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

(** The existence of identity elements. *)
Lemma join_ex_identities {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: forall a,
{e : A & identity e /\ joins e a}.