
theorem ODP:
  for a,b be odd Integer holds min (Parity(a-b),Parity(a+b)) <= 2
  proof
    let a,b be odd Integer;
    A0: |.a.| in NAT & |.b.| in NAT by INT_1:3; then
    reconsider k = |.a.| as odd Nat;
    reconsider l = |.b.| as odd Nat by A0;
    per cases;
    suppose
      k = l; then
      Parity |.|.a.| - |.b.|.| = 0 by Def1; then
      Parity |.a-b.| = 0 or Parity |.a+b.| = 0 by ABS1; then
      Parity (a-b) = 0 or Parity (a+b) = 0 by PMP;
      hence thesis by XXREAL_0:def 9;
    end;
    suppose
      k <> l;
      hence thesis by MPO;
    end;
  end;
