
theorem ADA: for a be Integer holds 2*(Parity a) divides Parity (a + Parity a)
  & 2*(Parity a) divides Parity (a - Parity a)
  proof
    let a be Integer;
    per cases;
    suppose
      a is zero; then
      Parity a = 0 by Def1;
      hence thesis;
    end;
    suppose not a is zero; then
      reconsider a as non zero Integer;
      2 divides Parity ((Oddity a) + 1) & 2 divides Parity ((Oddity a) - 1)
        by ABIAN:def 1; then
      2*(Parity a) divides (Parity a)*Parity ((Oddity a) + 1) &
        2*(Parity a) divides (Parity a)*Parity ((Oddity a) - 1) by NEWTON03:5;
      hence thesis by PGP;
    end;
  end;
