Proposition 4.4.3.label Let $X,Y$ be sets, $\cali{E}\suf 2^{X},\cali{F}\suf 2^{Y}$ be elementary families, then the collection of rectangles
\begin{align*}\cali{R}(\cali{E},\cali{F})=\curl{E\ti F:E\in\cali{E},F\in\cali{F}}\end{align*}
is an elementary family.
Proof. (P1): $\emp=\emp\ti\emp$
(P2): For any $A\ti B,C\ti D\in\cali{R}(\cali{E},\cali{F})$,
\begin{align*}(A\ti B)\cap (C\ti D)=(A\cap C)\ti (B\ti D)\in\cali{R}(\cali{E},\cali{F})\end{align*}
(E): For any $A\ti B,C\ti D\in\cali{R}(\cali{E},\cali{F})$,
\begin{align*}(A\ti B)\del (C\ti D)=(A\del C)\ti (B\del D)\sqcup(A\del C)\ti (B\cap D)\sqcup (A\cap C)\ti (B\del D)\end{align*}
Let $\curl{A_j}_{1}^{n}\suf\cali{E}$ such that $A\del C=\dcups{n}{j=1}A_{j}$ and $\curl{B_i}_{1}^{m}\suf\cali{F}$ such that $B\del D=\dcups{m}{i=1}B_{i}$, then
\begin{align*}(A\del C)\ti (B\del D)&=\dcups{n}{j=1}\dcups{m}{i=1}A_{j}\ti B_{i}\\ (A\del C)\ti (B\cap D)&=\dcups{n}{j=1}A_{j}\ti (B\cap D)\\ (A\cap C)\ti (B\del D)&=\dcups{m}{i=1}(A\cap C)\ti B_{i}\end{align*}
are all finite disjoint unions of elements of $\cali{R}(\cali{E},\cali{F})$. Therefore $(A\ti B)\del (C\ti D)\in\cali{R}(\cali{E},\cali{F})$.$\square$
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