 reserve R for finite Approximation_Space;
 reserve X,Y,Z for Subset of R;

theorem CMapMap:
  for f being preRIF of R holds
    CMap CMap f = f
  proof
    let f be preRIF of R;
    set g = CMap f;
    for x being Element of [:bool the carrier of R, bool the carrier of R:]
      holds
        (CMap g).x = f.x
    proof
      let x be Element of [:bool the carrier of R, bool the carrier of R:];
      reconsider x1 = x`1, x2 = x`2 as Subset of R;
      (CMap g).x = (CMap g).(x1,x2) by MCART_1:21
                .= 1 - g.(x1,x2) by CDef
                .= 1 - (1 - f.(x1,x2)) by CDef
                .= f.x by MCART_1:21;
      hence thesis;
    end;
    hence thesis;
  end;
