Section ring.
Variable S : Set.
Variable A : S -> Prop.
Variable Add Mult : S -> S -> S.
Variable O I : S.
Variable Opp : S -> S.
Variable v : S -> nat.
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Definition is_ring :=
commutativity S Add /\
is_group S A Add O Opp /\
intern S A Mult /\ associativity S Mult /\ distributivity S Add Mult.
intros.
elim H; intros; elim H2; intros; elim H3; intros; elim H4; intros.
elim H6; intros; elim H8; intros; elim H10; intros; elim H13; intros.
clear H2 H4 H5 H6 H8 H9 H10 H11 H13 H14.
(* H3: is_group, H7: intern Mult, H0: (A x), H15: (A O), H16: rest of neutral, H12: distributivity *)
split.
(* x*0 = 0 *)
apply
(regular_l S A Add O Opp H3 (Mult x O) O (H7 x O H0 H15) H15
(Mult x O) (H7 x O H0 H15)).
elim (H16 (Mult x O) (H7 x O H0 H15)); intros; rewrite H2.
elim (H12 x O O); intros; elim H6.
elim (H16 O H15); intros; rewrite H8; reflexivity.
(* 0*x = 0 *)
apply
(regular_l S A Add O Opp H3 (Mult O x) O (H7 O x H15 H0) H15
(Mult O x) (H7 O x H15 H0)).
elim (H16 (Mult O x) (H7 O x H15 H0)); intros; rewrite H2.
elim (H12 O O x); intros; elim H5.
elim (H16 O H15); intros; rewrite H8; reflexivity.
Qed.

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