Sum of Summations equals Summation of Sum/Infinite Sequence

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

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

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

Let $\displaystyle \sum_{\map R i} b_i$ and $\displaystyle \sum_{\map R i} c_i$ be convergent.

Then:
 * $\displaystyle \sum_{\map R i} \paren {b_i + c_i} = \sum_{\map R i} b_i + \sum_{\map R i} c_i$