Relation Segment of Auxiliary Relation is Ideal

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

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

Let $x \in S$.

Then
 * $x^R$ is ideal in $L$

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

Non-empty
By definition of auxiliary relation:
 * $\left({\bot, x}\right) \in R$

where $\bot$ denotes the smallest element in $L$.

By definition of relation segment:
 * $\bot \in x^R$

Thus by definition:
 * $x^R$ is non-empty.

Directed
Let $y, z \in x^R$.

By definition of relation segment:
 * $\left({y, x}\right) \in R$ and $\left({z, x}\right) \in R$

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

By definition of relation segment:
 * $y \vee z \in x^R$

Thus by Join Succeeds Operands:
 * $y \preceq y \vee z$ and $z \preceq y \vee z$

Thus by definition:
 * $x^R$ is directed.

Lower
Let $y \in x^R, z \in S$ such that
 * $z \preceq y$

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

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

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

Thus by definition of relation segment:
 * $z \in x^R$

Thus by definition:
 * $x^R$ is lower set.

Thus by definition:
 * $x^R$ is ideal in $L$.