
theorem
  for a,b be Integer holds
    Parity (a + b) = Parity(a gcd b)*Parity ((a+b)/(a gcd b))
   proof
    let a,b be Integer;
    per cases;
    suppose
      a = 0 & b = 0; then
      Parity (a gcd b) = 0 & Parity (a+b) = 0 by Def1;
      hence thesis;
    end;
    suppose
      A1: a <> 0 or b <> 0;
      Parity (((a+b)/(a gcd b))*(a gcd b))
        = Parity ((a+b)/(a gcd b)) * Parity(a gcd b) by ILP;
      hence thesis by A1,XCMPLX_1:87;
    end;
  end;
