# Product of Products

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## Theorem

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

Let $\displaystyle \prod_{R \paren i} x_i$ denote a product over $R$.

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

Then:

$\displaystyle \prod_{R \paren i} \paren {b_i c_i} = \paren {\prod_{R \paren i} b_i} \paren {\prod_{R \paren i} c_i}$

## Proof

Let $b_i =: a_{i 1}$ and $c_i =: a_{i 2}$.

Then:

 $\displaystyle \prod_{R \paren i} \paren {b_i c_i}$ $=$ $\displaystyle \prod_{R \paren i} \paren {a_{i 1} a_{i 2} }$ by definition $\displaystyle$ $=$ $\displaystyle \prod_{R \paren i} \paren {\prod_{1 \mathop \le j \mathop \le 2} a_{i j} }$ Definition of Product by Propositional Function $\displaystyle$ $=$ $\displaystyle \prod_{1 \mathop \le j \mathop \le 2} \paren {\prod_{R \paren i} a_{i j} }$ Exchange of Order of Product $\displaystyle$ $=$ $\displaystyle \paren {\prod_{R \paren i} a_{i 1} } \paren {\prod_{R \paren i} a_{i 2} }$ Definition of Product by Propositional Function $\displaystyle$ $=$ $\displaystyle \paren {\prod_{R \paren i} b_i} \paren {\prod_{R \paren i} c_i}$ by definition

$\blacksquare$