Preceding implies Inclusion of Segments of Auxiliary Relation

Theorem
Let $\left({S, \vee, \preceq}\right)$ be a bounded below join semilattice.

Let $R$ be an auxiliary relation on $S$.

Let $x, y \in S$ such that
 * $x \preceq y$

Then
 * $x^R \subseteq y^R$

where $x^R$ denotes the $R$-segment of $x$.

Proof
Let $a \in x^R$.

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

By definition of reflexivity:
 * $a \preceq a$

By definition of auxiliary relation:
 * $\left({a, y}\right) \in R$

Thus by definition of $R$-segment of $y$:
 * $a \in y^R$