Intersection of Relation Segments of Approximating Relations equals Way Below Closure

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a bounded below meet-continuous lattice.

Let $\mathit{App}\left({L}\right)$ be the set of all auxiliary approximating relations on $S$.

Then
 * $\displaystyle \bigcap \left\{ {x^{\mathcal R}: \mathcal R \in \mathit{App}\left({L}\right)}\right\} = x^\ll$