(** 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 a' b b' : A}, join_sub a a' -> join_sub b b' -> joins a' b' -> joins a b.