
theorem
  for a be even Integer, b be odd Integer holds
    a div b is odd iff a mod b is odd
  proof
    let a be even Integer, b be odd Integer;
    thus a div b is odd implies a mod b is odd
    proof
      assume a div b is odd; then
      a - b*(a div b) is odd;
      hence thesis by INT_1:def 10;
    end;
    assume a mod b is odd; then
    a - b*(a div b) is odd by INT_1:def 10;
    hence thesis;
  end;
