(* distinct_faces_Br1_aux_bis is defined *)
(* NEW, OK: *)

Lemma pre_ring3_Br1_aux:
forall(m:fmap)(x x' xs xs':dart)(l:list),
inv_hmap m -> planar m ->
let l1 := cons x x' (cons xs xs' l) in
double_link_list m l1 ->
pre_ring0 m l1 ->
face_adjacent m x x' xs xs' ->
pre_ring3 m l1 ->
pre_ring3 (Br1 m x x') (cons xs xs' l).