(* TO BE A DOUBLE-LINK: *) 

Definition double_link(m:fmap)(x x':dart):Prop:=
  x <> x' /\ expe m x x'.

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.