Permutation of Indices of Product

Theorem
Let $R: \Z \to \set {\mathrm T, \mathrm F}$ be a propositional function on the set of integers.

Let the fiber of truth of $R$ be finite.

Then:
 * $\displaystyle \prod_{\map R j} a_j = \prod_{\map R {\map \pi j} } a_{\map \pi j}$

where:
 * $\displaystyle \prod_{\map R j} a_j$ denotes the product over $a_j$ for all $j$ that satisfy the propositional function $\map R j$
 * $\pi$ is a permutation on the fiber of truth of $R$.