Permutation of Indices of Summation/Infinite Series

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

Let $\ds \sum_{\map R i} a_i$ be absolutely convergent.

Then:
 * $\ds \sum_{\map R j} a_j = \sum_{\map R {\map \pi j} } a_{\map \pi j}$

where:
 * $\ds \sum_{\map R j} a_j$ denotes the summation over $a_j$ for all $j$ that satisfy the propositional function $\map R j$
 * $\pi$ 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.