If Compact Between then Way Below

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

Let $x, k, y \in S$ such that
 * $x \preceq k$ and $k \preceq y$ and $k \in K\left({L}\right)$

where $K\left({L}\right)$ denotes the compact subset of $L$.

Then $x \ll y$

where $\ll$ denotes the way below relation.

Proof
By definition of compact subset:
 * $k$ is compact.

By definition of compact:
 * $k \ll k$

Thus by Preceding and Way Below implies Way Below:
 * $x \ll y$