Definition P2_bis
 (m:fmap)(z:dart)(s:set):Prop:=
  (card s <= ndN m ->
    card (Iter_rem_aux m z s) <= ndN m)%nat.

(* Specifications: *)
Lemma card_rem_aux_bis:forall(m:fmap)(z:dart)(s:set), (card s <= ndN m -> card (Iter_rem_aux m z s) <= ndN m)%nat.
Lemma upb_pos_aux:forall(m:fmap)(z:dart), exd m z -> (card (Iter_rem m z) < ndN m)%nat.
Theorem upb_pos:forall(m:fmap)(z:dart), exd m z -> (0 < Iter_upb m z)%nat.