(* Proof of lem_fun_inj_dom_img_is_bij                                     *)
unfold fun_inj, fun_bij in |- *; intros.
elim H; clear H Af; intros; elim H0; clear H0; intros; clear H0; elim H1;
clear H1; intros; clear H0.
split;
[ auto with zfc
| split; [ reflexivity | split; [ reflexivity | auto with zfc ] ] ].
Qed.
(* End of proof of lem_fun_inj_dom_img_is_bij                              *)

(* Specifications: *)
(***************************************************************************)
(* Si f est une bijection alors fun_recip de fun_recip de f est egale a f  *)
(***************************************************************************)
Lemma lem_fun_recip_involutive_under_bij :
forall (f : E) (Af : fun_ f) (Abijf : fun_bij f (dom f) (Img f))
(Arec : fun_ (fun_recip f Af)), fun_recip (fun_recip f Af) Arec = f.