Product of Products over Overlapping Domains

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

Let $\ds \prod_{\map R i} x_i$ denote a product over $R$.

Then:
 * $\ds \prod_{\map R j} a_j \prod_{\map S j} a_j = \paren {\prod_{\map R j \mathop \lor \map S j} a_j} \paren {\prod_{\map R j \mathop \land \map S j} a_j}$

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

Proof
Let:
 * $A := \set {j \in \Z: \map R j}$
 * $B := \set {j \in \Z: \map S j}$

The result then follows from Cardinality of Set Union.