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 being Path of a,b st a,b are_connected holds

f * P is Path of f . a,f . b

let f be continuous Function of S,T; :: thesis: for a, b being Point of S

for P being Path of a,b st a,b are_connected holds

f * P is Path of f . a,f . b

let a, b be Point of S; :: thesis: for P being Path of a,b st a,b are_connected holds

f * P is Path of f . a,f . b

let P be Path of a,b; :: thesis: ( a,b are_connected implies f * P is Path of f . a,f . b )

assume A1: a,b are_connected ; :: thesis: f * P is Path of f . a,f . b

A2: (f * P) . 1 = f . (P . j1) by FUNCT_2:15

.= f . b by A1, BORSUK_2:def 2 ;

A3: (f * P) . 0 = f . (P . j0) by FUNCT_2:15

.= f . a by A1, BORSUK_2:def 2 ;

( P is continuous & f . a,f . b are_connected ) by A1, Th23, BORSUK_2:def 2;

hence f * P is Path of f . a,f . b by A3, A2, BORSUK_2:def 2; :: thesis: verum

for a, b being Point of S

for P being Path of a,b st a,b are_connected holds

f * P is Path of f . a,f . b

let f be continuous Function of S,T; :: thesis: for a, b being Point of S

for P being Path of a,b st a,b are_connected holds

f * P is Path of f . a,f . b

let a, b be Point of S; :: thesis: for P being Path of a,b st a,b are_connected holds

f * P is Path of f . a,f . b

let P be Path of a,b; :: thesis: ( a,b are_connected implies f * P is Path of f . a,f . b )

assume A1: a,b are_connected ; :: thesis: f * P is Path of f . a,f . b

A2: (f * P) . 1 = f . (P . j1) by FUNCT_2:15

.= f . b by A1, BORSUK_2:def 2 ;

A3: (f * P) . 0 = f . (P . j0) by FUNCT_2:15

.= f . a by A1, BORSUK_2:def 2 ;

( P is continuous & f . a,f . b are_connected ) by A1, Th23, BORSUK_2:def 2;

hence f * P is Path of f . a,f . b by A3, A2, BORSUK_2:def 2; :: thesis: verum