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.
Definition is_ring :=
commutativity S Add /\
is_group S A Add O Opp /\
intern S A Mult /\ associativity S Mult /\ distributivity S Add Mult.

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