simple induction q.
intro.
absurd (0 = n).
apply lt_O_neq; auto with arith.
rewrite (mult_n_O m); auto with arith.
auto with arith.
Qed.

Lemma lt_quotient2_n :
forall n : nat, 0 < n -> forall q : nat, n = 2 * q -> q < n.