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

Lemma split_identity{A} {JA: Join A}{PA: Perm_alg A}{SA: Sep_alg A}:
forall a b c, join a b c -> identity c -> identity a.