Restriction of Restriction is Restriction

Theorem
Let $\RR$ be a relation on $S \times T$.

Let $X \subseteq S$, $Y \subseteq T$.

Let $W \subseteq X$, $V \subseteq Y$.

Then:
 * $\paren{\RR {\restriction_{X \times Y} } } {\restriction_{W \times V} } = \RR {\restriction_{W \times V} }$

That is, the restriction of $\RR {\restriction_{X \times Y} }$ to $W \times V$ is the restriction of $\RR$ to $W \times V$

Proof
From Cartesian Product of Subsets:
 * $W \times V \subseteq X \times Y$

We have: