
theorem OMO:
  for a be Integer holds |.Oddity a.| = Oddity |.a.|
  proof
    let a be Integer;
    per cases;
    suppose a = 0;
      hence thesis;
    end;
    suppose a <> 0; then
      reconsider a as non zero Integer;
      |.Oddity a.|*|.Parity a.| = |.(Oddity a)*(Parity a).| by COMPLEX1:65
      .= (Oddity |.a.|)*(Parity |.a.|)
      .= (Oddity |.a.|)*|.Parity a.| by PMP;
      hence thesis by XCMPLX_1:5;
    end;
  end;
