Relation Segment is Increasing

Theorem
Let $S$ be a set.

Let $\mathcal R, \mathcal Q$ be relation on $S$ such that
 * $\mathcal R \subseteq \mathcal Q$

Let $x \in S$.

Then
 * $x^{\mathcal R} \subseteq x^{\mathcal Q}$

where $x^{\mathcal R}$ denotes the $\mathcal R$-segment of $x$.

Proof
Let $y \in x^{\mathcal R}$

By definition of $\mathcal R$-segment:
 * $\left({y, x}\right) \in \mathcal R$

By definition of subset:
 * $\left({y, x}\right) \in \mathcal Q$

Thus by definition of $\mathcal Q$-segment:
 * $y \in x^{\mathcal Q}$