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.