intros u v.
generalize u; clear u; simple induction u.
auto.
intros x w Hyp inmon_x_w.
replace (cut1 (Append (cons x w) v)) with (cons x (cut1 (Append w v))).
replace (Append (cons x w) (cut1 v)) with (cons x (Append w (cut1 v))).
apply cons_cons.
trivial.
apply Hyp.
apply inmonoid_cons_inv with x; assumption.
trivial.
apply sym_equal.
unfold Append in |- *.
simpl in |- *.
apply cut1_cons.
apply inmonoid_cons_inv2 with w.
trivial.
Qed.
Axiom
cut2_cons :
forall (A : Elt) (v : Word), dans A X -> cut2 (cons A v) = cut2 v.

Lemma cut2_cons_N :
forall (A : Elt) (v : Word), ~ dans A X -> cut2 (cons A v) = cons A v.