;
then x in the carrier of (I[01] | C) by PRE_TOPC:8;
then x in dom h by FUNCT_2:def_1;
then HH . x = h . x by A16, FUNCT_4:13
.= x - (1 / 4) by A29, Th33
.= 3RP . x by A29, Def7 ;
hence HH . x = 3RP . x ; ::_thesis: verum
end;
supposeA30: x > 3 / 4 ; ::_thesis: HH . x = 3RP . x
then x in [.(3 / 4),1.] by A20, XXREAL_1:1;
then A31: x in the carrier of (I[01] | C) by PRE_TOPC:8;
then x in dom h by FUNCT_2:def_1;
then HH . x = h . x by A16, FUNCT_4:13
.= (2 * x) - 1 by A6, A31
.= 3RP . x by A30, Def7 ;
hence HH . x = 3RP . x ; ::_thesis: verum
end;
end;
end;
hence 3RP is continuous by A17, FUNCT_2:63; ::_thesis: verum
end;
end;
theorem Th49: :: BORSUK_6:49
( 3RP . 0 = 0 & 3RP . 1 = 1 )
proof
0 is Point of I[01] by BORSUK_1:43;
hence 3RP . 0 = (1 / 2) * 0 by Def7
.= 0 ;
::_thesis: 3RP . 1 = 1
1 is Point of I[01] by BORSUK_1:43;
hence 3RP . 1 = (2 * 1) - 1 by Def7
.= 1 ;
::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
let P be Path of a,b; ::_thesis: for Q being constant Path of b,b st a,b are_connected holds
RePar (P,1RP) = P + Q
let Q be constant Path of b,b; ::_thesis: ( a,b are_connected implies RePar (P,1RP) = P + Q )
set f = RePar (P,1RP);
set g = P + Q;
assume A1: a,b are_connected ; ::_thesis: RePar (P,1RP) = P + Q
A2: b,b are_connected ;
for p being Element of I[01] holds (RePar (P,1RP)) . p = (P + Q) . p
proof
0 in the carrier of I[01] by BORSUK_1:43;
then A3: 0 in dom Q by FUNCT_2:def_1;
let p be Element of I[01]; ::_thesis: (RePar (P,1RP)) . p = (P + Q) . p
p in the carrier of I[01] ;
then A4: p in dom 1RP by FUNCT_2:def_1;
A5: (RePar (P,1RP)) . p = (P * 1RP) . p by A1, Def4, Th47
.= P . (1RP . p) by A4, FUNCT_1:13 ;
percases ( p <= 1 / 2 or p > 1 / 2 ) ;
supposeA6: p <= 1 / 2 ; ::_thesis: (RePar (P,1RP)) . p = (P + Q) . p
then (RePar (P,1RP)) . p = P . (2 * p) by A5, Def5
.= (P + Q) . p by A1, A6, BORSUK_2:def_5 ;
hence (RePar (P,1RP)) . p = (P + Q) . p ; ::_thesis: verum
end;
supposeA7: p > 1 / 2 ; ::_thesis: (RePar (P,1RP)) . p = (P + Q) . p
then (2 * p) - 1 is Point of I[01] by Th4;
then (2 * p) - 1 in the carrier of I[01] ;
then A8: (2 * p) - 1 in dom Q by FUNCT_2:def_1;
(RePar (P,1RP)) . p = P . 1 by A5, A7, Def5
.= b by A1, BORSUK_2:def_2
.= Q . 0 by A2, BORSUK_2:def_2
.= Q . ((2 * p) - 1) by A3, A8, FUNCT_1:def_10
.= (P + Q) . p by A1, A7, BORSUK_2:def_5 ;
hence (RePar (P,1RP)) . p = (P + Q) . p ; ::_thesis: verum
end;
end;
end;
hence RePar (P,1RP) = P + Q by FUNCT_2:63; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
let P be Path of a,b; ::_thesis: for Q being constant Path of a,a st a,b are_connected holds
RePar (P,2RP) = Q + P
let Q be constant Path of a,a; ::_thesis: ( a,b are_connected implies RePar (P,2RP) = Q + P )
assume A1: a,b are_connected ; ::_thesis: RePar (P,2RP) = Q + P
set f = RePar (P,2RP);
set g = Q + P;
A2: a,a are_connected ;
for p being Element of I[01] holds (RePar (P,2RP)) . p = (Q + P) . p
proof
0 in the carrier of I[01] by BORSUK_1:43;
then A3: 0 in dom Q by FUNCT_2:def_1;
let p be Element of I[01]; ::_thesis: (RePar (P,2RP)) . p = (Q + P) . p
p in the carrier of I[01] ;
then A4: p in dom 2RP by FUNCT_2:def_1;
A5: (RePar (P,2RP)) . p = (P * 2RP) . p by A1, Def4, Th48
.= P . (2RP . p) by A4, FUNCT_1:13 ;
percases ( p <= 1 / 2 or p > 1 / 2 ) ;
supposeA6: p <= 1 / 2 ; ::_thesis: (RePar (P,2RP)) . p = (Q + P) . p
then 2 * p is Point of I[01] by Th3;
then 2 * p in the carrier of I[01] ;
then A7: 2 * p in dom Q by FUNCT_2:def_1;
(RePar (P,2RP)) . p = P . 0 by A5, A6, Def6
.= a by A1, BORSUK_2:def_2
.= Q . 0 by A2, BORSUK_2:def_2
.= Q . (2 * p) by A3, A7, FUNCT_1:def_10
.= (Q + P) . p by A1, A6, BORSUK_2:def_5 ;
hence (RePar (P,2RP)) . p = (Q + P) . p ; ::_thesis: verum
end;
supposeA8: p > 1 / 2 ; ::_thesis: (RePar (P,2RP)) . p = (Q + P) . p
then (RePar (P,2RP)) . p = P . ((2 * p) - 1) by A5, Def6
.= (Q + P) . p by A1, A8, BORSUK_2:def_5 ;
hence (RePar (P,2RP)) . p = (Q + P) . p ; ::_thesis: verum
end;
end;
end;
hence RePar (P,2RP) = Q + P by FUNCT_2:63; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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)
let a, b, c, d be Point of T; ::_thesis: 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)
let P be Path of a,b; ::_thesis: 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)
let Q be Path of b,c; ::_thesis: 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)
let R be Path of c,d; ::_thesis: ( a,b are_connected & b,c are_connected & c,d are_connected implies RePar (((P + Q) + R),3RP) = P + (Q + R) )
assume that
A1: a,b are_connected and
A2: b,c are_connected and
A3: c,d are_connected ; ::_thesis: RePar (((P + Q) + R),3RP) = P + (Q + R)
set F = (P + Q) + R;
set f = RePar (((P + Q) + R),3RP);
set g = P + (Q + R);
A4: a,c are_connected by A1, A2, Th42;
A5: b,d are_connected by A2, A3, Th42;
for x being Point of I[01] holds (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x
proof
let x be Point of I[01]; ::_thesis: (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x
x in the carrier of I[01] ;
then A6: x in dom 3RP by FUNCT_2:def_1;
A7: (RePar (((P + Q) + R),3RP)) . x = (((P + Q) + R) * 3RP) . x by A3, A4, Def4, Th42, Th49
.= ((P + Q) + R) . (3RP . x) by A6, FUNCT_1:13 ;
percases ( x <= 1 / 2 or ( x > 1 / 2 & x <= 3 / 4 ) or x > 3 / 4 ) ;
supposeA8: x <= 1 / 2 ; ::_thesis: (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x
reconsider y = (1 / 2) * x as Point of I[01] by Th6;
(1 / 2) * x <= (1 / 2) * (1 / 2) by A8, XREAL_1:64;
then A9: y <= 1 / 2 by XXREAL_0:2;
reconsider z = 2 * y as Point of I[01] ;
(RePar (((P + Q) + R),3RP)) . x = ((P + Q) + R) . y by A7, A8, Def7
.= (P + Q) . z by A3, A4, A9, BORSUK_2:def_5
.= P . (2 * x) by A1, A2, A8, BORSUK_2:def_5
.= (P + (Q + R)) . x by A1, A5, A8, BORSUK_2:def_5 ;
hence (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x ; ::_thesis: verum
end;
supposeA10: ( x > 1 / 2 & x <= 3 / 4 ) ; ::_thesis: (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x
then A11: (1 / 2) - (1 / 4) <= x - (1 / 4) by XREAL_1:9;
A12: x - (1 / 4) <= (3 / 4) - (1 / 4) by A10, XREAL_1:9;
then x - (1 / 4) <= 1 by XXREAL_0:2;
then reconsider y = x - (1 / 4) as Point of I[01] by A11, BORSUK_1:43;
reconsider z = 2 * y as Point of I[01] by A12, Th3;
A13: 2 * y >= 2 * (1 / 4) by A11, XREAL_1:64;
reconsider w = (2 * x) - 1 as Point of I[01] by A10, Th4;
2 * x <= 2 * (3 / 4) by A10, XREAL_1:64;
then A14: (2 * x) - 1 <= (3 / 2) - 1 by XREAL_1:9;
(RePar (((P + Q) + R),3RP)) . x = ((P + Q) + R) . y by A7, A10, Def7
.= (P + Q) . z by A3, A4, A12, BORSUK_2:def_5
.= Q . ((2 * z) - 1) by A1, A2, A13, BORSUK_2:def_5
.= Q . (2 * w)
.= (Q + R) . w by A2, A3, A14, BORSUK_2:def_5
.= (P + (Q + R)) . x by A1, A5, A10, BORSUK_2:def_5 ;
hence (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x ; ::_thesis: verum
end;
supposeA15: x > 3 / 4 ; ::_thesis: (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x
then reconsider w = (2 * x) - 1 as Point of I[01] by Th4, XXREAL_0:2;
2 * x > 2 * (3 / 4) by A15, XREAL_1:68;
then A16: (2 * x) - 1 > (2 * (3 / 4)) - 1 by XREAL_1:14;
reconsider y = (2 * x) - 1 as Point of I[01] by A15, Th4, XXREAL_0:2;
A17: x > 1 / 2 by A15, XXREAL_0:2;
(RePar (((P + Q) + R),3RP)) . x = ((P + Q) + R) . y by A7, A15, Def7
.= R . ((2 * y) - 1) by A3, A4, A16, BORSUK_2:def_5
.= (Q + R) . w by A2, A3, A16, BORSUK_2:def_5
.= (P + (Q + R)) . x by A1, A5, A17, BORSUK_2:def_5 ;
hence (RePar (((P + Q) + R),3RP)) . x = (P + (Q + R)) . x ; ::_thesis: verum
end;
end;
end;
hence RePar (((P + Q) + R),3RP) = P + (Q + R) by FUNCT_2:63; ::_thesis: verum
end;
begin
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
defpred S1[ set ] means ex a, b being Point of I[01] st
( $1 = [a,b] & b <= 1 - (2 * a) );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in the carrier of [:I[01],I[01]:] & S1[x] ) ) from XBOOLE_0:sch_1();
X c= the carrier of [:I[01],I[01]:]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in the carrier of [:I[01],I[01]:] )
assume x in X ; ::_thesis: x in the carrier of [:I[01],I[01]:]
hence x in the carrier of [:I[01],I[01]:] by A1; ::_thesis: verum
end;
then reconsider X = X as Subset of [:I[01],I[01]:] ;
take X ; ::_thesis: for x being set holds
( x in X iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) )
thus for x being set holds
( x in X iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) by A1; ::_thesis: verum
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
let X1, X2 be Subset of [:I[01],I[01]:]; ::_thesis: ( ( for x being set holds
( x in X1 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) ) & ( for x being set holds
( x in X2 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) ) implies X1 = X2 )
assume that
A2: for x being set holds
( x in X1 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) and
A3: for x being set holds
( x in X2 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) ) ; ::_thesis: X1 = X2
X1 = X2
proof
thus X1 c= X2 :: according to XBOOLE_0:def_10 ::_thesis: X2 c= X1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X1 or x in X2 )
assume x in X1 ; ::_thesis: x in X2
then ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) by A2;
hence x in X2 by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X2 or x in X1 )
assume x in X2 ; ::_thesis: x in X1
then ex a, b being Point of I[01] st
( x = [a,b] & b <= 1 - (2 * a) ) by A3;
hence x in X1 by A2; ::_thesis: verum
end;
hence X1 = X2 ; ::_thesis: verum
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
defpred S1[ set ] means ex a, b being Point of I[01] st
( $1 = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in the carrier of [:I[01],I[01]:] & S1[x] ) ) from XBOOLE_0:sch_1();
X c= the carrier of [:I[01],I[01]:]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in the carrier of [:I[01],I[01]:] )
assume x in X ; ::_thesis: x in the carrier of [:I[01],I[01]:]
hence x in the carrier of [:I[01],I[01]:] by A1; ::_thesis: verum
end;
then reconsider X = X as Subset of [:I[01],I[01]:] ;
take X ; ::_thesis: for x being set holds
( x in X iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) )
thus for x being set holds
( x in X iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) by A1; ::_thesis: verum
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
let X1, X2 be Subset of [:I[01],I[01]:]; ::_thesis: ( ( for x being set holds
( x in X1 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 X2 iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) ) implies X1 = X2 )
assume that
A2: for x being set holds
( x in X1 iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) and
A3: for x being set holds
( x in X2 iff ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) ) ; ::_thesis: X1 = X2
X1 = X2
proof
thus X1 c= X2 :: according to XBOOLE_0:def_10 ::_thesis: X2 c= X1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X1 or x in X2 )
assume x in X1 ; ::_thesis: x in X2
then ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) by A2;
hence x in X2 by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X2 or x in X1 )
assume x in X2 ; ::_thesis: x in X1
then ex a, b being Point of I[01] st
( x = [a,b] & b >= 1 - (2 * a) & b >= (2 * a) - 1 ) by A3;
hence x in X1 by A2; ::_thesis: verum
end;
hence X1 = X2 ; ::_thesis: verum
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
defpred S1[ set ] means ex a, b being Point of I[01] st
( $1 = [a,b] & b <= (2 * a) - 1 );
consider X being set such that
A1: for x being set holds
( x in X iff ( x in the carrier of [:I[01],I[01]:] & S1[x] ) ) from XBOOLE_0:sch_1();
