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

Lemma join_hom2_comparable {A}{B}{C}
`{Perm_alg A}{SA: Sep_alg A}`{Perm_alg B}{SB: Sep_alg B}`{Perm_alg C}{SC: Sep_alg C}:
forall (g: A -> B -> C) a1 a2 b1 b2,
comparable a1 a2
-> comparable b1 b2
-> join_hom2 g
-> comparable (g a1 b1) (g a2 b2).