
theorem
  for a,b be Integer holds (Oddity a) gcd (Oddity b) = Oddity (a gcd b)
  proof
    let a,b be Integer;
    per cases;
    suppose
      A1: a <> 0 & b <> 0; then
      reconsider a as non zero Integer;
      reconsider b as non zero Integer by A1;
      A2: Parity((Oddity a) gcd (Oddity b))  = 1 gcd 1 by NAT_2:def 1;
      Oddity (a gcd b) = Oddity (((Parity a)*(Oddity a)) gcd
        ((Parity b)*(Oddity b)))
      .= Oddity ((((Parity a) gcd ((Parity b)*(Oddity b))))*(((Oddity a) gcd
        ((Parity b)*(Oddity b))))) by OPC,NEWTON03:35
      .= Oddity ((((Parity a) gcd (Parity b)))*((Parity a) gcd (Oddity b))*
      (((Oddity a) gcd ((Parity b)*(Oddity b))))) by OPC,NEWTON03:35
      .= Oddity ((((Parity a) gcd (Parity b)))*1*
      (((Oddity a) gcd ((Parity b)*(Oddity b))))) by OPC
      .= Oddity ((((Parity a) gcd (Parity b)))*
      (((Oddity a) gcd (Parity b))* ((Oddity a) gcd (Oddity b))))
        by OPC,NEWTON03:35
      .= Oddity ((((Parity a) gcd (Parity b)))*(1* ((Oddity a)
        gcd (Oddity b)))) by OPC
      .= (Oddity ((Parity a) gcd (Parity b)))*(Oddity ((Oddity a)
        gcd (Oddity b))) by ILO
      .= Oddity (Parity (a gcd b))*Oddity ((Oddity a) gcd (Oddity b)) by PGG
      .= 1*Oddity ((Oddity a) gcd (Oddity b)) by OPA;
      hence thesis by A2;
    end;
    suppose a = 0; then
      Oddity (a gcd b) = Oddity |.b.| &
        (Oddity a) gcd (Oddity b) = |.Oddity b.| by INT_2:12,NEWTON02:3;
      hence thesis by OMO;
    end;
    suppose b = 0; then
      Oddity (a gcd b) = Oddity |.a.| &
        (Oddity a) gcd (Oddity b) = |.Oddity a.| by INT_2:12,NEWTON02:3;
      hence thesis by OMO;
    end;
  end;
