
theorem
  for a be Nat holds Parity (a - Parity a) >= 2*Parity a or a = Parity a
  proof
    let a be Nat;
    per cases;
    suppose a = Parity a;
      hence thesis;
    end;
    suppose
      A0: a <> Parity a; then
      reconsider a as non zero Nat by Def1;
      A1: Parity(a - Parity a) = (Parity ((Oddity a) - 1)) * (Parity a)
        by PGP;
      a = (Oddity a)*(Parity a); then
      A2: (Oddity a) - 1 <> 0 by A0;
      Parity ((Oddity a)-1) <> 1;
      hence thesis by A1,A2,XREAL_1:64,NAT_1:23;
    end;
  end;
