Product of Products

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: