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_fpm (A B: Type){JB: Join B} {PB: Perm_alg B}{PosB : Pos_alg B}
{CrB: Cross_alg B}:   Cross_alg (fpm A B) .