Relative Pseudocomplement and Shift Mapping form Galois Connection in Brouwerian Lattice

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

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

Let $g, d: S \to S$ be mappings such that
 * $\forall s \in S: g\left({s}\right) = a \to s$

and
 * $\forall s \in S: d\left({s}\right) = a \wedge s$

Then $\left({g, d}\right)$ is Galois connection.

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

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

By Galois Connection is Expressed by Maximum:
 * $\forall s \in S: g'\left({s}\right) = \max\left({d^{-1}\left[{s^\preceq}\right]}\right)$

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

By definition of lower closure of element:
 * $\forall s \in S: g'\left({s}\right) = \max\left({\left\{ {x \in S: d\left({x}\right) \preceq s}\right\} }\right)$

By definition of $d$:
 * $\forall s \in S: g'\left({s}\right) = \max\left({\left\{ {x \in S: a \wedge x \preceq s}\right\} }\right)$

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

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

Hence
 * $\left({g, d}\right)$ is Galois connection