Sum of Summations equals Summation of Sum/Infinite Sequence

Theorem
Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be a propositional function 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 $R$ be infinite.

Let $\displaystyle \sum_{R \left({i}\right)} b_i$ and $\displaystyle \sum_{R \left({i}\right)} c_i$ be convergent.

Then:
 * $\displaystyle \sum_{R \left({i}\right)} \left({b_i + c_i}\right) = \left({\sum_{R \left({i}\right)} b_i + \sum_{R \left({i}\right)} c_i}\right)$

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

Then: