elim H0; intros; elim H2; intros.
elim (H4 O (div_O_O H) (div_O_O H)); intros; elim H6; intros.
elim H8; intros. exact H9.
elim H9; intros; elim H10; reflexivity.
Qed.

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