Definition CoPrime (a b : nat) := gcd (Z_of_nat a) (Z_of_nat b) 1.
Definition prod : forall (n : nat) (x : nat -> nat), nat.
intros.
induction n as [| n Hrecn].
exact 1.
exact (x n * Hrecn).
Defined.
Definition factorial (n : nat) : nat := prod n S.

Lemma coPrimeProd :
forall (n : nat) (x : nat -> nat),
(forall z1 z2 : nat,
z1 < S n -> z2 < S n -> z1 <> z2 -> CoPrime (x z1) (x z2)) ->
(forall z : nat, z < S n -> x z > 0) -> CoPrime (prod n x) (x n).