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