Exchange of Order of Product

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Theorem

Let $R: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ and $S: \Z \to \left\{ {\mathrm T, \mathrm F}\right\}$ be propositional functions on the set of integers.

Let $\displaystyle \prod_{R \left({i}\right)} x_i$ denote a product over $R$.


Let the fiber of truth of both $R$ and $S$ be finite.


Then:

$\displaystyle \prod_{R \left({i}\right)} \prod_{S \left({j}\right)} a_{i j} = \prod_{S \left({j}\right)} \prod_{R \left({i}\right)} a_{i j}$


Proof


Also known as

The word interchange can often be seen for exchange.


Sources