Sum of Summations equals Summation of Sum/Infinite Sequence

Theorem

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

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

Then: