Permutation of Indices of Summation

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 finite.

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 $R$.

Also known as
The operation of permutation of indices of a summation can be seen referred to as a permutation of the range.

However, as the term range is ambiguous in the literature, and as its use here is not strictly accurate (it is the fiber of truth of $R$, not its range, which is being permuted, its use on is discouraged.