X c= the carrier of [:I[01],I[01]:]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X or x in the carrier of [:I[01],I[01]:] )
assume x in X ; ::_thesis: x in the carrier of [:I[01],I[01]:]
hence x in the carrier of [:I[01],I[01]:] by A1; ::_thesis: verum
end;
then reconsider X = X as Subset of [:I[01],I[01]:] ;
take X ; ::_thesis: for x being set holds
( x in X iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) )
thus for x being set holds
( x in X iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) by A1; ::_thesis: verum
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
let X1, X2 be Subset of [:I[01],I[01]:]; ::_thesis: ( ( for x being set holds
( x in X1 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) ) & ( for x being set holds
( x in X2 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) ) implies X1 = X2 )
assume that
A2: for x being set holds
( x in X1 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) and
A3: for x being set holds
( x in X2 iff ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) ) ; ::_thesis: X1 = X2
X1 = X2
proof
thus X1 c= X2 :: according to XBOOLE_0:def_10 ::_thesis: X2 c= X1
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X1 or x in X2 )
assume x in X1 ; ::_thesis: x in X2
then ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) by A2;
hence x in X2 by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in X2 or x in X1 )
assume x in X2 ; ::_thesis: x in X1
then ex a, b being Point of I[01] st
( x = [a,b] & b <= (2 * a) - 1 ) by A3;
hence x in X1 by A2; ::_thesis: verum
end;
hence X1 = X2 ; ::_thesis: verum
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
set P = { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ;
thus IAA c= { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } c= IAA
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in IAA or x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } )
assume A1: x in IAA ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) }
then reconsider x9 = x as Point of [:I[01],I[01]:] ;
consider a, b being Point of I[01] such that
A2: x = [a,b] and
A3: b <= 1 - (2 * a) by A1, Def8;
( x9 `1 = a & x9 `2 = b ) by A2, MCART_1:7;
hence x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } or x in IAA )
assume x in { p where p is Point of [:I[01],I[01]:] : p `2 <= 1 - (2 * (p `1)) } ; ::_thesis: x in IAA
then consider p being Point of [:I[01],I[01]:] such that
A4: p = x and
A5: p `2 <= 1 - (2 * (p `1)) ;
x in the carrier of [:I[01],I[01]:] by A4;
then x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A6: x = [(x `1),(x `2)] by MCART_1:21;
( p `1 is Point of I[01] & p `2 is Point of I[01] ) by Th27;
hence x in IAA by A4, A5, A6, Def8; ::_thesis: verum
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
set P = { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } ;
thus IBB c= { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } c= IBB
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in IBB or x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } )
assume A1: x in IBB ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) }
then reconsider x9 = x as Point of [:I[01],I[01]:] ;
consider a, b being Point of I[01] such that
A2: x = [a,b] and
A3: ( b >= 1 - (2 * a) & b >= (2 * a) - 1 ) by A1, Def9;
( x9 `1 = a & x9 `2 = b ) by A2, MCART_1:7;
hence x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } or x in IBB )
assume x in { p where p is Point of [:I[01],I[01]:] : ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) } ; ::_thesis: x in IBB
then consider p being Point of [:I[01],I[01]:] such that
A4: p = x and
A5: ( p `2 >= 1 - (2 * (p `1)) & p `2 >= (2 * (p `1)) - 1 ) ;
x in the carrier of [:I[01],I[01]:] by A4;
then x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A6: x = [(x `1),(x `2)] by MCART_1:21;
( p `1 is Point of I[01] & p `2 is Point of I[01] ) by Th27;
hence x in IBB by A4, A5, A6, Def9; ::_thesis: verum
end;
theorem Th55: :: BORSUK_6:55
ICC = { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 }
proof
set P = { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } ;
thus ICC c= { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } c= ICC
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ICC or x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } )
assume A1: x in ICC ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 }
then reconsider x9 = x as Point of [:I[01],I[01]:] ;
consider a, b being Point of I[01] such that
A2: x = [a,b] and
A3: b <= (2 * a) - 1 by A1, Def10;
( x9 `1 = a & x9 `2 = b ) by A2, MCART_1:7;
hence x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } by A3; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } or x in ICC )
assume x in { p where p is Point of [:I[01],I[01]:] : p `2 <= (2 * (p `1)) - 1 } ; ::_thesis: x in ICC
then consider p being Point of [:I[01],I[01]:] such that
A4: p = x and
A5: p `2 <= (2 * (p `1)) - 1 ;
x in the carrier of [:I[01],I[01]:] by A4;
then x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A6: x = [(x `1),(x `2)] by MCART_1:21;
( p `1 is Point of I[01] & p `2 is Point of I[01] ) by Th27;
hence x in ICC by A4, A5, A6, Def10; ::_thesis: verum
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
thus (IAA \/ IBB) \/ ICC c= [:[.0,1.],[.0,1.]:] by Th1; :: according to XBOOLE_0:def_10 ::_thesis: [:[.0,1.],[.0,1.]:] c= (IAA \/ IBB) \/ ICC
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in [:[.0,1.],[.0,1.]:] or x in (IAA \/ IBB) \/ ICC )
assume A1: x in [:[.0,1.],[.0,1.]:] ; ::_thesis: x in (IAA \/ IBB) \/ ICC
then reconsider q = x `1 , p = x `2 as Point of I[01] by BORSUK_1:40, MCART_1:10;
A2: x = [q,p] by A1, MCART_1:21;
( x in IAA or x in IBB or x in ICC )
proof
percases ( p >= 1 - (2 * q) or p < 1 - (2 * q) ) ;
supposeA3: p >= 1 - (2 * q) ; ::_thesis: ( x in IAA or x in IBB or x in ICC )
now__::_thesis:_(_x_in_IAA_or_x_in_IBB_or_x_in_ICC_)
percases ( p >= (2 * q) - 1 or p < (2 * q) - 1 ) ;
suppose p >= (2 * q) - 1 ; ::_thesis: ( x in IAA or x in IBB or x in ICC )
hence ( x in IAA or x in IBB or x in ICC ) by A2, A3, Def9; ::_thesis: verum
end;
suppose p < (2 * q) - 1 ; ::_thesis: ( x in IAA or x in IBB or x in ICC )
hence ( x in IAA or x in IBB or x in ICC ) by A2, Def10; ::_thesis: verum
end;
end;
end;
hence ( x in IAA or x in IBB or x in ICC ) ; ::_thesis: verum
end;
suppose p < 1 - (2 * q) ; ::_thesis: ( x in IAA or x in IBB or x in ICC )
hence ( x in IAA or x in IBB or x in ICC ) by A2, Def8; ::_thesis: verum
end;
end;
end;
then ( x in IAA \/ IBB or x in ICC ) by XBOOLE_0:def_3;
hence x in (IAA \/ IBB) \/ ICC by XBOOLE_0:def_3; ::_thesis: verum
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
set KK = { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) } ;
thus IAA /\ IBB c= { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) } c= IAA /\ IBB
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in IAA /\ IBB or x in { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) } )
assume A1: x in IAA /\ IBB ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) }
then x in IAA by XBOOLE_0:def_4;
then consider p being Point of [:I[01],I[01]:] such that
A2: x = p and
A3: p `2 <= 1 - (2 * (p `1)) by Th53;
x in IBB by A1, XBOOLE_0:def_4;
then ex q being Point of [:I[01],I[01]:] st
( x = q & q `2 >= 1 - (2 * (q `1)) & q `2 >= (2 * (q `1)) - 1 ) by Th54;
then p `2 = 1 - (2 * (p `1)) by A2, A3, XXREAL_0:1;
hence x in { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) } by A2; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) } or x in IAA /\ IBB )
assume x in { p where p is Point of [:I[01],I[01]:] : p `2 = 1 - (2 * (p `1)) } ; ::_thesis: x in IAA /\ IBB
then consider p being Point of [:I[01],I[01]:] such that
A4: p = x and
A5: p `2 = 1 - (2 * (p `1)) ;
x in the carrier of [:I[01],I[01]:] by A4;
then A6: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A7: ( x = [(p `1),(p `2)] & p `1 in the carrier of I[01] ) by A4, MCART_1:10, MCART_1:21;
A8: p `2 in the carrier of I[01] by A4, A6, MCART_1:10;
then A9: x in IAA by A5, A7, Def8;
1 - (2 * (p `1)) >= 0 by A5, A8, BORSUK_1:43;
then 0 + (2 * (p `1)) <= 1 by XREAL_1:19;
then (2 * (p `1)) / 2 <= 1 / 2 by XREAL_1:72;
then ( (2 * (p `1)) - 1 <= 0 & 0 <= 1 - (2 * (p `1)) ) by XREAL_1:217, XREAL_1:218;
then x in IBB by A5, A7, A8, Def9;
hence x in IAA /\ IBB by A9, XBOOLE_0:def_4; ::_thesis: verum
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
set KK = { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 } ;
thus ICC /\ IBB c= { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 } :: according to XBOOLE_0:def_10 ::_thesis: { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 } c= ICC /\ IBB
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in ICC /\ IBB or x in { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 } )
assume A1: x in ICC /\ IBB ; ::_thesis: x in { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 }
then x in ICC by XBOOLE_0:def_4;
then consider p being Point of [:I[01],I[01]:] such that
A2: x = p and
A3: p `2 <= (2 * (p `1)) - 1 by Th55;
x in IBB by A1, XBOOLE_0:def_4;
then ex q being Point of [:I[01],I[01]:] st
( x = q & q `2 >= 1 - (2 * (q `1)) & q `2 >= (2 * (q `1)) - 1 ) by Th54;
then p `2 = (2 * (p `1)) - 1 by A2, A3, XXREAL_0:1;
hence x in { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 } by A2; ::_thesis: verum
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 } or x in ICC /\ IBB )
assume x in { p where p is Point of [:I[01],I[01]:] : p `2 = (2 * (p `1)) - 1 } ; ::_thesis: x in ICC /\ IBB
then consider p being Point of [:I[01],I[01]:] such that
A4: p = x and
A5: p `2 = (2 * (p `1)) - 1 ;
x in the carrier of [:I[01],I[01]:] by A4;
then A6: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A7: ( x = [(p `1),(p `2)] & p `1 in the carrier of I[01] ) by A4, MCART_1:10, MCART_1:21;
A8: p `2 in the carrier of I[01] by A4, A6, MCART_1:10;
then A9: x in ICC by A5, A7, Def10;
(2 * (p `1)) - 1 >= 0 by A5, A8, BORSUK_1:43;
then 2 * (p `1) >= 0 + 1 by XREAL_1:19;
then (2 * (p `1)) / 2 >= 1 / 2 by XREAL_1:72;
then ( (2 * (p `1)) - 1 >= 0 & 0 >= 1 - (2 * (p `1)) ) by XREAL_1:219, XREAL_1:220;
then x in IBB by A5, A7, A8, Def9;
hence x in ICC /\ IBB by A9, XBOOLE_0:def_4; ::_thesis: verum
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
set GG = [:I[01],I[01]:];
let x be Point of [:I[01],I[01]:]; ::_thesis: ( x in IAA implies x `1 <= 1 / 2 )
assume x in IAA ; ::_thesis: x `1 <= 1 / 2
then consider a, b being Point of I[01] such that
A1: x = [a,b] and
A2: b <= 1 - (2 * a) by Def8;
b >= 0 by BORSUK_1:43;
then 1 >= 0 + (2 * a) by A2, XREAL_1:19;
then A3: 1 / 2 >= (2 * a) / 2 by XREAL_1:72;
x = [(x `1),(x `2)] by A1, MCART_1:8;
hence x `1 <= 1 / 2 by A1, A3, XTUPLE_0:1; ::_thesis: verum
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
set GG = [:I[01],I[01]:];
let x be Point of [:I[01],I[01]:]; ::_thesis: ( x in ICC implies x `1 >= 1 / 2 )
assume x in ICC ; ::_thesis: x `1 >= 1 / 2
then consider a, b being Point of I[01] such that
A1: x = [a,b] and
A2: b <= (2 * a) - 1 by Def10;
b >= 0 by BORSUK_1:43;
then 0 + 1 <= 2 * a by A2, XREAL_1:19;
then A3: 1 / 2 <= (2 * a) / 2 by XREAL_1:72;
x = [(x `1),(x `2)] by A1, MCART_1:8;
hence x `1 >= 1 / 2 by A1, A3, XTUPLE_0:1; ::_thesis: verum
end;
theorem Th61: :: BORSUK_6:61
for x being Point of I[01] holds [0,x] in IAA
proof
let x be Point of I[01]; ::_thesis: [0,x] in IAA
( 0 is Point of I[01] & x <= 1 - (2 * 0) ) by BORSUK_1:43;
hence [0,x] in IAA by Def8; ::_thesis: verum
end;
