Section PROD'.
Variable A : Type.
Variable J_A: Join A.
Variable saA : Perm_alg A.
Variable B: Type.
Variable J_B: Join B.
Variable saB : Perm_alg B.
Variable agB : ageable B.
Variable asb : Age_alg B.

Lemma join_level {A}{JA: Join A}{PA: Perm_alg A}{AG: ageable A}{AgeA: Age_alg A}:
forall x y z, join x y z -> level x = level z /\ level y = level z.