intros n m Hle; elim (Qpositive_le_total n m); auto.
intros H; elim Hle; auto.
Qed.

(* Specifications: *)
Lemma Qpositive_le_noneq_explicit: forall qp qp', Qpositive_le qp qp' -> ~ qp=qp' ->
let (p, q) := Qpositive_i qp in
let (p', q') := Qpositive_i qp' in (p * q' < p' * q)%nat.