Sum of Summations over Overlapping Domains/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.

Then provided that any $3$ of the $4$ summations converge:


 * $\displaystyle \sum_{R \left({j}\right)} a_j + \sum_{S \left({j}\right)} a_j = \sum_{R \left({j}\right) \mathop \lor S \left({j}\right)} a_j + \sum_{R \left({j}\right) \mathop \land S \left({j}\right)} a_j$

where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.