theorem Th62: :: BORSUK_6:62
for s being set st [0,s] in IBB holds
s = 1
proof
let s be set ; ::_thesis: ( [0,s] in IBB implies s = 1 )
assume [0,s] in IBB ; ::_thesis: s = 1
then consider a, b being Point of I[01] such that
A1: [0,s] = [a,b] and
A2: b >= 1 - (2 * a) and
b >= (2 * a) - 1 by Def9;
A3: b <= 1 by BORSUK_1:43;
( a = 0 & b = s ) by A1, XTUPLE_0:1;
hence s = 1 by A2, A3, XXREAL_0:1; ::_thesis: verum
end;
theorem Th63: :: BORSUK_6:63
for s being set st [s,1] in ICC holds
s = 1
proof
let s be set ; ::_thesis: ( [s,1] in ICC implies s = 1 )
assume [s,1] in ICC ; ::_thesis: s = 1
then consider a, b being Point of I[01] such that
A1: [s,1] = [a,b] and
A2: b <= (2 * a) - 1 by Def10;
b = 1 by A1, XTUPLE_0:1;
then 1 + 1 <= 2 * a by A2, XREAL_1:19;
then A3: 2 / 2 <= (2 * a) / 2 by XREAL_1:72;
( a <= 1 & a = s ) by A1, BORSUK_1:43, XTUPLE_0:1;
hence s = 1 by A3, XXREAL_0:1; ::_thesis: verum
end;
theorem Th64: :: BORSUK_6:64
[0,1] in IBB
proof
A1: ( 1 >= 1 - (2 * 0) & 1 >= (2 * 0) - 1 ) ;
( 0 is Point of I[01] & 1 is Point of I[01] ) by BORSUK_1:43;
hence [0,1] in IBB by A1, Def9; ::_thesis: verum
end;
theorem Th65: :: BORSUK_6:65
for x being Point of I[01] holds [x,1] in IBB
proof
let x be Point of I[01]; ::_thesis: [x,1] in IBB
x <= 1 by BORSUK_1:43;
then 2 * x <= 2 * 1 by XREAL_1:64;
then A1: ( 1 is Point of I[01] & (2 * x) - 1 <= (2 * 1) - 1 ) by BORSUK_1:43, XREAL_1:13;
x >= 0 by BORSUK_1:43;
then 1 - (2 * x) <= 1 - (2 * 0) by XREAL_1:13;
hence [x,1] in IBB by A1, Def9; ::_thesis: verum
end;
theorem Th66: :: BORSUK_6:66
( [(1 / 2),0] in ICC & [1,1] in ICC )
proof
A1: 0 <= (2 * (1 / 2)) - 1 ;
( 1 / 2 is Point of I[01] & 0 is Point of I[01] ) by BORSUK_1:43;
hence [(1 / 2),0] in ICC by A1, Def10; ::_thesis: [1,1] in ICC
( 1 is Point of I[01] & 1 <= (2 * 1) - 1 ) by BORSUK_1:43;
hence [1,1] in ICC by Def10; ::_thesis: verum
end;
theorem Th67: :: BORSUK_6:67
[(1 / 2),0] in IBB
proof
A1: 0 <= 1 - (2 * (1 / 2)) ;
( 1 / 2 is Point of I[01] & 0 is Point of I[01] ) by BORSUK_1:43;
hence [(1 / 2),0] in IBB by A1, Def9; ::_thesis: verum
end;
theorem Th68: :: BORSUK_6:68
for x being Point of I[01] holds [1,x] in ICC
proof
let x be Point of I[01]; ::_thesis: [1,x] in ICC
( 1 is Point of I[01] & x <= (2 * 1) - 1 ) by BORSUK_1:43;
hence [1,x] in ICC by Def10; ::_thesis: verum
end;
theorem Th69: :: BORSUK_6:69
for x being Point of I[01] st x >= 1 / 2 holds
[x,0] in ICC
proof
let x be Point of I[01]; ::_thesis: ( x >= 1 / 2 implies [x,0] in ICC )
assume x >= 1 / 2 ; ::_thesis: [x,0] in ICC
then 2 * x >= 2 * (1 / 2) by XREAL_1:64;
then A1: (2 * x) - 1 >= (2 * (1 / 2)) - 1 by XREAL_1:13;
0 is Point of I[01] by BORSUK_1:43;
hence [x,0] in ICC by A1, Def10; ::_thesis: verum
end;
theorem Th70: :: BORSUK_6:70
for x being Point of I[01] st x <= 1 / 2 holds
[x,0] in IAA
proof
let x be Point of I[01]; ::_thesis: ( x <= 1 / 2 implies [x,0] in IAA )
assume x <= 1 / 2 ; ::_thesis: [x,0] in IAA
then 2 * x <= 2 * (1 / 2) by XREAL_1:64;
then A1: 1 - (2 * x) >= 1 - (2 * (1 / 2)) by XREAL_1:13;
0 is Point of I[01] by BORSUK_1:43;
hence [x,0] in IAA by A1, Def8; ::_thesis: verum
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
let x be Point of I[01]; ::_thesis: ( x < 1 / 2 implies ( not [x,0] in IBB & not [x,0] in ICC ) )
assume A1: x < 1 / 2 ; ::_thesis: ( not [x,0] in IBB & not [x,0] in ICC )
thus not [x,0] in IBB ::_thesis: not [x,0] in ICC
proof
assume [x,0] in IBB ; ::_thesis: contradiction
then consider a, b being Point of I[01] such that
A2: [x,0] = [a,b] and
A3: b >= 1 - (2 * a) and
b >= (2 * a) - 1 by Def9;
( x = a & b = 0 ) by A2, XTUPLE_0:1;
then 0 + (2 * x) >= 1 by A3, XREAL_1:20;
then (2 * x) / 2 >= 1 / 2 by XREAL_1:72;
hence contradiction by A1; ::_thesis: verum
end;
not [x,0] in ICC
proof
assume [x,0] in ICC ; ::_thesis: contradiction
then consider a, b being Point of I[01] such that
A4: [x,0] = [a,b] and
A5: b <= (2 * a) - 1 by Def10;
( x = a & b = 0 ) by A4, XTUPLE_0:1;
then 0 + 1 <= 2 * x by A5, XREAL_1:19;
then 1 / 2 <= (2 * x) / 2 by XREAL_1:72;
hence contradiction by A1; ::_thesis: verum
end;
hence not [x,0] in ICC ; ::_thesis: verum
end;
theorem Th72: :: BORSUK_6:72
IAA /\ ICC = {[(1 / 2),0]}
proof
thus IAA /\ ICC c= {[(1 / 2),0]} :: according to XBOOLE_0:def_10 ::_thesis: {[(1 / 2),0]} c= IAA /\ ICC
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in IAA /\ ICC or x in {[(1 / 2),0]} )
assume A1: x in IAA /\ ICC ; ::_thesis: x in {[(1 / 2),0]}
then reconsider y = x as Point of [:I[01],I[01]:] ;
x in IAA by A1, XBOOLE_0:def_4;
then A2: y `1 <= 1 / 2 by Th59;
A3: x in ICC by A1, XBOOLE_0:def_4;
then y `1 >= 1 / 2 by Th60;
then A4: y `1 = 1 / 2 by A2, XXREAL_0:1;
y in the carrier of [:I[01],I[01]:] ;
then A5: y in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
A6: y `2 is Point of I[01] by Th27;
ex q being Point of [:I[01],I[01]:] st
( q = y & q `2 <= (2 * (q `1)) - 1 ) by A3, Th55;
then y `2 = 0 by A4, A6, BORSUK_1:43;
then y = [(1 / 2),0] by A5, A4, MCART_1:21;
hence x in {[(1 / 2),0]} by TARSKI:def_1; ::_thesis: verum
end;
1 / 2 is Point of I[01] by BORSUK_1:43;
then ( [(1 / 2),0] in IAA & [(1 / 2),0] in ICC ) by Th69, Th70;
then [(1 / 2),0] in IAA /\ ICC by XBOOLE_0:def_4;
hence {[(1 / 2),0]} c= IAA /\ ICC by ZFMISC_1:31; ::_thesis: verum
end;
Lm1: for x, y being Point of I[01] holds [x,y] in the carrier of [:I[01],I[01]:]
;
begin
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
let T be non empty TopSpace; ::_thesis: 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
let a, b, c, d be Point of T; ::_thesis: 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
let P be Path of a,b; ::_thesis: 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
let Q be Path of b,c; ::_thesis: 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
let R be Path of c,d; ::_thesis: ( a,b are_connected & b,c are_connected & c,d are_connected implies (P + Q) + R,P + (Q + R) are_homotopic )
assume that
A1: ( a,b are_connected & b,c are_connected ) and
A2: c,d are_connected ; ::_thesis: (P + Q) + R,P + (Q + R) are_homotopic
( a,c are_connected & RePar (((P + Q) + R),3RP) = P + (Q + R) ) by A1, A2, Th42, Th52;
hence (P + Q) + R,P + (Q + R) are_homotopic by A2, Th42, Th45, Th49; ::_thesis: verum
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
let X be non empty pathwise_connected TopSpace; ::_thesis: 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
let a1, b1, c1, d1 be Point of X; ::_thesis: 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
let P be Path of a1,b1; ::_thesis: for Q being Path of b1,c1
for R being Path of c1,d1 holds (P + Q) + R,P + (Q + R) are_homotopic
let Q be Path of b1,c1; ::_thesis: for R being Path of c1,d1 holds (P + Q) + R,P + (Q + R) are_homotopic
let R be Path of c1,d1; ::_thesis: (P + Q) + R,P + (Q + R) are_homotopic
A1: c1,d1 are_connected by BORSUK_2:def_3;
( a1,b1 are_connected & b1,c1 are_connected ) by BORSUK_2:def_3;
hence (P + Q) + R,P + (Q + R) are_homotopic by A1, Th73; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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
let a, b, c be Point of T; ::_thesis: 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
set BB = [:I[01],I[01]:];
reconsider R1 = L[01] (0,(1 / 2),0,1) as continuous Function of (Closed-Interval-TSpace (0,(1 / 2))),I[01] by Th34, TOPMETR:20;
let P1, P2 be Path of a,b; ::_thesis: 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
let Q1, Q2 be Path of b,c; ::_thesis: ( a,b are_connected & b,c are_connected & P1,P2 are_homotopic & Q1,Q2 are_homotopic implies P1 + Q1,P2 + Q2 are_homotopic )
assume that
A1: ( a,b are_connected & b,c are_connected ) and
A2: P1,P2 are_homotopic and
A3: Q1,Q2 are_homotopic ; ::_thesis: P1 + Q1,P2 + Q2 are_homotopic
reconsider R2 = L[01] ((1 / 2),1,0,1) as continuous Function of (Closed-Interval-TSpace ((1 / 2),1)),I[01] by Th34, TOPMETR:20;
A4: 1 is Point of I[01] by BORSUK_1:43;
A5: 0 is Point of I[01] by BORSUK_1:43;
then reconsider A01 = [.0,1.] as non empty Subset of I[01] by A4, BORSUK_4:24;
A6: 1 / 2 is Point of I[01] by BORSUK_1:43;
then reconsider B01 = [.0,(1 / 2).] as non empty Subset of I[01] by A5, BORSUK_4:24;
reconsider N2 = [:[.(1 / 2),1.],[.0,1.]:] as non empty compact Subset of [:I[01],I[01]:] by A5, A4, A6, Th9;
reconsider N1 = [:[.0,(1 / 2).],[.0,1.]:] as non empty compact Subset of [:I[01],I[01]:] by A5, A4, A6, Th9;
set T1 = [:I[01],I[01]:] | N1;
set T2 = [:I[01],I[01]:] | N2;
A01 = [#] I[01] by BORSUK_1:40;
then A7: I[01] = I[01] | A01 by TSEP_1:93;
set f1 = [:R1,(id I[01]):];
set g1 = [:R2,(id I[01]):];
reconsider f1 = [:R1,(id I[01]):] as continuous Function of [:(Closed-Interval-TSpace (0,(1 / 2))),I[01]:],[:I[01],I[01]:] ;
reconsider g1 = [:R2,(id I[01]):] as continuous Function of [:(Closed-Interval-TSpace ((1 / 2),1)),I[01]:],[:I[01],I[01]:] ;
A8: dom g1 = the carrier of [:(Closed-Interval-TSpace ((1 / 2),1)),I[01]:] by FUNCT_2:def_1
.= [: the carrier of (Closed-Interval-TSpace ((1 / 2),1)), the carrier of I[01]:] by BORSUK_1:def_2 ;
reconsider B02 = [.(1 / 2),1.] as non empty Subset of I[01] by A4, A6, BORSUK_4:24;
consider f being Function of [:I[01],I[01]:],T such that
A9: f is continuous and
A10: for s being Point of I[01] holds
( f . (s,0) = P1 . s & f . (s,1) = P2 . s & ( for t being Point of I[01] holds
( f . (0,t) = a & f . (1,t) = b ) ) ) by A2, BORSUK_2:def_7;
Closed-Interval-TSpace (0,(1 / 2)) = I[01] | B01 by TOPMETR:24;
then [:I[01],I[01]:] | N1 = [:(Closed-Interval-TSpace (0,(1 / 2))),I[01]:] by A7, BORSUK_3:22;
then reconsider K1 = f * f1 as continuous Function of ([:I[01],I[01]:] | N1),T by A9;
consider g being Function of [:I[01],I[01]:],T such that
A11: g is continuous and
A12: for s being Point of I[01] holds
( g . (s,0) = Q1 . s & g . (s,1) = Q2 . s & ( for t being Point of I[01] holds
( g . (0,t) = b & g . (1,t) = c ) ) ) by A3, BORSUK_2:def_7;
Closed-Interval-TSpace ((1 / 2),1) = I[01] | B02 by TOPMETR:24;
then [:I[01],I[01]:] | N2 = [:(Closed-Interval-TSpace ((1 / 2),1)),I[01]:] by A7, BORSUK_3:22;
then reconsider K2 = g * g1 as continuous Function of ([:I[01],I[01]:] | N2),T by A11;
A13: dom K2 = the carrier of [:(Closed-Interval-TSpace ((1 / 2),1)),I[01]:] by FUNCT_2:def_1
.= [: the carrier of (Closed-Interval-TSpace ((1 / 2),1)), the carrier of I[01]:] by BORSUK_1:def_2 ;
A14: for p being set st p in ([#] ([:I[01],I[01]:] | N1)) /\ ([#] ([:I[01],I[01]:] | N2)) holds
K1 . p = K2 . p
proof
A15: R2 . (1 / 2) = 0 by Th33;
let p be set ; ::_thesis: ( p in ([#] ([:I[01],I[01]:] | N1)) /\ ([#] ([:I[01],I[01]:] | N2)) implies K1 . p = K2 . p )
A16: R1 . (1 / 2) = 1 by Th33;
assume p in ([#] ([:I[01],I[01]:] | N1)) /\ ([#] ([:I[01],I[01]:] | N2)) ; ::_thesis: K1 . p = K2 . p
then p in [:{(1 / 2)},[.0,1.]:] by Th29;
then consider x, y being set such that
A17: x in {(1 / 2)} and
A18: y in [.0,1.] and
A19: p = [x,y] by ZFMISC_1:def_2;
A20: y in the carrier of I[01] by A18, TOPMETR:18, TOPMETR:20;
reconsider y = y as Point of I[01] by A18, TOPMETR:18, TOPMETR:20;
A21: y in dom (id I[01]) by A20, FUNCT_2:def_1;
A22: x = 1 / 2 by A17, TARSKI:def_1;
then x in [.(1 / 2),1.] by XXREAL_1:1;
then A23: x in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then p in [: the carrier of (Closed-Interval-TSpace ((1 / 2),1)), the carrier of I[01]:] by A19, A20, ZFMISC_1:87;
then p in the carrier of [:(Closed-Interval-TSpace ((1 / 2),1)),I[01]:] by BORSUK_1:def_2;
then A24: p in dom g1 by FUNCT_2:def_1;
x in [.0,(1 / 2).] by A22, XXREAL_1:1;
then A25: x in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by TOPMETR:18;
