(* A_1 on x0: *)

Lemma A_1_B_1_B_x0 : forall(m:fmap)(k:dim)(x0 y0:dart),
inv_qhmap m ->
exd m x0 -> exd m y0 -> y0<>x0 -> y0<>A m k x0 ->
A_1 (B_1 (B m k x0) k y0) k x0 =
if eq_dart_dec x0 (A m k x0) then nil
else A_1 m k x0.