Infimum of Open Set is Way Below Element in Complete Scott Topological Lattice

Theorem
Let $T = \left({S, \preceq, \tau}\right)$ be a complete topological lattice with Scott topology.

Let $X$ be an open subset of $S$,

Let $x \in X$.

Then $\inf X \ll x$

where $\ll$ denotes the way below relation.

Proof
By Open iff Upper and with Property (S) in Scott Topological Lattice:
 * $X$ is uooer and has property (S).

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

By definition of upper set:
 * $\sup D \in X$

By definition of property (S):
 * $\exists y \in D: \forall d \in D: y \preceq d \implies d \in X$

By definitions of infimum and complete lattice:
 * $\inf X$ is a lower bound for $X$.

By definition pf reflexivity:
 * $y \in X$

Thus by definition of lower bound:
 * $\exists d \in D: \inf X \preceq d$

Thus by definition pf way below relation:
 * $\inf X \ll x$