
theorem for a,b be Integer holds parity (a*b) = min (parity a, parity b)
  proof
    let a,b be Integer;
    min (parity a,parity b) = parity a or
      min (parity a,parity b) = parity b by XXREAL_0:def 9; then
    per cases by NAT_2:def 1;
    suppose
      B1: min(parity a, parity b) = 0; then
      a is even or b is even;
      hence thesis by B1;
    end;
    suppose
      B1: min (parity a, parity b) = 1; then
      a is odd & b is odd by XXREAL_0:def 9; then
      a*b is odd;
      hence thesis by B1;
    end;
  end;