then x in dom R1 by FUNCT_2:def_1;
then A26: [x,y] in [:(dom R1),(dom (id I[01])):] by A21, ZFMISC_1:87;
x in dom R2 by A23, FUNCT_2:def_1;
then A27: [x,y] in [:(dom R2),(dom (id I[01])):] by A21, ZFMISC_1:87;
p in [: the carrier of (Closed-Interval-TSpace (0,(1 / 2))), the carrier of I[01]:] by A19, A20, A25, ZFMISC_1:87;
then p in the carrier of [:(Closed-Interval-TSpace (0,(1 / 2))),I[01]:] by BORSUK_1:def_2;
then p in dom f1 by FUNCT_2:def_1;
then K1 . p = f . (f1 . (x,y)) by A19, FUNCT_1:13
.= f . ((R1 . x),((id I[01]) . y)) by A26, FUNCT_3:65
.= b by A10, A22, A16
.= g . ((R2 . x),((id I[01]) . y)) by A12, A22, A15
.= g . (g1 . (x,y)) by A27, FUNCT_3:65
.= K2 . p by A19, A24, FUNCT_1:13 ;
hence K1 . p = K2 . p ; ::_thesis: verum
end;
([#] ([:I[01],I[01]:] | N1)) \/ ([#] ([:I[01],I[01]:] | N2)) = [#] [:I[01],I[01]:] by Th28;
then consider h being Function of [:I[01],I[01]:],T such that
A28: h = K1 +* K2 and
A29: h is continuous by A14, BORSUK_2:1;
A30: dom f1 = the carrier of [:(Closed-Interval-TSpace (0,(1 / 2))),I[01]:] by FUNCT_2:def_1
.= [: the carrier of (Closed-Interval-TSpace (0,(1 / 2))), the carrier of I[01]:] by BORSUK_1:def_2 ;
A31: for s being Point of I[01] holds
( h . (s,0) = (P1 + Q1) . s & h . (s,1) = (P2 + Q2) . s )
proof
let s be Point of I[01]; ::_thesis: ( h . (s,0) = (P1 + Q1) . s & h . (s,1) = (P2 + Q2) . s )
A32: h . (s,1) = (P2 + Q2) . s
proof
percases ( s < 1 / 2 or s >= 1 / 2 ) ;
supposeA33: s < 1 / 2 ; ::_thesis: h . (s,1) = (P2 + Q2) . s
then A34: 2 * s is Point of I[01] by Th3;
A35: 1 in the carrier of I[01] by BORSUK_1:43;
then A36: 1 in dom (id I[01]) by FUNCT_2:def_1;
A37: s >= 0 by BORSUK_1:43;
then A38: R1 . s = (((1 - 0) / ((1 / 2) - 0)) * (s - 0)) + 0 by A33, Th35
.= 2 * s ;
s in [.0,(1 / 2).] by A33, A37, XXREAL_1:1;
then A39: s in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by TOPMETR:18;
then A40: [s,1] in dom f1 by A30, A35, ZFMISC_1:87;
s in dom R1 by A39, FUNCT_2:def_1;
then A41: [s,1] in [:(dom R1),(dom (id I[01])):] by A36, ZFMISC_1:87;
not s in [.(1 / 2),1.] by A33, XXREAL_1:1;
then not s in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then not [s,1] in dom K2 by A13, ZFMISC_1:87;
then h . (s,1) = K1 . (s,1) by A28, FUNCT_4:11
.= f . (f1 . (s,1)) by A40, FUNCT_1:13
.= f . ((R1 . s),((id I[01]) . 1)) by A41, FUNCT_3:65
.= f . ((2 * s),1) by A4, A38, FUNCT_1:18
.= P2 . (2 * s) by A10, A34 ;
hence h . (s,1) = (P2 + Q2) . s by A1, A33, BORSUK_2:def_5; ::_thesis: verum
end;
supposeA42: s >= 1 / 2 ; ::_thesis: h . (s,1) = (P2 + Q2) . s
A43: s <= 1 by BORSUK_1:43;
then A44: R2 . s = (((1 - 0) / (1 - (1 / 2))) * (s - (1 / 2))) + 0 by A42, Th35
.= (2 * s) - 1 ;
A45: (2 * s) - 1 is Point of I[01] by A42, Th4;
A46: 1 in the carrier of I[01] by BORSUK_1:43;
then A47: 1 in dom (id I[01]) by FUNCT_2:def_1;
s in [.(1 / 2),1.] by A42, A43, XXREAL_1:1;
then A48: s in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then A49: [s,1] in dom g1 by A8, A46, ZFMISC_1:87;
s in dom R2 by A48, FUNCT_2:def_1;
then A50: [s,1] in [:(dom R2),(dom (id I[01])):] by A47, ZFMISC_1:87;
[s,1] in dom K2 by A13, A48, A46, ZFMISC_1:87;
then h . (s,1) = K2 . (s,1) by A28, FUNCT_4:13
.= g . (g1 . (s,1)) by A49, FUNCT_1:13
.= g . ((R2 . s),((id I[01]) . 1)) by A50, FUNCT_3:65
.= g . (((2 * s) - 1),1) by A4, A44, FUNCT_1:18
.= Q2 . ((2 * s) - 1) by A12, A45 ;
hence h . (s,1) = (P2 + Q2) . s by A1, A42, BORSUK_2:def_5; ::_thesis: verum
end;
end;
end;
h . (s,0) = (P1 + Q1) . s
proof
percases ( s < 1 / 2 or s >= 1 / 2 ) ;
supposeA51: s < 1 / 2 ; ::_thesis: h . (s,0) = (P1 + Q1) . s
then A52: 2 * s is Point of I[01] by Th3;
A53: 0 in the carrier of I[01] by BORSUK_1:43;
then A54: 0 in dom (id I[01]) by FUNCT_2:def_1;
A55: s >= 0 by BORSUK_1:43;
then A56: R1 . s = (((1 - 0) / ((1 / 2) - 0)) * (s - 0)) + 0 by A51, Th35
.= 2 * s ;
s in [.0,(1 / 2).] by A51, A55, XXREAL_1:1;
then A57: s in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by TOPMETR:18;
then A58: [s,0] in dom f1 by A30, A53, ZFMISC_1:87;
s in dom R1 by A57, FUNCT_2:def_1;
then A59: [s,0] in [:(dom R1),(dom (id I[01])):] by A54, ZFMISC_1:87;
not s in [.(1 / 2),1.] by A51, XXREAL_1:1;
then not s in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then not [s,0] in dom K2 by A13, ZFMISC_1:87;
then h . (s,0) = K1 . (s,0) by A28, FUNCT_4:11
.= f . (f1 . (s,0)) by A58, FUNCT_1:13
.= f . ((R1 . s),((id I[01]) . 0)) by A59, FUNCT_3:65
.= f . ((2 * s),0) by A5, A56, FUNCT_1:18
.= P1 . (2 * s) by A10, A52 ;
hence h . (s,0) = (P1 + Q1) . s by A1, A51, BORSUK_2:def_5; ::_thesis: verum
end;
supposeA60: s >= 1 / 2 ; ::_thesis: h . (s,0) = (P1 + Q1) . s
A61: s <= 1 by BORSUK_1:43;
then A62: R2 . s = (((1 - 0) / (1 - (1 / 2))) * (s - (1 / 2))) + 0 by A60, Th35
.= (2 * s) - 1 ;
A63: (2 * s) - 1 is Point of I[01] by A60, Th4;
A64: 0 in the carrier of I[01] by BORSUK_1:43;
then A65: 0 in dom (id I[01]) by FUNCT_2:def_1;
s in [.(1 / 2),1.] by A60, A61, XXREAL_1:1;
then A66: s in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then A67: [s,0] in dom g1 by A8, A64, ZFMISC_1:87;
s in dom R2 by A66, FUNCT_2:def_1;
then A68: [s,0] in [:(dom R2),(dom (id I[01])):] by A65, ZFMISC_1:87;
[s,0] in dom K2 by A13, A66, A64, ZFMISC_1:87;
then h . (s,0) = K2 . (s,0) by A28, FUNCT_4:13
.= g . (g1 . (s,0)) by A67, FUNCT_1:13
.= g . ((R2 . s),((id I[01]) . 0)) by A68, FUNCT_3:65
.= g . (((2 * s) - 1),0) by A5, A62, FUNCT_1:18
.= Q1 . ((2 * s) - 1) by A12, A63 ;
hence h . (s,0) = (P1 + Q1) . s by A1, A60, BORSUK_2:def_5; ::_thesis: verum
end;
end;
end;
hence ( h . (s,0) = (P1 + Q1) . s & h . (s,1) = (P2 + Q2) . s ) by A32; ::_thesis: verum
end;
take h ; :: according to BORSUK_2:def_7 ::_thesis: ( h is continuous & ( for b1 being Element of the carrier of I[01] holds
( h . (b1,0) = (P1 + Q1) . b1 & h . (b1,1) = (P2 + Q2) . b1 & h . (0,b1) = a & h . (1,b1) = c ) ) )
for t being Point of I[01] holds
( h . (0,t) = a & h . (1,t) = c )
proof
let t be Point of I[01]; ::_thesis: ( h . (0,t) = a & h . (1,t) = c )
A69: dom K2 = the carrier of [:(Closed-Interval-TSpace ((1 / 2),1)),I[01]:] by FUNCT_2:def_1
.= [: the carrier of (Closed-Interval-TSpace ((1 / 2),1)), the carrier of I[01]:] by BORSUK_1:def_2 ;
t in the carrier of I[01] ;
then A70: t in dom (id I[01]) by FUNCT_2:def_1;
0 in [.0,(1 / 2).] by XXREAL_1:1;
then A71: 0 in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by TOPMETR:18;
then A72: [0,t] in dom f1 by A30, ZFMISC_1:87;
0 in dom R1 by A71, FUNCT_2:def_1;
then A73: [0,t] in [:(dom R1),(dom (id I[01])):] by A70, ZFMISC_1:87;
not 0 in [.(1 / 2),1.] by XXREAL_1:1;
then not 0 in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then not [0,t] in dom K2 by A69, ZFMISC_1:87;
hence h . (0,t) = K1 . (0,t) by A28, FUNCT_4:11
.= f . (f1 . (0,t)) by A72, FUNCT_1:13
.= f . ((R1 . 0),((id I[01]) . t)) by A73, FUNCT_3:65
.= f . ((R1 . 0),t) by FUNCT_1:18
.= f . (0,t) by Th33
.= a by A10 ;
::_thesis: h . (1,t) = c
t in the carrier of I[01] ;
then A74: t in dom (id I[01]) by FUNCT_2:def_1;
1 in [.(1 / 2),1.] by XXREAL_1:1;
then A75: 1 in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then 1 in dom R2 by FUNCT_2:def_1;
then A76: [1,t] in [:(dom R2),(dom (id I[01])):] by A74, ZFMISC_1:87;
1 in [.(1 / 2),1.] by XXREAL_1:1;
then 1 in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
then A77: [1,t] in dom g1 by A8, ZFMISC_1:87;
[1,t] in dom K2 by A69, A75, ZFMISC_1:87;
then h . (1,t) = K2 . (1,t) by A28, FUNCT_4:13
.= g . (g1 . (1,t)) by A77, FUNCT_1:13
.= g . ((R2 . 1),((id I[01]) . t)) by A76, FUNCT_3:65
.= g . ((R2 . 1),t) by FUNCT_1:18
.= g . (1,t) by Th33
.= c by A12 ;
hence h . (1,t) = c ; ::_thesis: verum
end;
hence ( h is continuous & ( for b1 being Element of the carrier of I[01] holds
( h . (b1,0) = (P1 + Q1) . b1 & h . (b1,1) = (P2 + Q2) . b1 & h . (0,b1) = a & h . (1,b1) = c ) ) ) by A29, A31; ::_thesis: verum
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
let X be non empty pathwise_connected TopSpace; ::_thesis: 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
let a1, b1, c1 be Point of X; ::_thesis: 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
let P1, P2 be Path of a1,b1; ::_thesis: for Q1, Q2 being Path of b1,c1 st P1,P2 are_homotopic & Q1,Q2 are_homotopic holds
P1 + Q1,P2 + Q2 are_homotopic
let Q1, Q2 be Path of b1,c1; ::_thesis: ( P1,P2 are_homotopic & Q1,Q2 are_homotopic implies P1 + Q1,P2 + Q2 are_homotopic )
( a1,b1 are_connected & b1,c1 are_connected ) by BORSUK_2:def_3;
hence ( P1,P2 are_homotopic & Q1,Q2 are_homotopic implies P1 + Q1,P2 + Q2 are_homotopic ) by Th75; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: for P, Q being Path of a,b st a,b are_connected & P,Q are_homotopic holds
- P, - Q are_homotopic
reconsider fF = id I[01] as continuous Function of I[01],I[01] ;
reconsider fB = L[01] (((0,1) (#)),((#) (0,1))) as continuous Function of I[01],I[01] by TOPMETR:20, TREAL_1:8;
let P, Q be Path of a,b; ::_thesis: ( a,b are_connected & P,Q are_homotopic implies - P, - Q are_homotopic )
assume A1: a,b are_connected ; ::_thesis: ( not P,Q are_homotopic or - P, - Q are_homotopic )
set F = [:fB,fF:];
A2: dom fB = the carrier of I[01] by FUNCT_2:def_1;
assume P,Q are_homotopic ; ::_thesis: - P, - Q are_homotopic
then consider f being Function of [:I[01],I[01]:],T such that
A3: f is continuous and
A4: for s being Point of I[01] holds
( f . (s,0) = P . s & f . (s,1) = Q . s & ( for t being Point of I[01] holds
( f . (0,t) = a & f . (1,t) = b ) ) ) by BORSUK_2:def_7;
reconsider ff = f * [:fB,fF:] as Function of [:I[01],I[01]:],T ;
take ff ; :: according to BORSUK_2:def_7 ::_thesis: ( ff is continuous & ( for b1 being Element of the carrier of I[01] holds
( ff . (b1,0) = (- P) . b1 & ff . (b1,1) = (- Q) . b1 & ff . (0,b1) = b & ff . (1,b1) = a ) ) )
thus ff is continuous by A3; ::_thesis: for b1 being Element of the carrier of I[01] holds
( ff . (b1,0) = (- P) . b1 & ff . (b1,1) = (- Q) . b1 & ff . (0,b1) = b & ff . (1,b1) = a )
A5: 0 is Point of I[01] by BORSUK_1:43;
A6: for t being Point of I[01] holds
( ff . (0,t) = b & ff . (1,t) = a )
proof
A7: for t being Point of I[01]
for t9 being Real st t = t9 holds
fB . t = 1 - t9
proof
let t be Point of I[01]; ::_thesis: for t9 being Real st t = t9 holds
fB . t = 1 - t9
let t9 be Real; ::_thesis: ( t = t9 implies fB . t = 1 - t9 )
assume A8: t = t9 ; ::_thesis: fB . t = 1 - t9
reconsider ee = t as Point of (Closed-Interval-TSpace (0,1)) by TOPMETR:20;
A9: ( (0,1) (#) = 1 & (#) (0,1) = 0 ) by TREAL_1:def_1, TREAL_1:def_2;
fB . t = fB . ee
.= ((0 - 1) * t9) + 1 by A8, A9, TREAL_1:7
.= 1 - (1 * t9) ;
hence fB . t = 1 - t9 ; ::_thesis: verum
end;
then A10: fB . 0 = 1 - 0 by A5
.= 1 ;
1 is Point of I[01] by BORSUK_1:43;
then A11: fB . 1 = 1 - 1 by A7
.= 0 ;
let t be Point of I[01]; ::_thesis: ( ff . (0,t) = b & ff . (1,t) = a )
A12: dom fF = the carrier of I[01] by FUNCT_2:def_1;
t in the carrier of I[01] ;
then reconsider tt = t as Real by BORSUK_1:40;
A13: dom fB = the carrier of I[01] by FUNCT_2:def_1;
then A14: 0 in dom fB by BORSUK_1:43;
A15: dom [:fB,fF:] = [:(dom fB),(dom fF):] by FUNCT_3:def_8;
then A16: [0,t] in dom [:fB,fF:] by A12, A14, ZFMISC_1:87;
A17: 1 in dom fB by A13, BORSUK_1:43;
then A18: [1,t] in dom [:fB,fF:] by A12, A15, ZFMISC_1:87;
[:fB,fF:] . (1,t) = [(fB . 1),(fF . t)] by A12, A17, FUNCT_3:def_8
.= [0,tt] by A11, FUNCT_1:18 ;
then A19: ff . (1,t) = f . (0,t) by A18, FUNCT_1:13
.= a by A4 ;
[:fB,fF:] . (0,t) = [(fB . 0),(fF . t)] by A12, A14, FUNCT_3:def_8
.= [1,tt] by A10, FUNCT_1:18 ;
