(* NEW, OK!!: *)

Lemma pre_ring1_Br1: forall(m:fmap)(l:list),
inv_hmap m -> planar m ->
double_link_list m l ->  pre_ring0 m l -> pre_ring1 m l ->
let (x,x') := first l in
let y  := cA m zero x in
let y' := cA m zero x' in
~expf m y y' -> pre_ring1 (Br1 m x x') (tail l).