
theorem PPD:
  for a,b be odd Integer holds
  Parity (a+b) = 2 iff parity (a div 2) = parity (b div 2)
  proof
    let a,b be odd Integer;
    thus Parity (a+b) = 2 implies parity (a div 2) = parity (b div 2)
    proof
      assume Parity (a+b) = 2; then
      2*(Parity (((a div 2) + (b div 2) + 1))) = 2*1 by SPA; then
      (a div 2) + (b div 2) is even;
      hence thesis by EVP;
    end;
    assume parity (a div 2) = parity (b div 2); then
    a div 2 is odd iff b div 2 is odd; then
    2*(Parity (((a div 2) + (b div 2)) + 1)) = 2*1 by NAT_2:def 1;
    hence thesis by SPA;
  end;
