apply Rmult_lt_0_compat; auto.
replace 0%R with (-0)%R; auto with real.
Qed.

(* Specifications: *)
Theorem Rlt_sign_neg_pos: forall x y, (x < 0 -> 0 < y  -> 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.
Theorem Rlt_sign_neg_pos: forall x y, (x < 0 -> 0 < y  -> 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.