(*========================================================== CHANGING THE OPENING IN AN ORBIT BY L_B: exd, inv_hmap, planar, cA, cA_1, bottom, cF, cF_1, top, expf, expe, eqc... ==========================================================*)
(* OK! *)

(* Specifications: *)
Lemma cA_L_B_top_bot:forall(m:fmap)(k:dim)(x z:dart),
inv_hmap m -> succ m k x ->
cA (L (B m k x) k (top m k x) (bottom m k x)) k z =
cA m k z.
Lemma cA_L_B_top_bot_ter:forall(m:fmap)(k:dim)(x z:dart),
inv_hmap m -> succ m k x ->
let k1 := dim_opp k in
cA (L (B m k x) k (top m k x) (bottom m k x)) k1 z =
cA m k1 z.
Lemma A_L_B_top_bot:forall(m:fmap)(k:dim)(x z:dart),
inv_hmap m -> succ m k x ->
A (L (B m k x) k (top m k x) (bottom m k x)) k z =
if eq_dart_dec x z then nil
else if eq_dart_dec (top m k x) z then (bottom m k x)
else A m k z.
Lemma A_L_B_top_bot_ter:forall(m:fmap)(k:dim)(x z:dart),
inv_hmap m -> succ m k x -> let k1:=dim_opp k in
A (L (B m k x) k (top m k x) (bottom m k x)) k1 z =
A m k1 z.