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

(* The contrapositive of split_identity *)
Lemma join_nonidentity {A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}: forall a b c,
nonidentity a -> join a b c -> nonidentity c.