(** 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_unit_for {A} {J: Join A}{PA: Perm_alg A}: forall e a b, unit_for e a -> join_sub a b -> unit_for e b.