Brouwerian Lattice is Distributive

Theorem
Let $\struct {S, \preceq}$ be a Brouwerian lattice.

Then $\struct {S, \preceq}$ is a distributive lattice

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

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

Define a mapping $f: S \to S$:
 * $\forall s \in S: \map f s = x \wedge s$

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 $\set {y, z}$

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

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

Thus

Thus by definition:
 * $\struct {S, \preceq}$ is a distributive lattice