(* OK!! VERY IMPORTANT: *)

Theorem nc_Br1: forall (m:fmap)(x x':dart),
inv_hmap m -> planar m -> double_link m x x' ->
let y  := cA m zero x in
let y' := cA m zero x' in
nc (Br1 m x x') = nc m +
if expf_dec m y y' then 1 else 0.