Proposition 14.4.3. Let $X, Y$ be sets, $\ce \subset 2^{X}$, and $\cf \subset 2^{Y}$ be elementary families, then the collection of rectangles
\[\mathcal{R}(\ce, \cf) = \bracs{E \times F| E \in \ce, F \in \cf}\]
is an elementry family.
Proof. (P1): $\emptyset = \emptyset \times \emptyset$.
(P2): For any $A \times B, C \times D \in \mathcal{R}(\ce, \cf)$,
\[(A \times B) \cap (C \times D) = \underbrace{(A \cap C)}_{\in \ce}\times \underbrace{(B \times D)}_{\in \cf}\in \mathcal{R}(\ce, \cf)\]
(E): Let $A \times B, C \times D \in \mathcal{R}(\ce, \cf)$, then
\begin{align*}(A \times B) \setminus (C \times D)&= (A \setminus C) \times (B \setminus D) \sqcup (A \setminus C) \times (B \cap D) \\&\sqcup (A \cap C) \times (B \setminus D)\end{align*}
Let $\seqf{A_j}\subset \ce$ such that $A \setminus C = \bigsqcup_{j = 1}^{n} A_{j}$ and $\bracsn{B_j}_{1}^{n} \subset \cf$ such that $B \setminus D = \bigsqcup_{j = 1}^{m} B_{j}$, then
\begin{align*}(A \setminus C) \times (B \setminus D)&= \bigsqcup_{i = 1}^{n} \bigsqcup_{j = 1}^{m} A_{i} \times B_{j} \\ (A \setminus C) \times (B \cap D)&= \bigsqcup_{i = 1}^{n} A_{i} \times (B \cap D) \\ (A \cap C) \times (B \setminus D)&= \bigsqcup_{j = 1}^{m} (A \cap C) \times B_{j}\end{align*}
are all finite disjoint unions of elements of $\mathcal{R}(\ce, \cf)$. Therefore $(A \times B) \setminus (C \times D) \in \mathcal{R}(\ce, \cf)$.$\square$