Rules for Manipulating Summations

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

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