Brouwerian Lattice is Distributive

Theorem
Let $\left({S, \preceq}\right)$ be a Brouwerian lattice.

Then $\left({S, \preceq}\right)$ is a distributive lattice

Proof
Let $x, y, z \in S$.

By Brouwerian Lattice iff Shift Mapping is Lower Adjoint:
 * $\forall x \in S, f: S \to S: \left({\forall s \in S: f \left({s}\right) = x \wedge s}\right) \implies f$ is a lower adjoint

Define a mapping $f: S \to S$:
 * $\forall s \in S: f \left({s}\right) = x \wedge s$

Then:
 * $f$ is a lower adjoint

By Lower Adjoint Preserves All Suprema:
 * $f$ preserves all suprema

By definition of preserves all suprema:
 * $f$ preserves the supremum of $\left\{ {y, z}\right\}$

By definition of lattice:
 * $\left\{ {y, z}\right\}$ admits a supremum

By preserves the supremum of set:
 * $\sup \left({f^\to\left({\left\{ {y, z}\right\} }\right)}\right) = f \left({\sup \left\{ {y, z}\right\} }\right)$

Thus

Thus by definition:
 * $\left({S, \preceq}\right)$ is a distributive lattice