Definition constructive_join_sub {A} {JOIN: Join A} (w1 w3: A) := {w2 | join w1 w2 w3}.

Lemma cjoin_sub_irr {A} `{Perm_alg A}{CA: Canc_alg A}: forall {w1 w3: A} (j1 j2: constructive_join_sub w1 w3), proj1_sig j1 = proj1_sig j2.