(* NEW: OK!! *)

Lemma pre_ring2_Br1: forall(m:fmap)(l:list),
inv_hmap m -> planar m ->
double_link_list m l -> pre_ring0 m l ->
pre_ring1 m l -> pre_ring2 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_ring2 (Br1 m x x') (tail l).