Hint Resolve @join_joins @join_joins' @join_join_sub @join_join_sub'.
Definition nonidentity {A} `{Perm_alg A} (a: A) := ~(identity a).

(** If [a] is a subelement of [b], their units are equal. *)
Lemma join_sub_units_eq {A} {J: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A} :
forall {a b} ea eb,
join_sub a b ->
unit_for ea a ->
unit_for eb b ->
ea = eb.