Operator Generated by Closure System is Closure Operator

Theorem
Let $L = \left({X, \vee, \wedge, \preceq}\right)$ be a complete lattice.

Let $S = \left({T, \precsim}\right)$ be a closure system of $L$.

Then $\operatorname{operator}\left({S}\right)$ is closure operator.

Proof
Define $f = \operatorname{operator}\left({S}\right)$.

Inflationary
By Intersection is Subset:
 * $x^\succeq \cap T \subseteq x^\succeq$

By Infimum of Subset:
 * $\inf x^\succeq \preceq \inf\left({x^\succeq \cap T}\right)$

By Infimum of Upper Closure of Element:
 * $x \preceq \inf\left({x^\succeq \cap T}\right)$

Thus by definition of operator generated by $S$:
 * $x \preceq f\left({x}\right)$

Increasing
Let $x, y \in X$ such that
 * $x \preceq y$

By Upper Closure is Decreasing:
 * $y^\succeq \subseteq x^\succeq$

By Set Intersection Preserves Subsets/Corollary:
 * $y^\succeq \cap T \subseteq x^\succeq \cap T$

By Infimum of Subset:
 * $\inf\left({x^\succeq \cap T}\right) \preceq \inf\left({y^\succeq \cap T}\right)$

Thus by definition of operator generated by $S$:
 * $f\left({x}\right) \preceq f\left({y}\right)$

Idempotent
Let $x \in X$.

By Intersection is Subset:
 * $x^\succeq \cap T \subseteq T$

By definition of complete lattice:
 * $x^\succeq \cap T$ admits an infimum in $L$.

By definition of infima inheriting:
 * $x^\succeq \cap T$ admits an infimum in $S$ and $\inf_S\left({x^\succeq \cap T}\right) = \inf_L \left({x^\succeq \cap T}\right)$

Then by definition of operator generated by $S$:
 * $f\left({x}\right) \in T$

and
 * $f\left({f\left({x}\right)}\right) = \inf_L \left({f\left({x}\right)^\succeq \cap T}\right)$

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

By definition of intersection:
 * $f\left({x}\right) \in f\left({x}\right)^\succeq \cap T$

By definitions of infimum and lower bound:
 * $\inf_L \left({f\left({x}\right)^\succeq \cap T}\right) \preceq f\left({x}\right)$

By Intersection is Subset:
 * $f\left({x}\right)^\succeq \cap f\left({x}\right)^\succeq$

By Infimum of Subset:
 * $\inf_L \left({f\left({x}\right)^\succeq}\right) \preceq \inf_L \left({f\left({x}\right)^\succeq \cap T}\right)$

By Infimum of Upper Closure of Element:
 * $f\left({x}\right) \preceq \inf_L \left({f\left({x}\right)^\succeq \cap T}\right)$

Thus by definition of antisymmetry:
 * $f\left({f\left({x}\right)}\right) = f\left({x}\right)$

Hence $f$ is closure operator.