intros w w' w'' H H0.
apply Qpositive_plus_simpl with w'.
rewrite Qpositive_plus_assoc.
rewrite (Qpositive_plus_sym w'').
rewrite <- Qpositive_plus_assoc.
rewrite Qpositive_sub_correct2; auto with *.
rewrite Qpositive_sub_correct2; auto with *.
red in |- *; intros H1; elim (Qpositive_plus_diff w w'').
apply Qpositive_le_antisym.
apply Qpositive_plus_le.
rewrite H1; exact H0.
apply Qpositive_le_trans with w; auto.
apply Qpositive_plus_le.
Qed.

Theorem Qpositive_sub_sub_assoc :
forall w w' w'' : Qpositive,
w' <> w'' ->
Qpositive_le w'' w' ->
w <> Qpositive_sub w' w'' ->
Qpositive_le (Qpositive_sub w' w'') w ->
Qpositive_sub w (Qpositive_sub w' w'') =
Qpositive_sub (Qpositive_plus w w'') w'.