Exchange of Order of Summation/Infinite Series

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

Let $\displaystyle \sum_{R \left({i}\right)} x_i$ denote a summation over $R$.

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

Let:
 * $\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} \left\vert{a_{i j} }\right\vert$

exist.

Then:


 * $\displaystyle \sum_{R \left({i}\right)} \sum_{S \left({j}\right)} a_{i j} = \sum_{S \left({j}\right)} \sum_{R \left({i}\right)} a_{i j}$