let S, T be non empty TopSpace; :: thesis: for f being continuous Function of S,T
for a, b being Point of S
for P, Q being Path of a,b
for P1, Q1 being Path of f . a,f . b st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
P1,Q1 are_homotopic
let f be continuous Function of S,T; :: thesis: for a, b being Point of S
for P, Q being Path of a,b
for P1, Q1 being Path of f . a,f . b st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
P1,Q1 are_homotopic
let a, b be Point of S; :: thesis: for P, Q being Path of a,b
for P1, Q1 being Path of f . a,f . b st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
P1,Q1 are_homotopic
let P, Q be Path of a,b; :: thesis: for P1, Q1 being Path of f . a,f . b st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
P1,Q1 are_homotopic
let P1, Q1 be Path of f . a,f . b; :: thesis: ( P,Q are_homotopic & P1 = f * P & Q1 = f * Q implies P1,Q1 are_homotopic )
assume that
A1: P,Q are_homotopic and
A2: P1 = f * P and
A3: Q1 = f * Q ; :: thesis: P1,Q1 are_homotopic
set F = the Homotopy of P,Q;
take G = f * the Homotopy of P,Q; :: according to BORSUK_2:def 7 :: thesis: ( G is continuous & ( for b₁ being Element of the carrier of K523() holds
( G . (b₁,0) = P1 . b₁ & G . (b₁,1) = Q1 . b₁ & G . (0,b₁) = f . a & G . (1,b₁) = f . b ) ) )
the Homotopy of P,Q is continuous by A1, BORSUK_6:def 11;
hence G is continuous ; :: thesis: for b₁ being Element of the carrier of K523() holds
( G . (b₁,0) = P1 . b₁ & G . (b₁,1) = Q1 . b₁ & G . (0,b₁) = f . a & G . (1,b₁) = f . b )
let s be Point of I[01]; :: thesis: ( G . (s,0) = P1 . s & G . (s,1) = Q1 . s & G . (0,s) = f . a & G . (1,s) = f . b )
thus G . (s,0) = f . ( the Homotopy of P,Q . (s,j0)) by Lm1, BINOP_1:18
.= f . (P . s) by A1, BORSUK_6:def 11
.= P1 . s by A2, FUNCT_2:15 ; :: thesis: ( G . (s,1) = Q1 . s & G . (0,s) = f . a & G . (1,s) = f . b )
thus G . (s,1) = f . ( the Homotopy of P,Q . (s,j1)) by Lm1, BINOP_1:18
.= f . (Q . s) by A1, BORSUK_6:def 11
.= Q1 . s by A3, FUNCT_2:15 ; :: thesis: ( G . (0,s) = f . a & G . (1,s) = f . b )
thus G . (0,s) = f . ( the Homotopy of P,Q . (j0,s)) by Lm1, BINOP_1:18
.= f . a by A1, BORSUK_6:def 11 ; :: thesis: G . (1,s) = f . b
thus G . (1,s) = f . ( the Homotopy of P,Q . (j1,s)) by Lm1, BINOP_1:18
.= f . b by A1, BORSUK_6:def 11 ; :: thesis: verum