
theorem ILO:
  for a,b be Integer holds Oddity (a*b) = (Oddity a)*(Oddity b)
  proof
    let a,b be Integer;
    per cases;
    suppose
      a = 0 or b = 0;
      hence thesis;
    end;
    suppose
      B1: a <> 0 & b <> 0; then
      reconsider a as non zero Integer;
      reconsider b as non zero Integer by B1;
      (Oddity a)*(Oddity b)*Parity (a*b)
       = (Oddity a)*(Oddity b)*((Parity a)*(Parity b)) by ILP
      .= ((Oddity a)*(Parity a))*((Oddity b)*(Parity b))
      .= Oddity (a*b)*Parity (a*b);
      hence thesis by XCMPLX_1:5;
    end;
  end;
