Auxiliary Relation Image of Element is Upper Section

Theorem
Let $L = \struct {S, \preceq}$ be an ordered set.

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

Let $x \in S$.

Then $\map R x$ is an upper section

where $\map R x$ denotes the image of $x$ under $R$.

Proof
Let $a \in \map R x, b \in S$ such that
 * $a \preceq b$

By definition of $R$-image of element:
 * $\tuple {x, a} \in R$

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

By definition of auxiliary relation:
 * $\tuple {x, b} \in R$

Thus by definition of $R$-image of element:
 * $b \in \map R x$