Way Below has Strong Interpolation Property

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

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

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

Proof
By Way Below is Approximating Relation and Way Below Relation is Auxiliary Relation:
 * $\ll$ is an auxiliary approximating relation on $S$.

Thus by Auxiliary Approximating Relation has Interpolation Property:
 * $\exists y \in S: x \ll y \land y \ll z \land x \ne y$