Compact Closure is Increasing

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

Let $x, y \in S$ such that
 * $x \preceq y$

Then $x^{\mathrm{compact} } \subseteq y^{\mathrm{compact} }$

where $x^{\mathrm{compact} }$ denotes the compact closure of $x$.

Proof
Let $z \in x^{\mathrm{compact} }$

By definition of compact closure:
 * $z$ is a compact element and $z \preceq x$

By definition of transitivity:
 * $z \preceq y$

Thus by definition of compact closure:
 * $z \in y^{\mathrm{compact} }$