(* TO BE A DOUBLE-LINK: *) 

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

Lemma Br1_comm: forall (m:fmap)(x x':dart), inv_hmap m -> double_link m x x' -> Br1 m x x' = Br1 m x' x.