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 gcd_unicity_apart_unities :
is_unitary_commutative_ring ->
integrity ->
forall a b d1 d2 : S,
is_gcd a b d1 ->
is_gcd a b d2 ->
exists x : S, inversible S Mult I x /\ A x /\ d2 = Mult d1 x.