(*=======================================================  B, B_1 ON cA, cA_1  =========================================================*)
(* OK: VERY LONG *)

Lemma cA_cA_1_B : forall (m:fmap)(k:dim)(x z:dart),
inv_hmap m -> succ m k x ->
cA (B m k x) k z =
(if eq_dart_dec x z then bottom m k x
else if eq_dart_dec (top m k x) z then A m k x
else cA m k z)
/\  cA_1 (B m k x) k z =
(if eq_dart_dec (A m k x) z then top m k x
else if eq_dart_dec (bottom m k x) z then x
else cA_1 m k z).