Partition of Indexing Set induces Bijection on Family of Sets/Lemma

Theorem
Let $I$ be an indexing set.

Let $\family {S_\alpha}_{\alpha \mathop \in I}$ be a family of sets indexed by $I$.

Let $I = I_1 \cup I_2$ such that $I_1 \cap I_2 = \O$.

Then there exists a bijection:
 * $\ds \psi: \paren {\prod_{\alpha \mathop \in I_1} S_\alpha} \times \paren {\prod_{\alpha \mathop \in I_2} S_\alpha} \to \prod_{\alpha \mathop \in I} S_\alpha$