Closure is Closed

Theorem
Let $(S,\preceq)$ be an ordered set.

Let $\operatorname{cl}: S \to S$ be a closure operator.

Let $x \in S$.

Then $\operatorname{cl} (x)$ is a closed element with respect to $\operatorname{cl}$.

Proof
By the definition of closure operator, $\operatorname{cl}$ is idempotent.

Therefore $\operatorname{cl} \left({\operatorname{cl} (x)}\right) = \operatorname{cl} (x)$, so $\operatorname{cl} (x)$ is a closed element.