(* A, A_1 on l: *)

Theorem A_A_1_l :
forall (m:fmap)(k j:dim)(x y z : dart),
inv_hmap m -> prec_lh m k x y ->
(A (l m k x y) j z =
if eq_dim_dec j k
then if eq_dart_dec z x then y
else if eq_dart_dec z (A_1 m k y)
then A m k x
else A m k z
else A m j z)
/\ (A_1 (l m k x y) j z =
if eq_dim_dec j k
then if eq_dart_dec z y then x
else if eq_dart_dec z (A m k x)
then A_1 m k y
else A_1 m k z
else A_1 m j z).