Auxiliary Relation Image of Element is Upper Section

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

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

Let $x \in S$.

Then $^R x$ is an upper set

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

Proof
Let $a \in {}^R x, b \in S$ such that
 * $a \preceq b$

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

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

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

Thus by definition of $R$-cosegment:
 * $b \in {}^R x$