Filtered iff Directed in Dual Ordered Set

Theorem
Let $\left({S, \preceq_1}\right)$ be an ordered set.

Let $\left({S, \preceq_2}\right)$ be a dual ordered set of $\left({S, \preceq_1}\right)$

Let $X \subseteq S$.

Then $X$ is filtered in $\left({S, \preceq_1}\right)$ $X$ is directed in $\left({S, \preceq_2}\right)$

Proof
By Dual of Dual Ordering:
 * $\left({S, \preceq_1}\right)$ is dual of $\left({S, \preceq_2}\right)$

Thus by Directed iff Filtered in Dual Ordered Set:
 * $X$ is filtered in $\left({S, \preceq_1}\right)$ $X$ is directed in $\left({S, \preceq_2}\right)$