then ff . (0,t) = f . (1,t) by A16, FUNCT_1:13
.= b by A4 ;
hence ( ff . (0,t) = b & ff . (1,t) = a ) by A19; ::_thesis: verum
end;
A20: dom fF = the carrier of I[01] by FUNCT_2:def_1;
for s being Point of I[01] holds
( ff . (s,0) = (- P) . s & ff . (s,1) = (- Q) . s )
proof
let s be Point of I[01]; ::_thesis: ( ff . (s,0) = (- P) . s & ff . (s,1) = (- Q) . s )
A21: for t being Point of I[01]
for t9 being Real st t = t9 holds
fB . t = 1 - t9
proof
let t be Point of I[01]; ::_thesis: for t9 being Real st t = t9 holds
fB . t = 1 - t9
let t9 be Real; ::_thesis: ( t = t9 implies fB . t = 1 - t9 )
assume A22: t = t9 ; ::_thesis: fB . t = 1 - t9
reconsider ee = t as Point of (Closed-Interval-TSpace (0,1)) by TOPMETR:20;
A23: ( (0,1) (#) = 1 & (#) (0,1) = 0 ) by TREAL_1:def_1, TREAL_1:def_2;
fB . t = fB . ee
.= ((0 - 1) * t9) + 1 by A22, A23, TREAL_1:7
.= 1 - (1 * t9) ;
hence fB . t = 1 - t9 ; ::_thesis: verum
end;
s is Real by XREAL_0:def_1;
then A24: fB . s = 1 - s by A21;
A25: 1 is Point of I[01] by BORSUK_1:43;
A26: dom [:fB,fF:] = [:(dom fB),(dom fF):] by FUNCT_3:def_8;
A27: 1 in dom fF by A20, BORSUK_1:43;
then A28: [s,1] in dom [:fB,fF:] by A2, A26, ZFMISC_1:87;
A29: 0 in dom fF by A20, BORSUK_1:43;
then A30: [s,0] in dom [:fB,fF:] by A2, A26, ZFMISC_1:87;
A31: 1 - s is Point of I[01] by JORDAN5B:4;
[:fB,fF:] . (s,1) = [(fB . s),(fF . 1)] by A2, A27, FUNCT_3:def_8
.= [(1 - s),1] by A24, A25, FUNCT_1:18 ;
then A32: ff . (s,1) = f . ((1 - s),1) by A28, FUNCT_1:13
.= Q . (1 - s) by A4, A31
.= (- Q) . s by A1, BORSUK_2:def_6 ;
[:fB,fF:] . (s,0) = [(fB . s),(fF . 0)] by A2, A29, FUNCT_3:def_8
.= [(1 - s),0] by A5, A24, FUNCT_1:18 ;
then ff . (s,0) = f . ((1 - s),0) by A30, FUNCT_1:13
.= P . (1 - s) by A4, A31
.= (- P) . s by A1, BORSUK_2:def_6 ;
hence ( ff . (s,0) = (- P) . s & ff . (s,1) = (- Q) . s ) by A32; ::_thesis: verum
end;
hence for b1 being Element of the carrier of I[01] holds
( ff . (b1,0) = (- P) . b1 & ff . (b1,1) = (- Q) . b1 & ff . (0,b1) = b & ff . (1,b1) = a ) by A6; ::_thesis: verum
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
proof
let X be non empty pathwise_connected TopSpace; ::_thesis: 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
let a1, b1 be Point of X; ::_thesis: for P, Q being Path of a1,b1 st P,Q are_homotopic holds
- P, - Q are_homotopic
let P, Q be Path of a1,b1; ::_thesis: ( P,Q are_homotopic implies - P, - Q are_homotopic )
a1,b1 are_connected by BORSUK_2:def_3;
hence ( P,Q are_homotopic implies - P, - Q are_homotopic ) by Th77; ::_thesis: verum
end;
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: for P, Q, R being Path of a,b st P,Q are_homotopic & Q,R are_homotopic holds
P,R are_homotopic
1 / 2 in [.0,(1 / 2).] by XXREAL_1:1;
then A1: 1 / 2 in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by TOPMETR:18;
reconsider B02 = [.(1 / 2),1.] as non empty Subset of I[01] by BORSUK_1:40, XXREAL_1:1, XXREAL_1:34;
A2: 1 in [.0,1.] by XXREAL_1:1;
A3: 1 / 2 in [.(1 / 2),1.] by XXREAL_1:1;
then A4: 1 / 2 in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by TOPMETR:18;
[.0,(1 / 2).] c= the carrier of I[01] by BORSUK_1:40, XXREAL_1:34;
then A5: [:[.0,1.],[.0,(1 / 2).]:] c= [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:40, ZFMISC_1:96;
A6: the carrier of (Closed-Interval-TSpace (0,(1 / 2))) = [.0,(1 / 2).] by TOPMETR:18;
0 in [.0,(1 / 2).] by XXREAL_1:1;
then reconsider Ewa = [:[.0,1.],[.0,(1 / 2).]:] as non empty Subset of [:I[01],I[01]:] by A5, A2, BORSUK_1:def_2;
set T1 = [:I[01],I[01]:] | Ewa;
reconsider P2 = P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))) as continuous Function of (Closed-Interval-TSpace ((1 / 2),1)),I[01] by TOPMETR:20, TREAL_1:12;
reconsider P1 = P[01] (0,(1 / 2),((#) (0,1)),((0,1) (#))) as continuous Function of (Closed-Interval-TSpace (0,(1 / 2))),I[01] by TOPMETR:20, TREAL_1:12;
let P, Q, R be Path of a,b; ::_thesis: ( P,Q are_homotopic & Q,R are_homotopic implies P,R are_homotopic )
assume that
A7: P,Q are_homotopic and
A8: Q,R are_homotopic ; ::_thesis: P,R are_homotopic
consider f being Function of [:I[01],I[01]:],T such that
A9: f is continuous and
A10: for s being Point of I[01] holds
( f . (s,0) = P . s & f . (s,1) = Q . s & ( for t being Point of I[01] holds
( f . (0,t) = a & f . (1,t) = b ) ) ) by A7, BORSUK_2:def_7;
A11: the carrier of (Closed-Interval-TSpace ((1 / 2),1)) = [.(1 / 2),1.] by TOPMETR:18;
[.0,1.] c= the carrier of I[01] by BORSUK_1:40;
then reconsider A01 = [.0,1.] as non empty Subset of I[01] by XXREAL_1:1;
reconsider B01 = [.0,(1 / 2).] as non empty Subset of I[01] by BORSUK_1:40, XXREAL_1:1, XXREAL_1:34;
A12: the carrier of (Closed-Interval-TSpace ((1 / 2),1)) = [.(1 / 2),1.] by TOPMETR:18;
A01 = [#] I[01] by BORSUK_1:40;
then A13: I[01] = I[01] | A01 by TSEP_1:93;
[.(1 / 2),1.] c= the carrier of I[01] by BORSUK_1:40, XXREAL_1:34;
then A14: [:[.0,1.],[.(1 / 2),1.]:] c= [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:40, ZFMISC_1:96;
A15: 1 in the carrier of I[01] by BORSUK_1:43;
1 in [.(1 / 2),1.] by XXREAL_1:1;
then reconsider Ewa1 = [:[.0,1.],[.(1 / 2),1.]:] as non empty Subset of [:I[01],I[01]:] by A2, A14, BORSUK_1:def_2;
set T2 = [:I[01],I[01]:] | Ewa1;
set e1 = [:(id I[01]),P1:];
set e2 = [:(id I[01]),P2:];
A16: ( dom (id I[01]) = the carrier of I[01] & dom (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#)))) = the carrier of (Closed-Interval-TSpace ((1 / 2),1)) ) by FUNCT_2:def_1;
A17: rng [:(id I[01]),P2:] = [:(rng (id I[01])),(rng (P[01] ((1 / 2),1,((#) (0,1)),((0,1) (#))))):] by FUNCT_3:67;
consider g being Function of [:I[01],I[01]:],T such that
A18: g is continuous and
A19: for s being Point of I[01] holds
( g . (s,0) = Q . s & g . (s,1) = R . s & ( for t being Point of I[01] holds
( g . (0,t) = a & g . (1,t) = b ) ) ) by A8, BORSUK_2:def_7;
set f1 = f * [:(id I[01]),P1:];
set g1 = g * [:(id I[01]),P2:];
dom g = the carrier of [:I[01],I[01]:] by FUNCT_2:def_1
.= [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2 ;
then A20: dom (g * [:(id I[01]),P2:]) = dom [:(id I[01]),P2:] by A17, RELAT_1:27, TOPMETR:20, ZFMISC_1:96
.= [: the carrier of I[01], the carrier of (Closed-Interval-TSpace ((1 / 2),1)):] by A16, FUNCT_3:def_8 ;
Closed-Interval-TSpace ((1 / 2),1) = I[01] | B02 by TOPMETR:24;
then ( [:(id I[01]),P2:] is continuous Function of [:I[01],(Closed-Interval-TSpace ((1 / 2),1)):],[:I[01],I[01]:] & [:I[01],I[01]:] | Ewa1 = [:I[01],(Closed-Interval-TSpace ((1 / 2),1)):] ) by A13, BORSUK_3:22;
then reconsider g1 = g * [:(id I[01]),P2:] as continuous Function of ([:I[01],I[01]:] | Ewa1),T by A18;
Closed-Interval-TSpace (0,(1 / 2)) = I[01] | B01 by TOPMETR:24;
then ( [:(id I[01]),P1:] is continuous Function of [:I[01],(Closed-Interval-TSpace (0,(1 / 2))):],[:I[01],I[01]:] & [:I[01],I[01]:] | Ewa = [:I[01],(Closed-Interval-TSpace (0,(1 / 2))):] ) by A13, BORSUK_3:22;
then reconsider f1 = f * [:(id I[01]),P1:] as continuous Function of ([:I[01],I[01]:] | Ewa),T by A9;
A21: 1 is Point of I[01] by BORSUK_1:43;
A22: 0 is Point of I[01] by BORSUK_1:43;
then A23: [.0,1.] is compact Subset of I[01] by A21, BORSUK_4:24;
A24: 1 / 2 is Point of I[01] by BORSUK_1:43;
then [.0,(1 / 2).] is compact Subset of I[01] by A22, BORSUK_4:24;
then A25: Ewa is compact Subset of [:I[01],I[01]:] by A23, BORSUK_3:23;
[.(1 / 2),1.] is compact Subset of I[01] by A21, A24, BORSUK_4:24;
then A26: Ewa1 is compact Subset of [:I[01],I[01]:] by A23, BORSUK_3:23;
A27: dom [:(id I[01]),P1:] = the carrier of [:I[01],(Closed-Interval-TSpace (0,(1 / 2))):] by FUNCT_2:def_1
.= [: the carrier of I[01], the carrier of (Closed-Interval-TSpace (0,(1 / 2))):] by BORSUK_1:def_2 ;
A28: dom [:(id I[01]),P2:] = [:(dom (id I[01])),(dom P2):] by FUNCT_3:def_8;
A29: dom [:(id I[01]),P1:] = [:(dom (id I[01])),(dom P1):] by FUNCT_3:def_8;
A30: dom [:(id I[01]),P2:] = the carrier of [:I[01],(Closed-Interval-TSpace ((1 / 2),1)):] by FUNCT_2:def_1
.= [: the carrier of I[01], the carrier of (Closed-Interval-TSpace ((1 / 2),1)):] by BORSUK_1:def_2 ;
A31: ( [#] ([:I[01],I[01]:] | Ewa) = Ewa & [#] ([:I[01],I[01]:] | Ewa1) = Ewa1 ) by PRE_TOPC:def_5;
then A32: ([#] ([:I[01],I[01]:] | Ewa)) /\ ([#] ([:I[01],I[01]:] | Ewa1)) = [:[.0,1.],([.0,(1 / 2).] /\ [.(1 / 2),1.]):] by ZFMISC_1:99
.= [:[.0,1.],{(1 / 2)}:] by XXREAL_1:418 ;
A33: for p being set st p in ([#] ([:I[01],I[01]:] | Ewa)) /\ ([#] ([:I[01],I[01]:] | Ewa1)) holds
f1 . p = g1 . p
proof
let p be set ; ::_thesis: ( p in ([#] ([:I[01],I[01]:] | Ewa)) /\ ([#] ([:I[01],I[01]:] | Ewa1)) implies f1 . p = g1 . p )
assume p in ([#] ([:I[01],I[01]:] | Ewa)) /\ ([#] ([:I[01],I[01]:] | Ewa1)) ; ::_thesis: f1 . p = g1 . p
then consider x, y being set such that
A34: x in [.0,1.] and
A35: y in {(1 / 2)} and
A36: p = [x,y] by A32, ZFMISC_1:def_2;
x in { r where r is Real : ( 0 <= r & r <= 1 ) } by A34, RCOMP_1:def_1;
then A37: ex r1 being Real st
( r1 = x & 0 <= r1 & r1 <= 1 ) ;
then reconsider x9 = x as Point of I[01] by BORSUK_1:43;
A38: y = 1 / 2 by A35, TARSKI:def_1;
f1 . p = g1 . p
proof
1 / 2 in [.0,(1 / 2).] by XXREAL_1:1;
then reconsider y9 = 1 / 2 as Point of (Closed-Interval-TSpace (0,(1 / 2))) by TOPMETR:18;
set t9 = 1 / 2;
reconsider r1 = (#) (0,1), r2 = (0,1) (#) as Real by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5;
A39: P1 . y9 = (((r2 - r1) / ((1 / 2) - 0)) * (1 / 2)) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0)) by TREAL_1:11
.= (((1 - r1) / ((1 / 2) - 0)) * (1 / 2)) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0)) by TREAL_1:def_2
.= (((1 - 0) / ((1 / 2) - 0)) * (1 / 2)) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0)) by TREAL_1:def_1
.= (((1 - 0) / ((1 / 2) - 0)) * (1 / 2)) + ((((1 / 2) * 0) - (0 * 1)) / ((1 / 2) - 0)) by TREAL_1:def_1
.= 1 ;
reconsider y9 = 1 / 2 as Point of (Closed-Interval-TSpace ((1 / 2),1)) by A3, TOPMETR:18;
A40: P2 . y9 = (((r2 - r1) / (1 - (1 / 2))) * (1 / 2)) + (((1 * r1) - ((1 / 2) * r2)) / (1 - (1 / 2))) by TREAL_1:11
.= 0 by BORSUK_1:def_14, TREAL_1:5 ;
A41: x in the carrier of I[01] by A37, BORSUK_1:43;
then A42: [x,y] in dom [:(id I[01]),P2:] by A30, A4, A38, ZFMISC_1:87;
A43: [x,y] in dom [:(id I[01]),P1:] by A1, A27, A38, A41, ZFMISC_1:87;
then f1 . p = f . ([:(id I[01]),P1:] . (x,y)) by A36, FUNCT_1:13
.= f . (((id I[01]) . x),(P1 . y)) by A29, A43, FUNCT_3:65
.= f . (x9,1) by A38, A39, FUNCT_1:18
.= Q . x9 by A10
.= g . (x9,0) by A19
.= g . (((id I[01]) . x9),(P2 . y)) by A38, A40, FUNCT_1:18
.= g . ([:(id I[01]),P2:] . (x,y)) by A28, A42, FUNCT_3:65
.= g1 . p by A36, A42, FUNCT_1:13 ;
hence f1 . p = g1 . p ; ::_thesis: verum
end;
hence f1 . p = g1 . p ; ::_thesis: verum
end;
([#] ([:I[01],I[01]:] | Ewa)) \/ ([#] ([:I[01],I[01]:] | Ewa1)) = [:[.0,1.],([.0,(1 / 2).] \/ [.(1 / 2),1.]):] by A31, ZFMISC_1:97
.= [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:40, XXREAL_1:174
.= [#] [:I[01],I[01]:] by BORSUK_1:def_2 ;
then consider h being Function of [:I[01],I[01]:],T such that
A44: h = f1 +* g1 and
A45: h is continuous by A25, A26, A33, BORSUK_2:1;
A46: the carrier of (Closed-Interval-TSpace (0,(1 / 2))) = [.0,(1 / 2).] by TOPMETR:18;
A47: for t being Point of I[01] holds
( h . (0,t) = a & h . (1,t) = b )
proof
let t be Point of I[01]; ::_thesis: ( h . (0,t) = a & h . (1,t) = b )
percases ( t < 1 / 2 or t >= 1 / 2 ) ;
supposeA48: t < 1 / 2 ; ::_thesis: ( h . (0,t) = a & h . (1,t) = b )
reconsider r1 = (#) (0,1), r2 = (0,1) (#) as Real by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5;
A49: 0 <= t by BORSUK_1:43;
