(** 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_trans {A} {J: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: forall a b c, join_sub a b -> join_sub b c -> join_sub a c.