(** Elememt [a] is a subelement of [c] .  This relation forms a partial order. *)
  Definition join_sub {A} `{Join A} (a c : A) : Prop :=
    exists b, join a b c.

Lemma join_sub_same_identity {A} {J: Join A}{PA: Perm_alg A}: forall e e' a c, identity e -> unit_for e a -> identity e' -> unit_for e' c -> join_sub a c -> e = e'.