(* IDEM:*)

(* Specifications: *)
Lemma bottom_bottom : forall (m:fmap)(k:dim)(z:dart),
inv_hmap m ->
bottom m k (bottom m k z) = bottom m k z.
Lemma inj_F_1_predf :
forall m:fmap, inv_hmap m ->
inj_dart (predf m) (F_1 m).