
theorem Th51:
  for T, S, R being non empty TopSpace for f being Function of T,S
  , g being Function of S,T, h being Function of S, R st f*g = id S & h is
  being_homeomorphism holds (h*f)*(g*(h")) = id R
proof
  let T, S, R be non empty TopSpace;
  let f be Function of T,S, g be Function of S,T, h be Function of S, R such
  that
A1: f*g = id S and
A2: h is being_homeomorphism;
A3: h is one-to-one by A2,TOPS_2:def 5;
A4: rng h = [#]R by A2,TOPS_2:def 5;
  thus (h*f)*(g*(h")) = (h*f)*g*(h") by RELAT_1:36
    .= h*(id the carrier of S)*(h") by A1,RELAT_1:36
    .= h*(h") by FUNCT_2:17
    .= id R by A3,A4,TOPS_2:52;
end;
