(* Warning: This lemma is valid, but it's not clear that it's useful *)
Definition opposite_bij {A B} (b: bijection A B) : bijection B A :=
Bijection _ _ (bij_g _ _ b) (bij_f _ _ b) (bij_gf _ _ b) (bij_fg _ _ b).

Lemma Cross_bij' : forall A B JA JB bij,
@Cross_alg B JB ->
JB =  (Join_bij A JA B bij) ->
@Cross_alg A  JA.