
theorem PGP:
  for a be Integer holds
  Parity(a + Parity a) = (Parity ((Oddity a) + 1)) * (Parity a) &
  Parity(a - Parity a) = (Parity ((Oddity a) - 1)) * (Parity a)
  proof
    let a be Integer;
    per cases;
    suppose a is non zero; then
      reconsider a as non zero Integer;
      A1: Parity(a - Parity a) = Parity((Oddity a)*Parity a - 1*Parity a)
      .= Parity (((Oddity a) - 1) * Parity a)
      .= (Parity ((Oddity a) - 1)) * Parity (Parity a) by ILP
      .= (Parity ((Oddity a) - 1)) * (Parity a);
      Parity(a + Parity a) = Parity ((Oddity a)*(Parity a) + Parity a)
      .= Parity (((Oddity a) + 1) * (Parity a))
      .= (Parity ((Oddity a) + 1)) * Parity (Parity a) by ILP
      .= (Parity ((Oddity a) + 1)) * (Parity a);
      hence thesis by A1;
    end;
    suppose a is zero; then
      Parity (a + Parity a) = Parity 0 & Parity a = 0 by Def1;
      hence thesis;
    end;
  end;
