intros.
elim H; intros; elim H3; intros; elim H4; intros; elim H5; intros.
elim H7; intros; elim H9; intros; elim H11; intros; elim (H15 x H0); intros.
elim H17; intros; elim H19; intros.
clear H2 H3 H5 H6 H7 H9 H10 H11 H12 H14 H15 H16 H17 H19.
apply (opp_unicity S A Add O Opp H4 (Mult x y) (Mult (Opp x) y)).
unfold is_opposite in |- *; split.
exact (H8 x y H0 H1).
split.
exact (H8 (Opp x) y H18 H1).
elim (mult_O H y H1); intros; elim H3; clear H H0 H1 H2 H3 H4 H8 H18.
pattern O at 1 in |- *; elim H20; elim H21; clear H20 H21.
elim (H13 x (Opp x) y); intros; rewrite H; clear H H0.
elim (H13 (Opp x) x y); intros; rewrite H; auto.
Qed.

(*****************)
Lemma mult_opp_opp :
is_ring -> forall x y : S, A x -> A y -> Mult (Opp x) (Opp y) = Mult x y.