
theorem Th11:
  for X,Y being non empty TopSpace for f being continuous Function
  of Y,Y st f is idempotent holds oContMaps(f, X) 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 fX = oContMaps(f, X);
  now
    let g be Element of oContMaps(Y, X);
    reconsider h = g as continuous Function of Y,X by Th2;
    thus (fX*fX).g = fX.(fX.g) by FUNCT_2:15
      .= fX.(h*f) by Def3
      .= h*f*f by Def3
      .= h*(f*f) by RELAT_1:36
      .= h*f by A1,QUANTAL1:def 9
      .= fX.g by Def3;
  end;
  then fX*fX = fX by FUNCT_2:63;
  hence thesis by QUANTAL1:def 9;
end;
