(*============================================================ NEW RING SYNTHESIS AND...DISCRETE JORDAN CURVE THEOREM: OK =============================================================*)
(* NEW: OK: *)

Lemma ring_Br1_aux: forall(m:fmap)(l:list),
inv_hmap m -> planar m ->
let x:= fst (first l) in let x' := snd (first l) in
let y := cA m zero x in
let y' := cA m zero x' in
let m1 := Br1 m x x' in
~expf m y y' -> ring m l ->
(emptyl (tail l) \/ ring m1 (tail l)).