Brouwerian Lattice iff Meet-Continuous and Distributive

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

Then
 * $\mathscr S$ is a Brouwerian lattice


 * $\mathscr S$ is meet-continuous and distributive.

Sufficient Condition
Thus by Brouwerian Lattice is Distributive:
 * $\mathscr S$ is distributive.