End join_hom_fun.
Arguments lookup [Key A] _ _ _.

Lemma join_hom_comparable {A}{B}`{Perm_alg A}{SA: Sep_alg A}`{Perm_alg B}{SB:Sep_alg B}:
forall (f: A -> B) a1 a2, comparable a1 a2 -> join_hom f -> comparable (f a1) (f a2).