unfold cut1, cut2, cut in |- *.
elim (cut_spec w).
intros a temp; elim temp; clear temp.
intros b temp; elim temp; clear temp.
intros inm temp; elim temp.
simpl in |- *; auto.
Qed.

(* Specifications: *)
Lemma cut1_cons :
forall (a : Elt) (w : Word), dans a X -> cut1 (cons a w) = cons a (cut1 w).