intros x' y' H1 H2; inversion H2; auto with *.
apply Qgt_pos_pos.
red in |- *; intros H5; elim H3.
apply Qpositive_le_trans with x'; auto with *.
apply not_Qpositive_le_Qpositive_le'.
auto.
intros x' y' H1 H2; inversion H2; auto with *.
apply Qgt_neg_neg.
red in |- *; intros H5; elim H3.
apply Qpositive_le_trans with x'; auto with *.
apply not_Qpositive_le_Qpositive_le'.
auto.
intros x' H1; inversion H1; auto with *.
intros x' y' H1; inversion H1; auto with *.
intros x' H1; inversion H1; auto.
Qed.

Theorem Qpositive_le_plus_simpl :
forall x y z : Qpositive,
Qpositive_le (Qpositive_plus x z) (Qpositive_plus y z) -> Qpositive_le x y.