unfold extension in |- *.
intros e f x dans_x_e.
elim (extension_spec e f x).
intro.
tauto.
Qed.

(* Specifications: *)
Lemma extension_out :
forall (e : Ensf) (f : Elt -> Elt) (x : Elt),
~ dans x e -> extension e f x = x :>Elt.