Admitted.

(* Specifications: *)
Lemma expf_planar_L0_rcp: forall (m:fmap)(x y:dart),
inv_qhmap (L m di0 x y) -> planar (L m di0 x y) ->
let mc:= clos m in
(~eqc mc x y \/ expf mc (A_1 mc di1 x) y).
Lemma expf_planar_L1_rcp: forall (m:fmap)(x y:dart),
inv_qhmap (L m di1 x y) -> planar (L m di1 x y) ->
let mc:= clos m in
(~eqc mc x y \/ expf mc x (A mc di0 y)).
Theorem plf_planar_rcp: forall m:fmap,
inv_qhmap m -> genus m = 0 -> plf m.