Product of Products over Overlapping Domains

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 $R$.

Then:
 * $\displaystyle \prod_{R \left({j}\right)} a_j \prod_{S \left({j}\right)} a_j = \left({\prod_{R \left({j}\right) \mathop \lor S \left({j}\right)} a_j}\right) \left({\prod_{R \left({j}\right) \mathop \land S \left({j}\right)} a_j}\right)$

where $\lor$ and $\land$ signify logical disjunction and logical conjunction respectively.

Proof
Let:
 * $A := \left\{ {j \in \Z: R \left({j}\right)}\right\}$
 * $B := \left\{ {j \in \Z: S \left({j}\right)}\right\}$

The result then follows from Cardinality of Set Union.