
theorem
  for a,b be non zero Integer holds
    Parity (a+b) = Parity b iff Parity a > Parity b
  proof
    let a,b be non zero Integer;
    thus Parity (a+b) = Parity b implies Parity a > Parity b
    proof
      assume
      A1: Parity (a+b) = Parity b;then
      A2: Parity (a+b) + 0 < (Parity a) + (Parity b) by XREAL_1:6;
      per cases by XXREAL_0:1;
      suppose Parity a = Parity b;
        hence thesis by A1,A2,Def1,PEQ;
      end;
      suppose Parity b > Parity a;
        hence thesis by A1,PAP;
      end;
      suppose Parity b < Parity a;
        hence thesis;
      end;
    end;
    thus thesis by PAP;
  end;
