Continuous Lattice is Meet-Continuous

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

Then
 * $L$ is meet-continuous.

Proof
Let $x \in S$, $D$ be a directed subset of $S$ such that
 * $x \preceq \sup D$

Thus
 * $x = \sup \left\{ {x \wedge d: d \in D}\right\}$

Thus by Meet-Continuous iff if Element Precedes Supremum of Directed Subset then Element equals Supremum of Meet of Element by Directed Subset:
 * $L$ is meet-continuous.