Definition app_1_n {T:Type} {n:nat} (pred : arity T (S n)) (x : T) (cp : cartesianPower T n) : Prop.

(* Specifications: *)
Lemma app_app_1_n {T:Type} {n:nat} :
forall (pred : arity T (S (S n))) (x : T) (cpp : cartesianPower T (S n)) (cpt : cartesianPower T (S (S n))),
app pred cpt -> headCP cpt = x -> tailCP cpt = cpp ->
app_1_n pred x cpp.
Lemma app_1_n_app_eq {T:Type} {n:nat} :
forall (pred : arity T (S (S n))) (x : T) (cpp : cartesianPower T (S n)) (cpt : cartesianPower T (S (S n))),
headCP cpt = x -> tailCP cpt = cpp ->
(app pred cpt <-> app_1_n pred x cpp).