(* OK!!: *)

Lemma pre_ring1_Br1_aux:
forall(m:fmap)(x x':dart)(l:list),
inv_hmap m -> planar m ->
let y:= cA m zero x in
let y':= cA m zero x' in
double_link_list m (cons x x' l) ->
pre_ring0 m (cons x x' l) ->
pre_ring1 m l ->
~expf m y y' -> pre_ring1 (Br1 m x x') l.