apply div_unic_r with (a := x * y) (b := y) (q1 := q) (q2 := x);
auto with arith.
apply is_div_def; inversion_clear H'; auto with arith.
apply le_lt_trans with (m := r); auto with arith.
Qed.

Lemma div_ex :
forall a b : nat, b <> 0 -> exists q : _, (exists r : _, is_div a b q r).