intros x.
simple induction e; auto.
Qed.
Hint Resolve sup_add.
(*  Finalement inutile :  *)
(* Lemma diff_natural : (n,m:nat)~(<nat>n=m)->~(<Elt>(natural n)=(natural m)). Intros; Red; Intro. Absurd (<nat>n=m). Assumption. Replace n with (natural_inv (natural n)). 2:Auto. Replace m with (natural_inv (natural m)). 2:Auto. Elim H0. Auto. Save. *)
(*  Finalement inutile  *)
(* Lemma lt_diff : (n,m:nat)(lt m n)->~(<nat>n=m). Intros. Red. Intro. Cut (lt m n); Auto. Elim H0. Change ~(lt n n). Auto. Save. *)

Lemma elt_not_sym : forall a b : Elt, a <> b :>Elt -> b <> a :>Elt.
auto.