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

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