Lemma concat_prop_per_part : forall P : Predicate A, (forall (i : Nat) (Hi : i < n), Pred_fun P (v (Build_finiteT Hi))) -> (forall (j : nat) (Hj : j < m), Pred_fun P (w (Build_finiteT Hj))) -> forall (k : nat) (Hk : k < n + m), Pred_fun P ((v ++ w) (Build_finiteT Hk)).