(* pre_ring2_B is defined *)
(*============================================================ NEW pre_ring3 PROPERTIES: OK!! =============================================================*)
(* ANCIENT : USEFUL IN THE FOLLOWING Lemma *)

Lemma betweenf_expf1:forall(m:fmap)(z v t:dart),
inv_hmap m -> exd m z ->
betweenf m z v t -> (expf m v z /\ expf m v t).