
theorem Th20:
  for X,Y,Z being non empty TopSpace for f being continuous
Function of Y,Z st f is being_homeomorphism holds oContMaps(X, f) is isomorphic
proof
  let X,Y,Z be non empty TopSpace;
  let f be continuous Function of Y,Z;
  set XY = oContMaps(X,Y), XZ = oContMaps(X,Z);
  assume
A1: f is being_homeomorphism;
  then reconsider g = f" as continuous Function of Z,Y by TOPS_2:def 5;
  set Xf = oContMaps(X,f), Xg = oContMaps(X,g);
A2: f is one-to-one & rng f = [#]Z by A1,TOPS_2:def 5;
  now
    let a be Element of XZ;
    reconsider h = a as continuous Function of X,Z by Th2;
    thus (Xf*Xg).a = Xf.(Xg.a) by FUNCT_2:15
      .= Xf.(g*h) by Def2
      .= f*(g*h) by Def2
      .= f*g*h by RELAT_1:36
      .= (id the carrier of Z)*h by A2,TOPS_2:52
      .= a by FUNCT_2:17
      .= (id XZ).a;
  end;
  then
A3: Xf*Xg = id XZ by FUNCT_2:63;
A4: dom f = [#]Y by A1,TOPS_2:def 5;
  now
    let a be Element of XY;
    reconsider h = a as continuous Function of X,Y by Th2;
    thus (Xg*Xf).a = Xg.(Xf.a) by FUNCT_2:15
      .= Xg.(f*h) by Def2
      .= g*(f*h) by Def2
      .= g*f*h by RELAT_1:36
      .= (id the carrier of Y)*h by A2,A4,TOPS_2:52
      .= a by FUNCT_2:17
      .= (id XY).a;
  end;
  then
A5: Xg*Xf = id XY by FUNCT_2:63;
  Xf is monotone & Xg is monotone by Th8;
  hence thesis by A5,A3,YELLOW16:15;
end;
