
theorem
  for X,Y being non empty TopSpace for f being continuous Function of Y,
  Y st f is idempotent holds oContMaps(X, f) is idempotent
proof
  let X,Y be non empty TopSpace;
  let f be continuous Function of Y,Y such that
A1: f is idempotent;
  set Xf = oContMaps(X, f);
  now
    let g be Element of oContMaps(X, Y);
    reconsider h = g as continuous Function of X,Y by Th2;
    thus (Xf*Xf).g = Xf.(Xf.g) by FUNCT_2:15
      .= Xf.(f*h) by Def2
      .= f*(f*h) by Def2
      .= (f*f)*h by RELAT_1:36
      .= f*h by A1,QUANTAL1:def 9
      .= Xf.g by Def2;
  end;
  then Xf*Xf = Xf by FUNCT_2:63;
  hence thesis by QUANTAL1:def 9;
end;
