(** 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_joins {A} `{HA: Perm_alg A}: forall {a b}, join_sub a b -> joins a b -> joins a a.