intros w w'; cut (Qpositive_le' w w' \/ Qpositive_le' w' w).
intros H; elim H; intros H1;
generalize (Qpositive_le'_to_Qpositive_le _ _ H1);
auto.
unfold Qpositive_le' in |- *.
case (Qpositive_i w'); case (Qpositive_i w); intros p q p' q'.
elim (le_or_lt (p * q') (p' * q)); auto with arith.
Qed.

Theorem not_Qpositive_le_not_eq :
forall n m : Qpositive, ~ Qpositive_le n m -> n <> m.