
theorem
  for a,b be non zero Integer st a + b <> 0 holds
  Parity (a+b) = Parity a implies Parity a < Parity b
  proof
    let a,b be non zero Integer such that
    A0: a + b <> 0;
    assume
    A1: Parity (a+b) = Parity a;
    (Parity a)+(Parity b) > Parity a + 0 by XREAL_1:6; then
    Parity a <> Parity b by A0,A1,PEQ; then
    Parity a > Parity b or Parity a < Parity b by XXREAL_0:1;
    hence thesis by PAP,A1;
  end;
