Permutation of Indices of Summation/Infinite Series

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

Let $\displaystyle \sum_{R \left({i}\right)} a_i$ be absolutely convergent.

Then:
 * $\displaystyle \sum_{R \left({j}\right)} a_j = \sum_{R \left({\pi \left({j}\right)}\right)} a_{\pi \left({j}\right)}$

where:
 * $\displaystyle \sum_{R \left({j}\right)} a_j$ denotes the summation over $a_j$ for all $j$ that satisfy the propositional function $R \left({j}\right)$
 * $\pi$ is a permutation on the fiber of truth of $R$.

Proof
This is a restatemtent of Manipulation of Absolutely Convergent Series: Permutation in the context of summations.