
theorem
  for a,b be non zero Integer holds
    a gcd b = ((Parity a) gcd (Parity b))*((Oddity a) gcd (Oddity b))
  proof
    let a,b be non zero Integer;
    a gcd b = ((Parity a)*(Oddity a)) gcd b
    .= ((Parity a) gcd (Parity b)*(Oddity b))* ((Oddity a) gcd b)
      by NEWTON03:35,OPC
    .= ((Parity a) gcd (Parity b))*((Parity a) gcd (Oddity b))*
      ((Oddity a) gcd b) by NEWTON03:35,OPC
    .= ((Parity a) gcd (Parity b))*1*((Oddity a) gcd b) by OPC
    .= ((Parity a) gcd (Parity b))*((Oddity a) gcd ((Parity b)*(Oddity b)))
    .= ((Parity a) gcd (Parity b))*(((Oddity a) gcd ((Parity b)))*((Oddity a)
      gcd(Oddity b))) by NEWTON03:35,OPC
    .= ((Parity a) gcd (Parity b))*(1*((Oddity a) gcd (Oddity b))) by OPC;
    hence thesis;
  end;