then A50: t in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by A6, A48, XXREAL_1:1;
0 in the carrier of I[01] by BORSUK_1:43;
then A51: [0,t] in dom [:(id I[01]),P1:] by A27, A50, ZFMISC_1:87;
P1 . t = (((r2 - r1) / ((1 / 2) - 0)) * t) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0)) by A50, TREAL_1:11
.= (((1 - r1) / (1 / 2)) * t) + (((1 / 2) * r1) / (1 / 2)) by TREAL_1:def_2
.= (((1 - 0) / (1 / 2)) * t) + (((1 / 2) * r1) / (1 / 2)) by TREAL_1:def_1
.= ((1 / (1 / 2)) * t) + (((1 / 2) * 0) / (1 / 2)) by TREAL_1:def_1
.= 2 * t ;
then A52: P1 . t is Point of I[01] by A48, Th3;
not t in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by A11, A48, XXREAL_1:1;
then not [0,t] in dom g1 by A20, ZFMISC_1:87;
hence h . (0,t) = f1 . (0,t) by A44, FUNCT_4:11
.= f . ([:(id I[01]),P1:] . (0,t)) by A51, FUNCT_1:13
.= f . (((id I[01]) . 0),(P1 . t)) by A29, A51, FUNCT_3:65
.= f . (0,(P1 . t)) by A22, FUNCT_1:18
.= a by A10, A52 ;
::_thesis: h . (1,t) = b
t in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by A46, A48, A49, XXREAL_1:1;
then A53: [1,t] in dom [:(id I[01]),P1:] by A27, A15, ZFMISC_1:87;
P1 . t = (((r2 - r1) / ((1 / 2) - 0)) * t) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0)) by A50, TREAL_1:11
.= (((1 - r1) / (1 / 2)) * t) + (((1 / 2) * r1) / (1 / 2)) by TREAL_1:def_2
.= (((1 - 0) / (1 / 2)) * t) + (((1 / 2) * r1) / (1 / 2)) by TREAL_1:def_1
.= ((1 / (1 / 2)) * t) + (((1 / 2) * 0) / (1 / 2)) by TREAL_1:def_1
.= 2 * t ;
then A54: P1 . t is Point of I[01] by A48, Th3;
not t in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by A12, A48, XXREAL_1:1;
then not [1,t] in dom g1 by A20, ZFMISC_1:87;
hence h . (1,t) = f1 . (1,t) by A44, FUNCT_4:11
.= f . ([:(id I[01]),P1:] . (1,t)) by A53, FUNCT_1:13
.= f . (((id I[01]) . 1),(P1 . t)) by A29, A53, FUNCT_3:65
.= f . (1,(P1 . t)) by A21, FUNCT_1:18
.= b by A10, A54 ;
::_thesis: verum
end;
supposeA55: t >= 1 / 2 ; ::_thesis: ( h . (0,t) = a & h . (1,t) = b )
reconsider r1 = (#) (0,1), r2 = (0,1) (#) as Real by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5;
t <= 1 by BORSUK_1:43;
then A56: t in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by A11, A55, XXREAL_1:1;
then A57: [1,t] in dom [:(id I[01]),P2:] by A30, A15, ZFMISC_1:87;
P2 . t = (((r2 - r1) / (1 - (1 / 2))) * t) + (((1 * r1) - ((1 / 2) * r2)) / (1 - (1 / 2))) by A56, TREAL_1:11
.= (((1 - r1) / (1 / 2)) * t) + (((1 * r1) - ((1 / 2) * r2)) / (1 / 2)) by TREAL_1:def_2
.= (((1 - 0) / (1 / 2)) * t) + (((1 * r1) - ((1 / 2) * r2)) / (1 / 2)) by TREAL_1:def_1
.= (2 * t) + (((1 * 0) - ((1 / 2) * r2)) / (1 / 2)) by TREAL_1:def_1
.= (2 * t) + ((- ((1 / 2) * r2)) / (1 / 2))
.= (2 * t) + ((- ((1 / 2) * 1)) / (1 / 2)) by TREAL_1:def_2
.= (2 * t) - 1 ;
then A58: P2 . t is Point of I[01] by A55, Th4;
P2 . t = (((r2 - r1) / (1 - (1 / 2))) * t) + (((1 * r1) - ((1 / 2) * r2)) / (1 - (1 / 2))) by A56, TREAL_1:11
.= (((1 - r1) / (1 / 2)) * t) + (((1 * r1) - ((1 / 2) * r2)) / (1 / 2)) by TREAL_1:def_2
.= (((1 - 0) / (1 / 2)) * t) + (((1 * r1) - ((1 / 2) * r2)) / (1 / 2)) by TREAL_1:def_1
.= ((1 / (1 / 2)) * t) + (((1 * 0) - ((1 / 2) * r2)) / (1 / 2)) by TREAL_1:def_1
.= ((1 / (1 / 2)) * t) + (((1 * 0) - ((1 / 2) * 1)) / (1 / 2)) by TREAL_1:def_2
.= (2 * t) - 1 ;
then A59: P2 . t is Point of I[01] by A55, Th4;
A60: 0 in the carrier of I[01] by BORSUK_1:43;
then A61: [0,t] in dom [:(id I[01]),P2:] by A30, A56, ZFMISC_1:87;
[0,t] in dom g1 by A20, A60, A56, ZFMISC_1:87;
hence h . (0,t) = g1 . (0,t) by A44, FUNCT_4:13
.= g . ([:(id I[01]),P2:] . (0,t)) by A61, FUNCT_1:13
.= g . (((id I[01]) . 0),(P2 . t)) by A28, A61, FUNCT_3:65
.= g . (0,(P2 . t)) by A22, FUNCT_1:18
.= a by A19, A59 ;
::_thesis: h . (1,t) = b
[1,t] in dom g1 by A20, A15, A56, ZFMISC_1:87;
hence h . (1,t) = g1 . (1,t) by A44, FUNCT_4:13
.= g . ([:(id I[01]),P2:] . (1,t)) by A57, FUNCT_1:13
.= g . (((id I[01]) . 1),(P2 . t)) by A28, A57, FUNCT_3:65
.= g . (1,(P2 . t)) by A21, FUNCT_1:18
.= b by A19, A58 ;
::_thesis: verum
end;
end;
end;
for s being Point of I[01] holds
( h . (s,0) = P . s & h . (s,1) = R . s )
proof
reconsider r1 = (#) (0,1), r2 = (0,1) (#) as Real by BORSUK_1:def_14, BORSUK_1:def_15, TREAL_1:5;
let s be Point of I[01]; ::_thesis: ( h . (s,0) = P . s & h . (s,1) = R . s )
( 1 = (0,1) (#) & 1 = ((1 / 2),1) (#) ) by TREAL_1:def_2;
then A62: P2 . 1 = 1 by TREAL_1:13;
A63: the carrier of (Closed-Interval-TSpace ((1 / 2),1)) = [.(1 / 2),1.] by TOPMETR:18;
then A64: 1 in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by XXREAL_1:1;
then A65: [s,1] in dom [:(id I[01]),P2:] by A30, ZFMISC_1:87;
[s,1] in dom g1 by A20, A64, ZFMISC_1:87;
then A66: h . (s,1) = g1 . (s,1) by A44, FUNCT_4:13
.= g . ([:(id I[01]),P2:] . (s,1)) by A65, FUNCT_1:13
.= g . (((id I[01]) . s),(P2 . 1)) by A28, A65, FUNCT_3:65
.= g . (s,1) by A62, FUNCT_1:18
.= R . s by A19 ;
A67: 0 in the carrier of (Closed-Interval-TSpace (0,(1 / 2))) by A6, XXREAL_1:1;
then A68: P1 . 0 = (((r2 - r1) / ((1 / 2) - 0)) * 0) + ((((1 / 2) * r1) - (0 * r2)) / ((1 / 2) - 0)) by TREAL_1:11
.= (((1 / 2) * 0) - (0 * r2)) / ((1 / 2) - 0) by TREAL_1:def_1 ;
A69: [s,0] in dom [:(id I[01]),P1:] by A27, A67, ZFMISC_1:87;
not 0 in the carrier of (Closed-Interval-TSpace ((1 / 2),1)) by A63, XXREAL_1:1;
then not [s,0] in dom g1 by A20, ZFMISC_1:87;
then h . (s,0) = f1 . (s,0) by A44, FUNCT_4:11
.= f . ([:(id I[01]),P1:] . (s,0)) by A69, FUNCT_1:13
.= f . (((id I[01]) . s),(P1 . 0)) by A29, A69, FUNCT_3:65
.= f . (s,0) by A68, FUNCT_1:18
.= P . s by A10 ;
hence ( h . (s,0) = P . s & h . (s,1) = R . s ) by A66; ::_thesis: verum
end;
hence P,R are_homotopic by A45, A47, BORSUK_2:def_7; ::_thesis: verum
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
let P be Path of a,b; ::_thesis: for Q being constant Path of b,b st a,b are_connected holds
P + Q,P are_homotopic
let Q be constant Path of b,b; ::_thesis: ( a,b are_connected implies P + Q,P are_homotopic )
assume A1: a,b are_connected ; ::_thesis: P + Q,P are_homotopic
RePar (P,1RP) = P + Q by A1, Th50;
hence P + Q,P are_homotopic by A1, Th45, Th47; ::_thesis: verum
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
proof
let X be non empty pathwise_connected TopSpace; ::_thesis: 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
let a1, b1 be Point of X; ::_thesis: for P being Path of a1,b1
for Q being constant Path of b1,b1 holds P + Q,P are_homotopic
let P be Path of a1,b1; ::_thesis: for Q being constant Path of b1,b1 holds P + Q,P are_homotopic
let Q be constant Path of b1,b1; ::_thesis: P + Q,P are_homotopic
a1,b1 are_connected by BORSUK_2:def_3;
hence P + Q,P are_homotopic by Th80; ::_thesis: verum
end;
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
let P be Path of a,b; ::_thesis: for Q being constant Path of a,a st a,b are_connected holds
Q + P,P are_homotopic
let Q be constant Path of a,a; ::_thesis: ( a,b are_connected implies Q + P,P are_homotopic )
assume A1: a,b are_connected ; ::_thesis: Q + P,P are_homotopic
RePar (P,2RP) = Q + P by A1, Th51;
hence Q + P,P are_homotopic by A1, Th45, Th48; ::_thesis: verum
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
proof
let X be non empty pathwise_connected TopSpace; ::_thesis: 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
let a1, b1 be Point of X; ::_thesis: for P being Path of a1,b1
for Q being constant Path of a1,a1 holds Q + P,P are_homotopic
let P be Path of a1,b1; ::_thesis: for Q being constant Path of a1,a1 holds Q + P,P are_homotopic
let Q be constant Path of a1,a1; ::_thesis: Q + P,P are_homotopic
a1,b1 are_connected by BORSUK_2:def_3;
hence Q + P,P are_homotopic by Th82; ::_thesis: verum
end;
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
let T be non empty TopSpace; ::_thesis: 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
let a, b be Point of T; ::_thesis: 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
set S = [:I[01],I[01]:];
let P be Path of a,b; ::_thesis: for Q being constant Path of a,a st a,b are_connected holds
P + (- P),Q are_homotopic
let Q be constant Path of a,a; ::_thesis: ( a,b are_connected implies P + (- P),Q are_homotopic )
assume A1: a,b are_connected ; ::_thesis: P + (- P),Q are_homotopic
reconsider e2 = pr2 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:40;
set gg = (- P) * e2;
- P is continuous by A1, BORSUK_2:def_2;
then reconsider gg = (- P) * e2 as continuous Function of [:I[01],I[01]:],T ;
set S2 = [:I[01],I[01]:] | IBB;
reconsider g = gg | IBB as Function of ([:I[01],I[01]:] | IBB),T by PRE_TOPC:9;
reconsider g = g as continuous Function of ([:I[01],I[01]:] | IBB),T by TOPMETR:7;
A2: for x being Point of ([:I[01],I[01]:] | IBB) holds g . x = P . (1 - (x `2))
proof
let x be Point of ([:I[01],I[01]:] | IBB); ::_thesis: g . x = P . (1 - (x `2))
x in the carrier of ([:I[01],I[01]:] | IBB) ;
then A3: x in IBB by PRE_TOPC:8;
then A4: x in the carrier of [:I[01],I[01]:] ;
then A5: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A6: x = [(x `1),(x `2)] by MCART_1:21;
then A7: x `2 in the carrier of I[01] by A5, ZFMISC_1:87;
x `1 in the carrier of I[01] by A5, A6, ZFMISC_1:87;
then A8: e2 . ((x `1),(x `2)) = x `2 by A7, FUNCT_3:def_5;
A9: x in dom e2 by A4, FUNCT_2:def_1;
g . x = gg . x by A3, FUNCT_1:49
.= (- P) . (e2 . x) by A9, FUNCT_1:13
.= P . (1 - (x `2)) by A1, A6, A7, A8, BORSUK_2:def_6 ;
hence g . x = P . (1 - (x `2)) ; ::_thesis: verum
end;
set S3 = [:I[01],I[01]:] | ICC;
set S1 = [:I[01],I[01]:] | IAA;
reconsider e1 = pr1 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:39;
A10: a,a are_connected ;
then reconsider PP = P + (- P) as continuous Path of a,a by BORSUK_2:def_2;
set ff = PP * e1;
reconsider f = (PP * e1) | IAA as Function of ([:I[01],I[01]:] | IAA),T by PRE_TOPC:9;
reconsider f = f as continuous Function of ([:I[01],I[01]:] | IAA),T by TOPMETR:7;
set S12 = [:I[01],I[01]:] | (IAA \/ IBB);
reconsider S12 = [:I[01],I[01]:] | (IAA \/ IBB) as non empty SubSpace of [:I[01],I[01]:] ;
A11: the carrier of S12 = IAA \/ IBB by PRE_TOPC:8;
set hh = PP * e1;
reconsider h = (PP * e1) | ICC as Function of ([:I[01],I[01]:] | ICC),T by PRE_TOPC:9;
reconsider h = h as continuous Function of ([:I[01],I[01]:] | ICC),T by TOPMETR:7;
A12: for x being Point of ([:I[01],I[01]:] | ICC) holds h . x = (- P) . ((2 * (x `1)) - 1)
proof
let x be Point of ([:I[01],I[01]:] | ICC); ::_thesis: h . x = (- P) . ((2 * (x `1)) - 1)
x in the carrier of ([:I[01],I[01]:] | ICC) ;
then A13: x in ICC by PRE_TOPC:8;
then A14: x `1 >= 1 / 2 by Th60;
A15: x in the carrier of [:I[01],I[01]:] by A13;
then A16: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A17: x = [(x `1),(x `2)] by MCART_1:21;
then A18: x `1 in the carrier of I[01] by A16, ZFMISC_1:87;
x `2 in the carrier of I[01] by A16, A17, ZFMISC_1:87;
then A19: e1 . ((x `1),(x `2)) = x `1 by A18, FUNCT_3:def_4;
A20: x in dom e1 by A15, FUNCT_2:def_1;
h . x = (PP * e1) . x by A13, FUNCT_1:49
.= (P + (- P)) . (e1 . x) by A20, FUNCT_1:13
.= (- P) . ((2 * (x `1)) - 1) by A1, A17, A18, A19, A14, BORSUK_2:def_5 ;
hence h . x = (- P) . ((2 * (x `1)) - 1) ; ::_thesis: verum
end;
A21: for x being Point of ([:I[01],I[01]:] | IAA) holds f . x = P . (2 * (x `1))
proof
let x be Point of ([:I[01],I[01]:] | IAA); ::_thesis: f . x = P . (2 * (x `1))
x in the carrier of ([:I[01],I[01]:] | IAA) ;
then A22: x in IAA by PRE_TOPC:8;
then A23: x `1 <= 1 / 2 by Th59;
A24: x in the carrier of [:I[01],I[01]:] by A22;
then A25: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A26: x = [(x `1),(x `2)] by MCART_1:21;
then A27: x `1 in the carrier of I[01] by A25, ZFMISC_1:87;
