Since $X_1, \ldots, X_N$ are finite sets, then result
R1832: Cardinality of a finite cartesian product of finite sets shows that
\begin{equation}
|X| = \left| \prod_{n = 1}^N X_n \right| = \prod_{n = 1}^N K_n
\end{equation}
Notice that $\mathcal{R}$ is the
D80: Power set of $X$; this result is therefore now a consequence of
R1839: Cardinality of power set of a finite set. $\square$
