Definition lift_prod1  {A}{JA: Join A}{B} : (@lifted (A * B) (Join_prod A _ B (Join_equiv B))) -> (@lifted A _ * B).

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.