Definition lift_prod1  {A}{JA: Join A}{B} : (@lifted (A * B) (Join_prod A _ B (Join_equiv B))) -> (@lifted A _ * B).
intros [x Hx].
destruct x as [a b].
split; auto.
apply (mk_lifted a (lift_prod_aux1 Hx)).
Defined.

(* Specifications: *)
Lemma lift_prod_aux2 {A}{JA: Join A}{B}:
forall x,
nonunit (fst x) -> @nonunit (A * B) (Join_prod A JA B (Join_equiv B)) x.