(* SUPER OK!!: *)

Lemma pre_ring3_Br1: forall(m:fmap)(l:list),
inv_hmap m -> planar m ->
let (x,x') := first l in
double_link_list m l ->
pre_ring0 m l ->  pre_ring1 m l -> pre_ring3 m l ->
pre_ring3 (Br1 m x x') (tail l).