
theorem SPA: for a,b be odd Integer holds
  Parity (a+b) = 2*Parity ((a div 2) + (b div 2) + 1)
  proof
    let a,b be odd Integer;
    parity a = 1 & parity b = 1 by NAT_2:def 1; then
    Parity (a + b) = Parity ( 2*((a div 2) + (b div 2)) + 1 + 1) by SAB
    .= Parity (2*((a div 2) + (b div 2) + 1))
    .= (Parity 2)*Parity ((a div 2) + (b div 2) + 1) by ILP;
    hence thesis;
  end;
