Auxiliary Relation Image of Element is Upper Set

Theorem
Let $L = \left({S, \preceq}\right)$ be an ordered set.

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

Let $x \in S$.

Then $R\left({x}\right)$ is an upper set

where $R\left({x}\right)$ denotes the image of $x$ under $R$.

Proof
Let $a \in R\left({x}\right), b \in S$ such that
 * $a \preceq b$

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

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

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

Thus by definition of $R$-image of element:
 * $b \in R \left({x}\right)$