Instance Cross_prod : forall A B saA saB,
@Cross_alg A saA ->
@Cross_alg B saB ->
@Cross_alg (A * B) (Join_prod _ saA _ saB).

Lemma cross_split_fpm : forall A B
(JB: Join B) (PB: Perm_alg B)(SB : Sep_alg B)(CB: Disj_alg B)
(Bdec: forall x:B, {identity x}+{~identity x}) ,
Cross_alg B  ->
Cross_alg (fpm A (lifted JB)) .