(* A on xk := A m k x0 : *)

Lemma A_B_1_B_xk :
forall(m:fmap)(k:dim)(x0 y0:dart),
inv_qhmap m ->
exd m x0 -> exd m y0 ->
let xk := A m k x0 in
let yk := A_1 m k y0 in
x0 <> xk -> y0 <> xk ->
A (B_1 (B m k x0) k y0) k xk =
if eq_dart_dec xk yk then nil
else A m k xk.