
theorem
  for a,b be non zero Nat holds Parity (a+b) < (Parity a)+(Parity b) implies
  Parity (a+b) = min (Parity a, Parity b)
  proof
    let a,b be non zero Nat;
    assume Parity (a+b) < (Parity a)+(Parity b); then
    Parity a <> Parity b by PEQ; then
    per cases by XXREAL_0:1;
    suppose
      B1: Parity a > Parity b; then
      Parity (a+b) = Parity b by PAP;
      hence thesis by B1,XXREAL_0:def 9;
    end;
    suppose
      B1: Parity a < Parity b; then
      Parity (a+b) = Parity a by PAP;
      hence thesis by B1,XXREAL_0:def 9;
    end;
  end;
