(* TO BE A DOUBLE-LINK: *) 

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

Lemma cF_1_Br1:forall(m:fmap)(x x' z:dart), inv_hmap m -> double_link m x x' -> cF_1 (Br1 m x x') z = if eq_dart_dec x (cA m one z) then cA m zero x' else if eq_dart_dec x' (cA m one z) then cA m zero x else cF_1 m z.