Mapping is Continuous implies Mapping Preserves Filtered Infima in Lower Topological Lattice

Theorem
Let $T = \left({S, \preceq, \tau}\right)$ and $Q = \left({X, \preceq', \tau'}\right)$ be complete topological lattices with lower topologies.

Let $f: S \to X$ be a mapping such that
 * $f$ is a continuous mapping.

Then $f$ preserves filtered infima.

Proof
Define $B := \left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$

By definition of lower topology:
 * $B$ is an analytic sub-basis.

Let $F$ be a filtered subset of $S$ such that
 * $F$ admits an infimum in $T$.

Thus by definition of complete lattice:
 * $f\left[{F}\right]$ admits an infimum in $Q$.

We will prove that
 * $\forall A \in B: \inf F \in A \implies F \cap A \ne \varnothing$

Let $A \in B$ be such that
 * $\inf F \in A$

By definition of $B$:
 * $\exists x \in S: A = \complement_S\left({x^\succeq}\right)$

By definition of relative complement:
 * $\inf F \notin x^\succeq$

By definition of upper closure of element:
 * $x \npreceq \inf F$

By definition of infimum:
 * $x$ is not lower bound for $F$.

By definition of lower bound:
 * $\exists y \in F: x \npreceq y$

By definition of upper closure of element:
 * $y \notin x^\succeq$

By definition of relative complement:
 * $y \in A$

By definitions of intersection and non-empty set:
 * $F \cap A \ne \varnothing$

Then by If Infimum of Filtered Subset belongs to Element of Sub-Basis then Subset and Element Intersect implies Infimum of Subset belongs to Closure of Subset:
 * $\inf F \in F^-$

We will prove that
 * $f$ is an increasing mapping.

Let $x, y \in S$ be such that
 * $x \preceq y$

By definition of reflexivity:
 * $f\left({x}\right) \preceq' f\left({x}\right)$

By definition of upper closure of element:
 * $f\left({x}\right) \in \left({f\left({x}\right)}\right)^{\succeq'}$

By definition of preimage of set:
 * $x \in f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$

By Complement of Upper Closure of Element is Open in Lower Topology:
 * $\left({f\left({x}\right)}\right)^{\succeq'}$ is closed.

By Continuity by Closed Sets:
 * $f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$ is closed.

By Closed Subset is Upper in Lower Topology:
 * $f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$ is upper.

By definition of upper set:
 * $y \in f^{-1}\left[{\left({f\left({x}\right)}\right)^{\succeq'} }\right]$

By definition of preimage of set:
 * $f\left({y}\right) \in \left({f\left({x}\right)}\right)^{\succeq'}$

Thus by definition of upper closure of element:
 * $f\left({x}\right) \preceq' f\left({y}\right)$

We will prove that
 * $f\left({\inf F}\right)$ is lower bound for $f\left[{F}\right]$

By definition of infimum:
 * $f\left({\inf F}\right) \preceq' \inf\left({f\left[{F}\right]}\right)$

We will prove that
 * $F \subseteq f^{-1}\left[{\left({\inf\left({f\left[{F}\right]}\right)}\right)^{\succeq'} }\right]$