
theorem PMG:
  for a be odd Nat, b be non trivial odd Nat holds
    Parity (a+b) = min(Parity (a+1), Parity (b-1)) or
      Parity (a+b) >= 2*Parity (a+1)
  proof
    let a be odd Nat, b be non trivial odd Nat;
    per cases by XXREAL_0:1;
    suppose
      A1: Parity (a+1) = Parity (b-1); then
      Parity ((a+1)+(b-1)) >= Parity (a+1) + Parity (b-1) by PEQ;
      hence thesis by A1;
    end;
    suppose
      B1: Parity (a+1) > Parity (b-1); then
      Parity ((a+1)+(b-1)) = Parity (b-1) by PAP;
      hence thesis by B1,XXREAL_0:def 9;
    end;
    suppose
      B1: Parity (a+1) < Parity (b-1); then
      Parity ((a+1)+(b-1)) = Parity (a+1) by PAP;
      hence thesis by B1,XXREAL_0:def 9;
    end;
  end;
