If Compact Between then Way Below

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

Let $x, k, y \in S$ such that:
 * $x \preceq k$ and $k \preceq y$ and $k \in \map K L$

where $\map K L$ 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:
 * $x \ll y$