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

(** Joinable identities are unique. *)
Lemma identities_unique {A} `{HA: Perm_alg A} :
forall e1 e2,  identity e1 ->  identity e2 ->  joins e1 e2 ->  e1 = e2.