(* Second PART: *)

Lemma inv_hmap_l_2: forall (m:fmap)(k:dim)(x y:dart),
inv_hmap m -> prec_lh m k x y ->
(forall (x0 : dart) (k0 : dim),
exd (l m k x y) x0 ->
succ (l m k x y) k0 x0 /\ pred (l m k x y) k0 x0).