(*===========================================================
                expe AND bottom
============================================================*)

(* expo and between in an edge: *)

Definition expe(m:fmap)(z t:dart):=
  MA0.expo m z t.Definition betweene(m:fmap)(z v t:dart):=
  MA0.between m z v t.

Lemma A_1_L_B_top_bot:forall(m:fmap)(k:dim)(x z:dart), inv_hmap m -> succ m k x -> A_1 (L (B m k x) k (top m k x) (bottom m k x)) k z = if eq_dart_dec (A m k x) z then nil else if eq_dart_dec (bottom m k x) z then (top m k x) else A_1 m k z.