Permutation of Indices

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Theorem

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

Let the fiber of truth of $R$ be finite.

Let $\pi$ is a permutation on the fiber of truth of $R$.


Then:

Permutation of Indices of Summation

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


Permutation of Indices of Product

$\ds \prod_{\map R j} a_j = \prod_{\map R {\map \pi j} } a_{\map \pi j}$