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

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

Let $B$ be an analytic sub-basis of $T$.

Let $F$ be a filtered subset of $S$ such that
 * $\forall A \in B: \inf F \in A \implies F \cap A \ne \varnothing$

Then $\inf F \in F^-$

where $F^-$ denotes the topological closure of $F$.