Mapping Preserves Non-Empty Infima implies Mapping is Continuous 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
 * for all non-empty subsets $Y$ of $S$: $f$ preserves the infimum of $Y$.

Then $f$ is continuous mapping.

Proof
Define $B = \left\{ {\complement_X\left({x^{\succeq'}}\right): x \in X}\right\}$

We will prove that
 * $\forall A \in B: f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is closed.

Let $A \in B$.

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

By Relative Complement of Relative Complement:
 * $\complement_X\left({A}\right) = x^{\succeq'}$

By Infimum of Upper Closure of Element:
 * $\inf \left({\complement_X\left({A}\right)}\right) = x$

Suppose that the case: $f^{-1}\left[{\complement_X\left({A}\right)}\right] = \varnothing$ holds.

Thus by Empty Set is Closed in Topological Space:
 * $f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is closed.

Suppose that the case: $f^{-1}\left[{\complement_X\left({A}\right)}\right] \ne \varnothing$ holds.

By assumption:
 * $f$ preserves the infimum of $f^{-1}\left[{\complement_X\left({A}\right)}\right]$

By definitions of mapping preserves the infimum and complete lattice:
 * $f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) = \inf\left({f\left[{f^{-1}\left[{\complement_X\left({A}\right)}\right]}\right]}\right)$

By Image of Preimage under Mapping:
 * $f\left[{f^{-1}\left[{\complement_X\left({A}\right)}\right]}\right] \subseteq x^{\succeq'}$

By Infimum of Subset and definition of complete lattice:
 * $x \preceq' f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right)$

We will prove that
 * $f^{-1}\left[{\complement_X\left({A}\right)}\right] = {\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right)}^\succeq$

Let $a \in f^{-1}\left[{\complement_X\left({A}\right)}\right]$

By definitions of infimum and lower bound:
 * $\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right) \preceq a$

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

Let $a \in {\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right)}^\succeq$

By assumption:
 * $f$ preserves the infimum of $\left\{ {\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right), a}\right\}$

By definitions of mapping preserves the infimum and complete lattice:
 * $f\left({\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \wedge a}\right) = f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \wedge f\left({a}\right)$

By definition of upper closure of element:
 * $\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right) \preceq a$

By Meet Precedes Operands:
 * $f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) = f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \wedge f\left({a}\right)$

By Preceding iff Meet equals Less Operand:
 * $f\left({\inf \left({f^{-1}\left[{x^{\succeq'} }\right]}\right)}\right) \preceq' f\left({a}\right)$

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

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

Thus by definition of preimage of set:
 * $a \in f^{-1}\left[{\complement_X\left({A}\right)}\right]$

Thus by Complement of Upper Closure of Element is Open in Lower Topology:
 * $f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is closed.

We will prove that
 * $\forall A \in B: f^{-1}\left[{A}\right] \in \tau$

Let $A \in B$.

Then by previous:
 * $f^{-1}\left[{\complement_X\left({A}\right)}\right]$ is closed.

By Complement of Preimage equals Preimage of Complement
 * $f^{-1}\left[{\complement_X\left({A}\right)}\right] = \complement_S\left({f^{-1}\left[{A}\right]}\right)$

Thus by definition of closed set
 * $f^{-1}\left[{A}\right] \in \tau$

Thus Continuity Test using Sub-Basis:
 * $f$ is continuous mapping.