All Infima Preserving Mapping is Upper Adjoint of Galois Connection

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

Let $\struct {T, \precsim}$ be an ordered set.

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

Then there exists a mapping $d: T \to S$ such that $\tuple {g, d}$ is Galois connection and:
 * $\forall t \in T: \map d t = \map \min {g^{-1} \sqbrk {t^\succsim} }$

where:
 * $\min$ denotes the minimum
 * $g^{-1} \sqbrk {t^\succsim}$ denotes the image of $t^\succsim$ under relation $g^{-1}$
 * $t^\succsim$ denotes the upper closure of $t$

Lemma 1
Let us define a mapping $d: T \to S$ as:
 * $\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succsim} }$

Lemma 2
Thus by Galois Connection is Expressed by Minimum:
 * $\tuple {g, d}$ is a Galois connection.

Thus by lemma $2$:
 * $\forall t \in T: \map d t = \map \min {g^{-1} \sqbrk {t^\succsim} }$