Sum of Summations equals Summation of Sum

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

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

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

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

Proof
Let $b_i =: a_{i 1}$ and $c_i =: a_{i 2}$.

Then: