intros.
unfold xk in |- *; unfold yk in |- *.
split.
intro.
intro.
assert (y = A m k x).
apply A_1_A.
unfold inv_hmap in H; tauto.
tauto.
tauto.
symmetry  in |- *.
tauto.
tauto.
intro.
intro.
assert (x = A_1 m k y).
apply A_A_1.
unfold inv_hmap in H; tauto.
tauto.
tauto.
tauto.
rewrite H4 in H2.
tauto.
Qed.
(*======================================================== Correction of l for hmaps : PROOF OBLIGATION ========================================================*)
(* BY PARTS: *)
(* From qhmap, by B then by B_1, we arrive in qhmap: *)

Lemma inv_qhmap_l_11:
forall (m:fmap)(k:dim)(x y:dart),
inv_qhmap m -> prec_lh m k x y ->
inv_qhmap (B_1 (B m k x) k y).