Let $E_1, \dots, E_N$ each be a D17: Finite set such that

(i)	$\prod_{n = 1}^N E_n$ is the D326: Cartesian product of $E_1, \dots, E_N$

Since $E_1, \ldots, E_N$ are each finite, then \begin{equation} |E_1| = K_1, \quad \ldots, \quad |E_N| = K_N \end{equation} for some $K_1, \ldots, K_N \in \mathbb{N}$. Result R1832: Cardinality of a finite cartesian product of finite sets shows that \begin{equation} \left| \prod_{n = 1}^N E_n \right| = \prod_{n = 1}^N |E_n| = \prod_{n = 1}^N K_n < \infty \end{equation} $\square$