Definition integrity := forall a b : S, Mult a b = O -> {a = O} + {b = O}.
Definition is_unitary_commutative_ring :=
is_ring /\ commutativity S Mult /\ neutral S A Mult I.
(* Division *)
Definition divide (a b : S) :=
A a /\ A b /\ (b = O \/ a <> O /\ (exists q : S, A q /\ b = Mult a q)).

Lemma simplification_integrity :
is_unitary_commutative_ring ->
integrity -> forall a x : S, A a -> A x -> a <> O -> Mult a x = a -> x = I.