simpl in |- *.
intros x dans_x_X.
apply extension_out.
apply inter_dans with X; auto.
Qed.
Hint Resolve Id_injproducg1.

Lemma Id_injproducd2 : forall x : Elt, dans x X -> injproducd V2 x = x :>Elt.
unfold Id, injproducd in |- *.