Exchange of Order of Summation with Dependency on Both Indices

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

Let $S: \Z \times \Z \to \set {\T, \F}$ be a propositional function on the Cartesian product of the set of integers with itself.

Let $\ds \sum_{\map R i} x_i$ denote a summation over $R$.

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

Then:
 * $\ds \sum_{\map R i} \sum_{\map S {i, j} } a_{i j} = \sum_{\map {S'} j} \sum_{\map {R'} {i, j} } a_{i j}$

where:
 * $\map {S'} j$ denotes the propositional function:
 * there exists an $i$ such that both $\map R i$ and $\map S {i, j}$ hold
 * $\map {R'} {i, j}$ denotes the propositional function:
 * both $\map R i$ and $\map S {i, j}$ hold.