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) .