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
for F being Homotopy of P,Q st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
f * F is Homotopy of P1,Q1
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
for F being Homotopy of P,Q st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
f * F is Homotopy of P1,Q1
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
for F being Homotopy of P,Q st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
f * F is Homotopy of P1,Q1
let P, Q be Path of a,b; :: thesis: for P1, Q1 being Path of f . a,f . b
for F being Homotopy of P,Q st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
f * F is Homotopy of P1,Q1
let P1, Q1 be Path of f . a,f . b; :: thesis: for F being Homotopy of P,Q st P,Q are_homotopic & P1 = f * P & Q1 = f * Q holds
f * F is Homotopy of P1,Q1
let F be Homotopy of P,Q; :: thesis: ( P,Q are_homotopic & P1 = f * P & Q1 = f * Q implies f * F is Homotopy of P1,Q1 )
assume that
A1: P,Q are_homotopic and
A2: P1 = f * P and
A3: Q1 = f * Q ; :: thesis: f * F is Homotopy of P1,Q1
thus P1,Q1 are_homotopic by A1, A2, A3, Th27; :: according to BORSUK_6:def 11 :: thesis: ( f * F is continuous & ( for b₁ being Element of the carrier of K523() holds
( (f * F) . (b₁,0) = P1 . b₁ & (f * F) . (b₁,1) = Q1 . b₁ & (f * F) . (0,b₁) = f . a & (f * F) . (1,b₁) = f . b ) ) )
set G = f * F;
F is continuous by A1, BORSUK_6:def 11;
hence f * F is continuous ; :: thesis: for b₁ being Element of the carrier of K523() holds
( (f * F) . (b₁,0) = P1 . b₁ & (f * F) . (b₁,1) = Q1 . b₁ & (f * F) . (0,b₁) = f . a & (f * F) . (1,b₁) = f . b )
let s be Point of I[01]; :: thesis: ( (f * F) . (s,0) = P1 . s & (f * F) . (s,1) = Q1 . s & (f * F) . (0,s) = f . a & (f * F) . (1,s) = f . b )
thus (f * F) . (s,0) = f . (F . (s,j0)) by Lm1, BINOP_1:18
.= f . (P . s) by A1, BORSUK_6:def 11
.= P1 . s by A2, FUNCT_2:15 ; :: thesis: ( (f * F) . (s,1) = Q1 . s & (f * F) . (0,s) = f . a & (f * F) . (1,s) = f . b )
thus (f * F) . (s,1) = f . (F . (s,j1)) by Lm1, BINOP_1:18
.= f . (Q . s) by A1, BORSUK_6:def 11
.= Q1 . s by A3, FUNCT_2:15 ; :: thesis: ( (f * F) . (0,s) = f . a & (f * F) . (1,s) = f . b )
thus (f * F) . (0,s) = f . (F . (j0,s)) by Lm1, BINOP_1:18
.= f . a by A1, BORSUK_6:def 11 ; :: thesis: (f * F) . (1,s) = f . b
thus (f * F) . (1,s) = f . (F . (j1,s)) by Lm1, BINOP_1:18
.= f . b by A1, BORSUK_6:def 11 ; :: thesis: verum