Relation Segment is Increasing

Theorem
Let $S$ be a set.

Let $\RR, \QQ$ be relations on $S$ such that
 * $\RR \subseteq \QQ$

Let $x \in S$.

Then
 * $x^\RR \subseteq x^\QQ$

where $x^\RR$ denotes the $\RR$-segment of $x$.

Proof
Let $y \in x^\RR$.

By definition of $\RR$-segment:
 * $\tuple {y, x} \in \RR$

By definition of subset:
 * $\tuple {y, x} \in \QQ$

Thus by definition of $\QQ$-segment:
 * $y \in x^\QQ$