Segment of Auxiliary Relation is Subset of Lower Closure

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

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

Let $x \in S$.

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

where
 * $x^R$ denotes the $R$-segment of $x$,
 * $x^\preceq$ denotes the lower closure of $x$.

Proof
Let $a \in x^R$.

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

By definition of auxiliary relation:
 * $a \preceq x$

Thus by definition of lower closure of element:
 * $a \in x^\preceq$