
theorem
  for a,b be Integer holds Parity (a+b) > (Parity a)+(Parity b) implies
  Parity a = Parity b
  proof
    let a,b be Integer;
    per cases;
    suppose a is zero; then
      Parity (a+b) = Parity b & Parity a = 0 by Def1;
      hence thesis;
    end;
    suppose b is zero; then
      Parity (a+b) = Parity a & Parity b = 0 by Def1;
      hence thesis;
    end;
    suppose not a is zero & not b is zero;
      hence thesis by LEQ;
    end;
  end;
