:: 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 ) ) ) );