Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice

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

Let $a$ be an element of $S$.

Let $g, d: S \to S$ be mappings such that:
 * $\forall s \in S: \map g s = a \to s$

and
 * $\forall s \in S: \map d s = a \wedge s$

Then $\struct {g, d}$ is a Galois connection.

Proof
By Brouwerian Lattice iff Shift Mapping is Lower Adjoint:
 * $d$ is lower adjoint

By definition of lower adjoint:
 * $\exists g': S \to S: \struct {g', d}$ is Galois connection

By Galois Connection is Expressed by Maximum:
 * $\forall s \in S: \map {g'} s = \map \max {d^{-1} \sqbrk {s^\preceq} }$

By definition of image of set:
 * $\forall s \in S: \map {g'} s = \map \max {\set {x \in S: \map d x \in s^\preceq} }$

By definition of lower closure of element:
 * $\forall s \in S: \map {g'} s = \map \max {\set {x \in S: \map d x \preceq s} }$

By definition of $d$:
 * $\forall s \in S: \map {g'} s = \map \max {\set {x \in S: a \wedge x \preceq s} }$

By definition of relative pseudocomplement:
 * $\forall s \in S: \map {g'} s = a \to s = \map g s$

By Equality of Mappings:
 * $g = g'$

Hence
 * $\struct {g, d}$ is a Galois connection.