All Infima Preserving Mapping is Upper Adjoint of Galois Connection

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

Let $\left({T, \precsim}\right)$ be an ordered set.

Let $g: S \to T$ be all infima preserving mapping.

Then there exists a mapping $d: T \to S$ such that $\left({g, d}\right)$ is Galois connection and
 * $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$

where
 * $\min$ denotes the minimum
 * $g^{-1}\left[{t^\succsim}\right]$ denotes the image of $t^\succsim$ under relation $g^{-1}$
 * $t^\succsim$ denotes the upper closure of $t$

Proof
Define a mapping $d: T \to S$:
 * $\forall t \in T:d\left({t}\right) := \inf\left({g^{-1}\left[{t^\succsim}\right]}\right)$

We will prove as lemma 1 that
 * $g$ is an increasing mapping.

We will prove as lemma 2 that
 * $d$ is an increasing mapping.

We will prove as lemma 3 that
 * $\forall s \in S, t \in T: t \precsim g\left({s}\right) \iff d\left({t}\right) \preceq s$