x `2 in the carrier of I[01] by A25, A26, ZFMISC_1:87;
then A28: e1 . ((x `1),(x `2)) = x `1 by A27, FUNCT_3:def_4;
A29: x in dom e1 by A24, FUNCT_2:def_1;
f . x = (PP * e1) . x by A22, FUNCT_1:49
.= (P + (- P)) . (e1 . x) by A29, FUNCT_1:13
.= P . (2 * (x `1)) by A1, A26, A27, A28, A23, BORSUK_2:def_5 ;
hence f . x = P . (2 * (x `1)) ; ::_thesis: verum
end;
A30: for p being set st p in ([#] ([:I[01],I[01]:] | IAA)) /\ ([#] ([:I[01],I[01]:] | IBB)) holds
f . p = g . p
proof
let p be set ; ::_thesis: ( p in ([#] ([:I[01],I[01]:] | IAA)) /\ ([#] ([:I[01],I[01]:] | IBB)) implies f . p = g . p )
assume p in ([#] ([:I[01],I[01]:] | IAA)) /\ ([#] ([:I[01],I[01]:] | IBB)) ; ::_thesis: f . p = g . p
then A31: p in ([#] ([:I[01],I[01]:] | IAA)) /\ IBB by PRE_TOPC:def_5;
then A32: p in IAA /\ IBB by PRE_TOPC:def_5;
then consider r being Point of [:I[01],I[01]:] such that
A33: r = p and
A34: r `2 = 1 - (2 * (r `1)) by Th57;
A35: 2 * (r `1) = 1 - (r `2) by A34;
p in IAA by A32, XBOOLE_0:def_4;
then reconsider pp = p as Point of ([:I[01],I[01]:] | IAA) by PRE_TOPC:8;
p in IBB by A31, XBOOLE_0:def_4;
then A36: pp is Point of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
f . p = P . (2 * (pp `1)) by A21
.= g . p by A2, A33, A35, A36 ;
hence f . p = g . p ; ::_thesis: verum
end;
reconsider s3 = [#] ([:I[01],I[01]:] | ICC) as Subset of [:I[01],I[01]:] by PRE_TOPC:def_5;
A37: s3 = ICC by PRE_TOPC:def_5;
A38: ( [:I[01],I[01]:] | IAA is SubSpace of S12 & [:I[01],I[01]:] | IBB is SubSpace of S12 ) by TOPMETR:22, XBOOLE_1:7;
A39: [#] ([:I[01],I[01]:] | IBB) = IBB by PRE_TOPC:def_5;
A40: [#] ([:I[01],I[01]:] | IAA) = IAA by PRE_TOPC:def_5;
then reconsider s1 = [#] ([:I[01],I[01]:] | IAA), s2 = [#] ([:I[01],I[01]:] | IBB) as Subset of S12 by A11, A39, XBOOLE_1:7;
A41: s1 is closed by A40, TOPS_2:26;
A42: s2 is closed by A39, TOPS_2:26;
([#] ([:I[01],I[01]:] | IAA)) \/ ([#] ([:I[01],I[01]:] | IBB)) = [#] S12 by A11, A39, PRE_TOPC:def_5;
then consider fg being Function of S12,T such that
A43: fg = f +* g and
A44: fg is continuous by A30, A38, A41, A42, JGRAPH_2:1;
A45: [#] ([:I[01],I[01]:] | ICC) = ICC by PRE_TOPC:def_5;
A46: for p being set st p in ([#] S12) /\ ([#] ([:I[01],I[01]:] | ICC)) holds
fg . p = h . p
proof
let p be set ; ::_thesis: ( p in ([#] S12) /\ ([#] ([:I[01],I[01]:] | ICC)) implies fg . p = h . p )
[(1 / 2),0] in IBB /\ ICC by Th66, Th67, XBOOLE_0:def_4;
then A47: {[(1 / 2),0]} c= IBB /\ ICC by ZFMISC_1:31;
assume p in ([#] S12) /\ ([#] ([:I[01],I[01]:] | ICC)) ; ::_thesis: fg . p = h . p
then p in {[(1 / 2),0]} \/ (IBB /\ ICC) by A11, A45, Th72, XBOOLE_1:23;
then A48: p in IBB /\ ICC by A47, XBOOLE_1:12;
then p in ICC by XBOOLE_0:def_4;
then reconsider pp = p as Point of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
A49: p in IBB by A48, XBOOLE_0:def_4;
then A50: pp is Point of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
A51: ex q being Point of [:I[01],I[01]:] st
( q = p & q `2 = (2 * (q `1)) - 1 ) by A48, Th58;
then A52: (2 * (pp `1)) - 1 is Point of I[01] by Th27;
p in the carrier of ([:I[01],I[01]:] | IBB) by A49, PRE_TOPC:8;
then p in dom g by FUNCT_2:def_1;
then fg . p = g . p by A43, FUNCT_4:13
.= P . (1 - (pp `2)) by A2, A50
.= (- P) . ((2 * (pp `1)) - 1) by A1, A51, A52, BORSUK_2:def_6
.= h . p by A12 ;
hence fg . p = h . p ; ::_thesis: verum
end;
([#] S12) \/ ([#] ([:I[01],I[01]:] | ICC)) = (IAA \/ IBB) \/ ICC by A11, PRE_TOPC:def_5
.= [#] [:I[01],I[01]:] by Th56, BORSUK_1:40, BORSUK_1:def_2 ;
then consider H being Function of [:I[01],I[01]:],T such that
A53: H = fg +* h and
A54: H is continuous by A11, A44, A46, A37, JGRAPH_2:1;
A55: for s being Point of I[01] holds
( H . (s,0) = (P + (- P)) . s & H . (s,1) = Q . s )
proof
let s be Point of I[01]; ::_thesis: ( H . (s,0) = (P + (- P)) . s & H . (s,1) = Q . s )
thus H . (s,0) = (P + (- P)) . s ::_thesis: H . (s,1) = Q . s
proof
A56: [s,0] `1 = s ;
percases ( s < 1 / 2 or s = 1 / 2 or s > 1 / 2 ) by XXREAL_0:1;
supposeA57: s < 1 / 2 ; ::_thesis: H . (s,0) = (P + (- P)) . s
then not [s,0] in IBB by Th71;
then not [s,0] in the carrier of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
then A58: not [s,0] in dom g ;
[s,0] in IAA by A57, Th70;
then A59: [s,0] in the carrier of ([:I[01],I[01]:] | IAA) by PRE_TOPC:8;
not [s,0] in ICC by A57, Th71;
then not [s,0] in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then not [s,0] in dom h ;
then H . [s,0] = fg . [s,0] by A53, FUNCT_4:11
.= f . [s,0] by A43, A58, FUNCT_4:11
.= P . (2 * s) by A21, A56, A59
.= (P + (- P)) . s by A1, A57, BORSUK_2:def_5 ;
hence H . (s,0) = (P + (- P)) . s ; ::_thesis: verum
end;
supposeA60: s = 1 / 2 ; ::_thesis: H . (s,0) = (P + (- P)) . s
then A61: [s,0] in the carrier of ([:I[01],I[01]:] | ICC) by Th66, PRE_TOPC:8;
then [s,0] in dom h by FUNCT_2:def_1;
then H . [s,0] = h . [s,0] by A53, FUNCT_4:13
.= (- P) . ((2 * s) - 1) by A12, A56, A61
.= b by A1, A60, BORSUK_2:def_2
.= P . (2 * (1 / 2)) by A1, BORSUK_2:def_2
.= (P + (- P)) . s by A1, A60, BORSUK_2:def_5 ;
hence H . (s,0) = (P + (- P)) . s ; ::_thesis: verum
end;
supposeA62: s > 1 / 2 ; ::_thesis: H . (s,0) = (P + (- P)) . s
then [s,0] in ICC by Th69;
then A63: [s,0] in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then [s,0] in dom h by FUNCT_2:def_1;
then H . [s,0] = h . [s,0] by A53, FUNCT_4:13
.= (- P) . ((2 * s) - 1) by A12, A56, A63
.= (P + (- P)) . s by A1, A62, BORSUK_2:def_5 ;
hence H . (s,0) = (P + (- P)) . s ; ::_thesis: verum
end;
end;
end;
thus H . (s,1) = Q . s ::_thesis: verum
proof
A64: [s,1] `2 = 1 ;
A65: [s,1] `1 = s ;
A66: dom Q = the carrier of I[01] by FUNCT_2:def_1;
then A67: 0 in dom Q by BORSUK_1:43;
percases ( s <> 1 or s = 1 ) ;
supposeA68: s <> 1 ; ::_thesis: H . (s,1) = Q . s
[s,1] in IBB by Th65;
then A69: [s,1] in the carrier of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
then A70: [s,1] in dom g by FUNCT_2:def_1;
not [s,1] in ICC by A68, Th63;
then not [s,1] in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then not [s,1] in dom h ;
then H . [s,1] = fg . [s,1] by A53, FUNCT_4:11
.= g . [s,1] by A43, A70, FUNCT_4:13
.= P . (1 - 1) by A2, A64, A69
.= a by A1, BORSUK_2:def_2
.= Q . 0 by A10, BORSUK_2:def_2
.= Q . s by A66, A67, FUNCT_1:def_10 ;
hence H . (s,1) = Q . s ; ::_thesis: verum
end;
supposeA71: s = 1 ; ::_thesis: H . (s,1) = Q . s
then A72: [s,1] in the carrier of ([:I[01],I[01]:] | ICC) by Th66, PRE_TOPC:8;
then [s,1] in dom h by FUNCT_2:def_1;
then H . [s,1] = h . [s,1] by A53, FUNCT_4:13
.= (- P) . ((2 * s) - 1) by A12, A65, A72
.= a by A1, A71, BORSUK_2:def_2
.= Q . 0 by A10, BORSUK_2:def_2
.= Q . s by A66, A67, FUNCT_1:def_10 ;
hence H . (s,1) = Q . s ; ::_thesis: verum
end;
end;
end;
end;
for t being Point of I[01] holds
( H . (0,t) = a & H . (1,t) = a )
proof
let t be Point of I[01]; ::_thesis: ( H . (0,t) = a & H . (1,t) = a )
thus H . (0,t) = a ::_thesis: H . (1,t) = a
proof
0 in the carrier of I[01] by BORSUK_1:43;
then reconsider x = [0,t] as Point of [:I[01],I[01]:] by Lm1;
A73: x `2 = t by MCART_1:7;
x in IAA by Th61;
then A74: x is Point of ([:I[01],I[01]:] | IAA) by PRE_TOPC:8;
A75: x `1 = 0 by MCART_1:7;
then not x in ICC by Th60;
then not x in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then A76: not [0,t] in dom h ;
percases ( t <> 1 or t = 1 ) ;
suppose t <> 1 ; ::_thesis: H . (0,t) = a
then not x in IBB by Th62;
then not x in the carrier of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
then not x in dom g ;
then fg . [0,t] = f . [0,t] by A43, FUNCT_4:11
.= P . (2 * (x `1)) by A21, A74
.= a by A1, A75, BORSUK_2:def_2 ;
hence H . (0,t) = a by A53, A76, FUNCT_4:11; ::_thesis: verum
end;
supposeA77: t = 1 ; ::_thesis: H . (0,t) = a
then A78: x in the carrier of ([:I[01],I[01]:] | IBB) by Th64, PRE_TOPC:8;
then x in dom g by FUNCT_2:def_1;
then fg . [0,t] = g . [0,1] by A43, A77, FUNCT_4:13
.= P . (1 - (x `2)) by A2, A77, A78
.= a by A1, A73, A77, BORSUK_2:def_2 ;
hence H . (0,t) = a by A53, A76, FUNCT_4:11; ::_thesis: verum
end;
end;
end;
thus H . (1,t) = a ::_thesis: verum
proof
1 in the carrier of I[01] by BORSUK_1:43;
then reconsider x = [1,t] as Point of [:I[01],I[01]:] by Lm1;
A79: x `1 = 1 by MCART_1:7;
x in ICC by Th68;
then A80: x in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then A81: [1,t] in dom h by FUNCT_2:def_1;
h . [1,t] = (- P) . ((2 * (x `1)) - 1) by A12, A80
.= a by A1, A79, BORSUK_2:def_2 ;
hence H . (1,t) = a by A53, A81, FUNCT_4:13; ::_thesis: verum
end;
end;
hence P + (- P),Q are_homotopic by A54, A55, BORSUK_2:def_7; ::_thesis: verum
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
proof
let X be non empty pathwise_connected TopSpace; ::_thesis: 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
let a1, b1 be Point of X; ::_thesis: for P being Path of a1,b1
for Q being constant Path of a1,a1 holds P + (- P),Q are_homotopic
let P be Path of a1,b1; ::_thesis: for Q being constant Path of a1,a1 holds P + (- P),Q are_homotopic
let Q be constant Path of a1,a1; ::_thesis: P + (- P),Q are_homotopic
a1,b1 are_connected by BORSUK_2:def_3;
hence P + (- P),Q are_homotopic by Th84; ::_thesis: verum
end;
theorem Th86: :: BORSUK_6:86
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
let T be non empty TopSpace; ::_thesis: 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
let b, a be Point of T; ::_thesis: 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
set S = [:I[01],I[01]:];
let P be Path of b,a; ::_thesis: for Q being constant Path of a,a st b,a are_connected holds
(- P) + P,Q are_homotopic
let Q be constant Path of a,a; ::_thesis: ( b,a are_connected implies (- P) + P,Q are_homotopic )
assume A1: b,a are_connected ; ::_thesis: (- P) + P,Q are_homotopic
reconsider e2 = pr2 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:40;
set gg = P * e2;
P is continuous by A1, BORSUK_2:def_2;
then reconsider gg = P * e2 as continuous Function of [:I[01],I[01]:],T ;
set S2 = [:I[01],I[01]:] | IBB;
reconsider g = gg | IBB as Function of ([:I[01],I[01]:] | IBB),T by PRE_TOPC:9;
reconsider g = g as continuous Function of ([:I[01],I[01]:] | IBB),T by TOPMETR:7;
A2: for x being Point of ([:I[01],I[01]:] | IBB) holds g . x = (- P) . (1 - (x `2))
proof
let x be Point of ([:I[01],I[01]:] | IBB); ::_thesis: g . x = (- P) . (1 - (x `2))
x in the carrier of ([:I[01],I[01]:] | IBB) ;
then A3: x in IBB by PRE_TOPC:8;
then A4: x in the carrier of [:I[01],I[01]:] ;
then A5: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A6: x = [(x `1),(x `2)] by MCART_1:21;
then A7: x `2 in the carrier of I[01] by A5, ZFMISC_1:87;
then A8: 1 - (x `2) in the carrier of I[01] by JORDAN5B:4;
x `1 in the carrier of I[01] by A5, A6, ZFMISC_1:87;
then A9: e2 . ((x `1),(x `2)) = x `2 by A7, FUNCT_3:def_5;
A10: x in dom e2 by A4, FUNCT_2:def_1;
g . x = gg . ((x `1),(x `2)) by A3, A6, FUNCT_1:49
.= P . (1 - (1 - (x `2))) by A6, A9, A10, FUNCT_1:13
.= (- P) . (1 - (x `2)) by A1, A8, BORSUK_2:def_6 ;
hence g . x = (- P) . (1 - (x `2)) ; ::_thesis: verum
end;
set S3 = [:I[01],I[01]:] | ICC;
set S1 = [:I[01],I[01]:] | IAA;
reconsider e1 = pr1 ( the carrier of I[01], the carrier of I[01]) as continuous Function of [:I[01],I[01]:],I[01] by YELLOW12:39;
A11: a,a are_connected ;
then reconsider PP = (- P) + P as continuous Path of a,a by BORSUK_2:def_2;
set ff = PP * e1;
reconsider f = (PP * e1) | IAA as Function of ([:I[01],I[01]:] | IAA),T by PRE_TOPC:9;
reconsider f = f as continuous Function of ([:I[01],I[01]:] | IAA),T by TOPMETR:7;
set S12 = [:I[01],I[01]:] | (IAA \/ IBB);
