
theorem Th25:
  for S being non empty pathwise_connected TopSpace, T being non
  empty TopSpace, f being continuous Function of S,T, a, b being Point of S, P
  being Path of a,b holds f*P is Path of f.a,f.b
proof
  let S be non empty pathwise_connected TopSpace;
  let T be non empty TopSpace;
  let f be continuous Function of S,T;
  let a, b be Point of S;
  let P be Path of a,b;
A1: a,b are_connected by BORSUK_2:def 3;
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;
  f.a,f.b are_connected by A1,Th23;
  hence thesis by A3,A2,BORSUK_2:def 2;
end;
