apply Ropp_le_contravar; auto with real.
rewrite Ropp_0; auto with real.
Qed.

Theorem Rle_sign_pos_neg: forall x y: R, (0 <= x -> y <= 0  -> x * y <= 0)%R.
intros x y H1 H2; apply Rle_pos_neg; replace (- (x * y))%R with (x * (- y))%R; auto with real; try ring.