Definition cut (u : Word) :=
  let (a, s) := cut_spec u in (a, let (b, s2) := s in b).Definition cut1 (u : Word) := fst (cut u).Definition cut2 (u : Word) := snd (cut u).

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