
theorem
  for T being non empty TopSpace, f being Function of T, T st f*f = f
  holds corestr f * incl Image f = id Image f
proof
  let T be non empty TopSpace, f be Function of T, T such that
A1: f*f = f;
  set cf = corestr f, i = incl Image f;
  for x being object st x in the carrier of Image f holds (cf*i).x = (id
  Image f).x
  proof
A2: the carrier of Image f c= the carrier of T by BORSUK_1:1;
    let x be object;
    assume
A3: x in the carrier of Image f;
    the carrier of Image f = rng f by WAYBEL18:9;
    then
A4: ex y being object st y in dom f & f.y = x by A3,FUNCT_1:def 3;
    thus (cf*i).x = cf.(i.x) by A3,FUNCT_2:15
      .= cf.x by A3,A2,TMAP_1:84
      .= f.x by WAYBEL18:def 7
      .= x by A1,A4,FUNCT_2:15
      .= (id Image f).x by A3,FUNCT_1:18;
  end;
  hence thesis;
end;
