Definition join_hom2 := forall (a1 a2 a3: A)(b1 b2 b3: B),
      join a1 a2 a3
      -> join b1 b2 b3
      -> join (f a1 b1) (f a2 b2) (f a3 b3).(* Two-argument join homomorphisms with a dummy argument *)
  Definition join_hom2' := forall (a1 a2 a3: A)(b1 b2 b3: B),
      join a1 a2 a3
      -> join (f a1 b1) (f a2 b2) (f a3 b3).

Lemma join_hom_sum : join_hom (fun s: A+B => match s with | inl x => inl _ (f x) | inr y => inr _ (g y) end).