(*===========================================================
                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 planar_L_B_top_bot_0:forall(m:fmap)(x:dart), inv_hmap m -> succ m zero x -> planar m -> planar (L (B m zero x) zero (top m zero x) (bottom m zero x)).