elim H; intros; elim H1; intros; elim H2; intros; elim H5; intros.
elim H7; intros; elim H8; intros.
split. exact H10.
split. exact H10.
left; reflexivity.
Qed.

(**************)
Theorem div_add :
is_ring -> forall a b d : S, divide d a -> divide d b -> divide d (Add a b).