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 sa_distributive_bij : forall A B JA bij, @sa_distributive A JA -> @sa_distributive B (Join_bij A JA B bij).