Open iff Upper and with Property (S) in Scott Topological Lattice

Theorem
Let $T = \struct {S, \preceq, \tau}$ be an up-complete topological lattice.

Let $A$ be a subset of $S$.

Then $A$ is open $A$ is upper and with property (S).

Sufficient Condition
Let $A$ be open.

Thus by definition of Scott topology:
 * $A$ is an upper set.

Let $D$ be a directed subset of $S$ such that
 * $\sup D \in A$

By definition of Scott topology:
 * $A$ is inaccessible by directed suprema.

By definition of inaccessible by directed suprema:
 * $A \cap D \ne \O$

By definition of non-empty:
 * $\exists y: y \in A \cap D$

By definition of intersection:
 * $y \in A$ and $y \in D$

Thus $y \in D$.

Thus by definition of upper set:
 * $\forall x \in D: y \preceq x \implies x \in A$

Necessary Condition
Assume that
 * $A$ is upper and with property (S).

According to the definition of Scott topology it should be proved that
 * $A$ is upper and inaccessible by directed suprema.

Thus by assumption:
 * $A$ is upper.

Let $D$ be a directed subset of $S$ such that
 * $\sup D \in A$

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

By definition of reflexivity:
 * $y \preceq y$

Then
 * $y \in A$

By definition of intersection:
 * $y \in A \cap D$

Thus by definition of non-empty:
 * $A \cap D \ne \O$