Filtered iff Directed in Dual Ordered Set

Theorem
Let $\struct {S, \preceq_1}$ be an ordered set.

Let $\struct {S, \preceq_2}$ be a dual ordered set of $\struct {S, \preceq_1}$

Let $X \subseteq S$.

Then:
 * $X$ is filtered in $\struct {S, \preceq_1}$


 * $X$ is directed in $\struct {S, \preceq_2}$
 * $X$ is directed in $\struct {S, \preceq_2}$

Proof
By Dual of Dual Ordering:
 * $\struct {S, \preceq_1}$ is the dual of $\struct {S, \preceq_2}$.

Thus by Directed iff Filtered in Dual Ordered Set:
 * $X$ is filtered in $\struct {S, \preceq_1}$


 * $X$ is directed in $\struct {S, \preceq_2}$.
 * $X$ is directed in $\struct {S, \preceq_2}$.