Upper Way Below Open Subset Complement is Non Empty implies There Exists Maximal Element of Complement

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

Let $X$ be upper way below open subset of $S$.

Let $x \in S$ such that
 * $x \in \complement_S\left({X}\right)$

Then
 * $\exists m \in S: x \preceq m \land m = \max \complement_S\left({X}\right)$