Way Below has Interpolation Property

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

Let $x, z \in S$ such that
 * $x \ll z$

Then
 * $\exists y \in S: x \ll y \land y \ll z$

Proof
Case $x \ne z$:

By Way Below has Strong Interpolation Property:
 * $\exists y \in S: x \ll y \land y \ll z \land x \ne y$

Thus
 * $\exists y \in S: x \ll y \land y \ll z$

Case $x = z$:

Define $y = x$

Thus
 * $x \ll y \land y \ll z$