Way Above Closures Form Basis

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

Then $\left\{ {x^\gg: x \in S}\right\}$ is basis of $L$.

Proof
Define $B = \left\{ {x^\gg: x \in S}\right\}$

Thus by Way Above Closure is Open:
 * $B \subseteq \tau$

We will prove that
 * for all $x \in S$: there exists local basis $Q$ of $x$: $Q \subseteq B$

Let $x \in S$.

By Way Above Closures that Way Below Form Local Basis:
 * $Q := \left\{ {g^\gg: g \in S \land g \ll x}\right\}$ is local basis of $x$.

Thus by definition of subset:
 * $Q \subseteq B$

Thus by Characterization of Analytic Basis by Local Bases:
 * $B$ is basis of $L$.