Set of Upper Closures of Compact Elements is Basis implies Complete Scott Topological Lattice is Algebraic

Theorem
Let:
 * $L = \struct {S, \preceq, \tau}$

be a complete Scott Definition:Topological Lattice.

Let:
 * $\BB = \left\{ {x^\succeq: x \in K\left({L}\right)}\right\}$

be a basis of $L$.

where:
 * $x^\succeq$ denotes the upper closure of $x$,
 * $K\left({L}\right)$ denotes the compact subset of $L$.

Then:
 * $L$ is algebraic.

Proof
Thus by Compact Closure is Directed:
 * $\forall x \in S:x^{\mathrm{compact} }$ is directed

where:
 * $x^{\mathrm{compact} }$

denotes the compact closure of $x$.

Thus by definition of complete lattice:
 * $L$ is up-complete.

Let $x \in S$.

By definition of lower closure of element:
 * $x$ is upper closure for $x^\preceq$

By definition of supremum:
 * $\sup \left({x^\preceq}\right) \preceq x$

By Compact Closure is Intersection of Lower Closure and Compact Subset:
 * $x^{\mathrm{compact} } = x^\preceq \cap K\left({L}\right)$

By Intersection is Subset:
 * $x^{\mathrm{compact} } \subseteq x^\preceq$

By Supremum of Subset:
 * $\sup \left({x^{\mathrm{compact} } }\right) \preceq \sup \left({x^\preceq}\right)$

By definition of transitivity:
 * $\sup \left({x^{\mathrm{compact} } }\right) \preceq x$


 * $x \ne \sup \left({x^{\mathrm{compact} } }\right)$

We will prove that
 * $x \notin \left({\left({x^{\mathrm{compact} } }\right) }\right)^\preceq$


 * $x \in \left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq$

By definition of lower closure of element:
 * $x \preceq \sup \left({x^{\mathrm{compact} } }\right)$

Thus by definition of antisymmetry:
 * this contrasicts $x \ne \sup \left({x^{\mathrm{compact} } }\right)$

By definition of relative complement:
 * $x \in \complement_S\left({\left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq}\right)$

By Complement of Lower Closure of Element is Open in Scott Topological Ordered Set:
 * $\complement_S\left({\left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq}\right)$ is open.

By definition of basis:
 * $\complement_S\left({\left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq}\right) = \bigcup\left\{ {G \in \BB: G \subseteq \complement_S\left({\left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq}\right)}\right\}$

By definition of union
 * $\exists X \in \left\{ {G \in \BB: G \subseteq \complement_S\left({\left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq}\right)}\right\}: x \in X$

Then:
 * $X \in \BB \land X \subseteq \complement_S\left({\left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq}\right)$

By definition of $\BB$:
 * $\exists k \in K\left({L}\right): X = k^\succeq$

By definition of compact subset:
 * $k$ is compact.

By definition of upper closure of element:
 * $k \preceq x$

By definition of compact closure:
 * $k \in x^{\mathrm{compact} }$

By definitions of supremum and upper bound:
 * $k \preceq \sup \left({x^{\mathrm{compact} } }\right)$

By definition of upper closure of element:
 * $\sup \left({x^{\mathrm{compact} } }\right) \in X$

By gefinition of subset:
 * $\sup \left({x^{\mathrm{compact} } }\right) \in \complement_S\left({\left({\sup \left({x^{\mathrm{compact} } }\right) }\right)^\preceq}\right)$

By definition of difference:
 * $\sup \left({x^{\mathrm{compact} } }\right) \notin \left({\sup \left({x^{\mathrm{compact} } }\right)}\right)^\preceq$

Thus by definition of lower closure of element:
 * this contradicts $\sup \left({x^{\mathrm{compact} } }\right) \in \left({\sup \left({x^{\mathrm{compact} } }\right)}\right)^\preceq$