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

(** If a joins with b, then their units are equal. *)
Lemma joins_units_eq {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}{CA: Disj_alg A}:
forall {a b} ea eb,
joins a b ->
unit_for ea a ->
unit_for eb b ->
ea  = eb.