reconsider S12 = [:I[01],I[01]:] | (IAA \/ IBB) as non empty SubSpace of [:I[01],I[01]:] ;
A12: the carrier of S12 = IAA \/ IBB by PRE_TOPC:8;
set hh = PP * e1;
reconsider h = (PP * e1) | ICC as Function of ([:I[01],I[01]:] | ICC),T by PRE_TOPC:9;
reconsider h = h as continuous Function of ([:I[01],I[01]:] | ICC),T by TOPMETR:7;
A13: for x being Point of ([:I[01],I[01]:] | ICC) holds h . x = P . ((2 * (x `1)) - 1)
proof
let x be Point of ([:I[01],I[01]:] | ICC); ::_thesis: h . x = P . ((2 * (x `1)) - 1)
x in the carrier of ([:I[01],I[01]:] | ICC) ;
then A14: x in ICC by PRE_TOPC:8;
then A15: x `1 >= 1 / 2 by Th60;
A16: x in the carrier of [:I[01],I[01]:] by A14;
then A17: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A18: x = [(x `1),(x `2)] by MCART_1:21;
then A19: x `1 in the carrier of I[01] by A17, ZFMISC_1:87;
x `2 in the carrier of I[01] by A17, A18, ZFMISC_1:87;
then A20: e1 . ((x `1),(x `2)) = x `1 by A19, FUNCT_3:def_4;
A21: x in dom e1 by A16, FUNCT_2:def_1;
h . x = (PP * e1) . x by A14, FUNCT_1:49
.= ((- P) + P) . (e1 . ((x `1),(x `2))) by A18, A21, FUNCT_1:13
.= P . ((2 * (x `1)) - 1) by A1, A19, A20, A15, BORSUK_2:def_5 ;
hence h . x = P . ((2 * (x `1)) - 1) ; ::_thesis: verum
end;
A22: for x being Point of ([:I[01],I[01]:] | IAA) holds f . x = (- P) . (2 * (x `1))
proof
let x be Point of ([:I[01],I[01]:] | IAA); ::_thesis: f . x = (- P) . (2 * (x `1))
x in the carrier of ([:I[01],I[01]:] | IAA) ;
then A23: x in IAA by PRE_TOPC:8;
then A24: x `1 <= 1 / 2 by Th59;
A25: x in the carrier of [:I[01],I[01]:] by A23;
then A26: x in [: the carrier of I[01], the carrier of I[01]:] by BORSUK_1:def_2;
then A27: x = [(x `1),(x `2)] by MCART_1:21;
then A28: x `1 in the carrier of I[01] by A26, ZFMISC_1:87;
x `2 in the carrier of I[01] by A26, A27, ZFMISC_1:87;
then A29: e1 . ((x `1),(x `2)) = x `1 by A28, FUNCT_3:def_4;
A30: x in dom e1 by A25, FUNCT_2:def_1;
f . x = (PP * e1) . x by A23, FUNCT_1:49
.= ((- P) + P) . (e1 . x) by A30, FUNCT_1:13
.= (- P) . (2 * (x `1)) by A1, A27, A28, A29, A24, BORSUK_2:def_5 ;
hence f . x = (- P) . (2 * (x `1)) ; ::_thesis: verum
end;
A31: for p being set st p in ([#] ([:I[01],I[01]:] | IAA)) /\ ([#] ([:I[01],I[01]:] | IBB)) holds
f . p = g . p
proof
let p be set ; ::_thesis: ( p in ([#] ([:I[01],I[01]:] | IAA)) /\ ([#] ([:I[01],I[01]:] | IBB)) implies f . p = g . p )
assume p in ([#] ([:I[01],I[01]:] | IAA)) /\ ([#] ([:I[01],I[01]:] | IBB)) ; ::_thesis: f . p = g . p
then A32: p in ([#] ([:I[01],I[01]:] | IAA)) /\ IBB by PRE_TOPC:def_5;
then A33: p in IAA /\ IBB by PRE_TOPC:def_5;
then consider r being Point of [:I[01],I[01]:] such that
A34: r = p and
A35: r `2 = 1 - (2 * (r `1)) by Th57;
A36: 2 * (r `1) = 1 - (r `2) by A35;
p in IAA by A33, XBOOLE_0:def_4;
then reconsider pp = p as Point of ([:I[01],I[01]:] | IAA) by PRE_TOPC:8;
p in IBB by A32, XBOOLE_0:def_4;
then A37: pp is Point of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
f . p = (- P) . (2 * (pp `1)) by A22
.= g . p by A2, A34, A36, A37 ;
hence f . p = g . p ; ::_thesis: verum
end;
reconsider s3 = [#] ([:I[01],I[01]:] | ICC) as Subset of [:I[01],I[01]:] by PRE_TOPC:def_5;
A38: s3 = ICC by PRE_TOPC:def_5;
A39: ( [:I[01],I[01]:] | IAA is SubSpace of S12 & [:I[01],I[01]:] | IBB is SubSpace of S12 ) by TOPMETR:22, XBOOLE_1:7;
A40: [#] ([:I[01],I[01]:] | IBB) = IBB by PRE_TOPC:def_5;
A41: [#] ([:I[01],I[01]:] | IAA) = IAA by PRE_TOPC:def_5;
then reconsider s1 = [#] ([:I[01],I[01]:] | IAA), s2 = [#] ([:I[01],I[01]:] | IBB) as Subset of S12 by A12, A40, XBOOLE_1:7;
A42: s1 is closed by A41, TOPS_2:26;
A43: s2 is closed by A40, TOPS_2:26;
([#] ([:I[01],I[01]:] | IAA)) \/ ([#] ([:I[01],I[01]:] | IBB)) = [#] S12 by A12, A40, PRE_TOPC:def_5;
then consider fg being Function of S12,T such that
A44: fg = f +* g and
A45: fg is continuous by A31, A39, A42, A43, JGRAPH_2:1;
A46: [#] ([:I[01],I[01]:] | ICC) = ICC by PRE_TOPC:def_5;
A47: for p being set st p in ([#] S12) /\ ([#] ([:I[01],I[01]:] | ICC)) holds
fg . p = h . p
proof
let p be set ; ::_thesis: ( p in ([#] S12) /\ ([#] ([:I[01],I[01]:] | ICC)) implies fg . p = h . p )
[(1 / 2),0] in IBB /\ ICC by Th66, Th67, XBOOLE_0:def_4;
then A48: {[(1 / 2),0]} c= IBB /\ ICC by ZFMISC_1:31;
assume p in ([#] S12) /\ ([#] ([:I[01],I[01]:] | ICC)) ; ::_thesis: fg . p = h . p
then p in {[(1 / 2),0]} \/ (IBB /\ ICC) by A12, A46, Th72, XBOOLE_1:23;
then A49: p in IBB /\ ICC by A48, XBOOLE_1:12;
then p in ICC by XBOOLE_0:def_4;
then reconsider pp = p as Point of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
A50: ex q being Point of [:I[01],I[01]:] st
( q = p & q `2 = (2 * (q `1)) - 1 ) by A49, Th58;
pp `2 is Point of I[01] by A49, Th27;
then A51: 1 - (pp `2) in the carrier of I[01] by JORDAN5B:4;
A52: p in IBB by A49, XBOOLE_0:def_4;
then A53: pp is Point of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
p in the carrier of ([:I[01],I[01]:] | IBB) by A52, PRE_TOPC:8;
then p in dom g by FUNCT_2:def_1;
then fg . p = g . p by A44, FUNCT_4:13
.= (- P) . (1 - (pp `2)) by A2, A53
.= P . (1 - (1 - (pp `2))) by A1, A51, BORSUK_2:def_6
.= h . p by A13, A50 ;
hence fg . p = h . p ; ::_thesis: verum
end;
([#] S12) \/ ([#] ([:I[01],I[01]:] | ICC)) = [#] [:I[01],I[01]:] by A12, A46, Th56, BORSUK_1:40, BORSUK_1:def_2;
then consider H being Function of [:I[01],I[01]:],T such that
A54: H = fg +* h and
A55: H is continuous by A12, A45, A47, A38, JGRAPH_2:1;
A56: for s being Point of I[01] holds
( H . (s,0) = ((- P) + P) . s & H . (s,1) = Q . s )
proof
let s be Point of I[01]; ::_thesis: ( H . (s,0) = ((- P) + P) . s & H . (s,1) = Q . s )
thus H . (s,0) = ((- P) + P) . s ::_thesis: H . (s,1) = Q . s
proof
A57: [s,0] `1 = s ;
percases ( s < 1 / 2 or s = 1 / 2 or s > 1 / 2 ) by XXREAL_0:1;
supposeA58: s < 1 / 2 ; ::_thesis: H . (s,0) = ((- P) + P) . s
then not [s,0] in IBB by Th71;
then not [s,0] in the carrier of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
then A59: not [s,0] in dom g ;
[s,0] in IAA by A58, Th70;
then A60: [s,0] in the carrier of ([:I[01],I[01]:] | IAA) by PRE_TOPC:8;
not [s,0] in ICC by A58, Th71;
then not [s,0] in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then not [s,0] in dom h ;
then H . [s,0] = fg . [s,0] by A54, FUNCT_4:11
.= f . [s,0] by A44, A59, FUNCT_4:11
.= (- P) . (2 * s) by A22, A57, A60
.= ((- P) + P) . s by A1, A58, BORSUK_2:def_5 ;
hence H . (s,0) = ((- P) + P) . s ; ::_thesis: verum
end;
supposeA61: s = 1 / 2 ; ::_thesis: H . (s,0) = ((- P) + P) . s
then A62: [s,0] in the carrier of ([:I[01],I[01]:] | ICC) by Th66, PRE_TOPC:8;
then [s,0] in dom h by FUNCT_2:def_1;
then H . [s,0] = h . [s,0] by A54, FUNCT_4:13
.= P . ((2 * s) - 1) by A13, A57, A62
.= b by A1, A61, BORSUK_2:def_2
.= (- P) . (2 * (1 / 2)) by A1, BORSUK_2:def_2
.= ((- P) + P) . s by A1, A61, BORSUK_2:def_5 ;
hence H . (s,0) = ((- P) + P) . s ; ::_thesis: verum
end;
supposeA63: s > 1 / 2 ; ::_thesis: H . (s,0) = ((- P) + P) . s
then [s,0] in ICC by Th69;
then A64: [s,0] in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then [s,0] in dom h by FUNCT_2:def_1;
then H . [s,0] = h . [s,0] by A54, FUNCT_4:13
.= P . ((2 * s) - 1) by A13, A57, A64
.= ((- P) + P) . s by A1, A63, BORSUK_2:def_5 ;
hence H . (s,0) = ((- P) + P) . s ; ::_thesis: verum
end;
end;
end;
thus H . (s,1) = Q . s ::_thesis: verum
proof
A65: [s,1] `2 = 1 ;
A66: [s,1] `1 = s ;
A67: dom Q = the carrier of I[01] by FUNCT_2:def_1;
then A68: 0 in dom Q by BORSUK_1:43;
percases ( s <> 1 or s = 1 ) ;
supposeA69: s <> 1 ; ::_thesis: H . (s,1) = Q . s
[s,1] in IBB by Th65;
then A70: [s,1] in the carrier of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
then A71: [s,1] in dom g by FUNCT_2:def_1;
not [s,1] in ICC by A69, Th63;
then not [s,1] in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then not [s,1] in dom h ;
then H . [s,1] = fg . [s,1] by A54, FUNCT_4:11
.= g . [s,1] by A44, A71, FUNCT_4:13
.= (- P) . (1 - 1) by A2, A65, A70
.= a by A1, BORSUK_2:def_2
.= Q . 0 by A11, BORSUK_2:def_2
.= Q . s by A67, A68, FUNCT_1:def_10 ;
hence H . (s,1) = Q . s ; ::_thesis: verum
end;
supposeA72: s = 1 ; ::_thesis: H . (s,1) = Q . s
then A73: [s,1] in the carrier of ([:I[01],I[01]:] | ICC) by Th66, PRE_TOPC:8;
then [s,1] in dom h by FUNCT_2:def_1;
then H . [s,1] = h . [s,1] by A54, FUNCT_4:13
.= P . ((2 * s) - 1) by A13, A66, A73
.= a by A1, A72, BORSUK_2:def_2
.= Q . 0 by A11, BORSUK_2:def_2
.= Q . s by A67, A68, FUNCT_1:def_10 ;
hence H . (s,1) = Q . s ; ::_thesis: verum
end;
end;
end;
end;
for t being Point of I[01] holds
( H . (0,t) = a & H . (1,t) = a )
proof
let t be Point of I[01]; ::_thesis: ( H . (0,t) = a & H . (1,t) = a )
thus H . (0,t) = a ::_thesis: H . (1,t) = a
proof
0 in the carrier of I[01] by BORSUK_1:43;
then reconsider x = [0,t] as Point of [:I[01],I[01]:] by Lm1;
A74: x `2 = t by MCART_1:7;
x in IAA by Th61;
then A75: x is Point of ([:I[01],I[01]:] | IAA) by PRE_TOPC:8;
A76: x `1 = 0 by MCART_1:7;
then not x in ICC by Th60;
then not x in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then A77: not [0,t] in dom h ;
percases ( t <> 1 or t = 1 ) ;
suppose t <> 1 ; ::_thesis: H . (0,t) = a
then not x in IBB by Th62;
then not x in the carrier of ([:I[01],I[01]:] | IBB) by PRE_TOPC:8;
then not x in dom g ;
then fg . [0,t] = f . [0,t] by A44, FUNCT_4:11
.= (- P) . (2 * (x `1)) by A22, A75
.= a by A1, A76, BORSUK_2:def_2 ;
hence H . (0,t) = a by A54, A77, FUNCT_4:11; ::_thesis: verum
end;
supposeA78: t = 1 ; ::_thesis: H . (0,t) = a
then A79: x in the carrier of ([:I[01],I[01]:] | IBB) by Th64, PRE_TOPC:8;
then x in dom g by FUNCT_2:def_1;
then fg . [0,t] = g . [0,1] by A44, A78, FUNCT_4:13
.= (- P) . (1 - (x `2)) by A2, A78, A79
.= a by A1, A74, A78, BORSUK_2:def_2 ;
hence H . (0,t) = a by A54, A77, FUNCT_4:11; ::_thesis: verum
end;
end;
end;
thus H . (1,t) = a ::_thesis: verum
proof
1 in the carrier of I[01] by BORSUK_1:43;
then reconsider x = [1,t] as Point of [:I[01],I[01]:] by Lm1;
A80: x `1 = 1 by MCART_1:7;
x in ICC by Th68;
then A81: x in the carrier of ([:I[01],I[01]:] | ICC) by PRE_TOPC:8;
then A82: [1,t] in dom h by FUNCT_2:def_1;
h . [1,t] = P . ((2 * (x `1)) - 1) by A13, A81
.= a by A1, A80, BORSUK_2:def_2 ;
hence H . (1,t) = a by A54, A82, FUNCT_4:13; ::_thesis: verum
end;
end;
hence (- P) + P,Q are_homotopic by A55, A56, BORSUK_2:def_7; ::_thesis: verum
end;
theorem :: BORSUK_6:87
for X being non empty pathwise_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
let X be non empty pathwise_connected TopSpace; ::_thesis: 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
let b1, a1 be Point of X; ::_thesis: for P being Path of b1,a1
for Q being constant Path of a1,a1 holds (- P) + P,Q are_homotopic
let P be Path of b1,a1; ::_thesis: for Q being constant Path of a1,a1 holds (- P) + P,Q are_homotopic
let Q be constant Path of a1,a1; ::_thesis: (- P) + P,Q are_homotopic
b1,a1 are_connected by BORSUK_2:def_3;
hence (- P) + P,Q are_homotopic by Th86; ::_thesis: verum
end;
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
let T be non empty TopSpace; ::_thesis: for a being Point of T
for P, Q being constant Path of a,a holds P,Q are_homotopic
let a be Point of T; ::_thesis: for P, Q being constant Path of a,a holds P,Q are_homotopic
let P, Q be constant Path of a,a; ::_thesis: P,Q are_homotopic
( P = I[01] --> a & Q = I[01] --> a ) by BORSUK_2:5;
hence P,Q are_homotopic by BORSUK_2:12; ::_thesis: verum
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, BORSUK_2:def_7;
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 ) ) ) );