Definition nthCP {T:Type} {m:nat} (n : nat) (cp : cartesianPower T m) (Default : T) := nth (n-1) (CPToList cp) Default.

Lemma nthFirst {T:Type} {n:nat} : forall (cp : cartesianPower T (S n)) (t Default : T), t = nthCP 1 cp Default -> t = headCP cp.