Exchange of Order of Product

Theorem
Let $R: \Z \to \set {\T, \F}$ and $S: \Z \to \set {\T, \F}$ be propositional functions on the set of integers.

Let $\ds \prod_{\map R i} x_i$ denote a product over $R$.

Let the fiber of truth of both $R$ and $S$ be finite.

Then:
 * $\ds \prod_{\map R i} \prod_{\map S j} a_{i j} = \prod_{\map S j} \prod_{\map R i} a_{i j}$

Also known as
The word interchange can often be seen for exchange.