Upper Adjoint of Galois Connection is Surjection implies Lower Adjoint at Element is Minimum of Preimage of Singleton of Element

Theorem
Let $L = \left({S, \preceq}\right), R = \left({T, \precsim}\right)$ be ordered sets.

Ley $g:S \to T, d:T \to S$ be mappings such that
 * $\left({g, d}\right)$ is Galois connection

and
 * $g$ is a surjection/

Then
 * $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{\left\{ {t}\right\} }\right]}\right)$

Proof
By definition of Galois connection:
 * $g$ is an increasing mapping.

Let $t \in T$.

By definition of surjection:
 * $\operatorname{Im}\left({g}\right) = T$

By Image of Preimage under Mapping/Corollary:
 * $g\left[{g^{-1}\left[{t^\succeq}\right]}\right] - t^\succeq$

By Galois Connection is Expressed by Minimum:
 * $d\left({t}\right) = \min\left({g^{-1}\left[{t^\succeq}\right]}\right)$

By definition of minimum:
 * $d\left({t}\right) = \inf\left({g^{-1}\left[{t^\succeq}\right]}\right)$ and $d\left({t}\right) \in g^{-1}\left[{t^\succeq}\right]$

By definition of image of set:
 * $g\left({d\left({t}\right)}\right) \in g\left[{g^{-1}\left[{t^\succeq}\right]}\right]$

By definition of upper closure of element:
 * $t \precsim g\left({d\left({t}\right)}\right)$

By definition of minimum:
 * $g^{-1}\left[{t^\succeq}\right]$ admits an infimum.

By definition of infimum:
 * $d\left({t}\right)$ is lower bound for $g^{-1}\left[{t^\succeq}\right]$

By definition of surjection:
 * $\exists s \in S: t = g\left({s}\right)$

By definition of singleton:
 * $t \in \left\{ {t}\right\}$

By Set is Subset of Upper Closure
 * $\left\{ {t}\right\} \subseteq \left\{ {t}\right\}^\succeq$

By Upper Closure of Singleton:
 * $\left\{ {t}\right\}^\succeq = t^\succeq$

By definition of image of set:
 * $s \in g^{-1}\left[{t^\succeq}\right]$

By definition of lower bound:
 * $d\left({t}\right) \preceq s$

By definition of increasing mapping:
 * $g\left({d\left({t}\right)}\right) \precsim t$

By definition of antisymmetry:
 * $g\left({d\left({t}\right)}\right) = t$

By definition of preimage of set:
 * $d\left({t}\right) \in g^{-1}\left[{\left\{ {t}\right\} }\right]$

By Image of Subset under Relation is Subset of Image/Corollary 3:
 * $g^{-1} \left[{\left\{ {t}\right\} }\right] \subseteq g^{-1} \left[{t^\succeq}\right]$

We will prove that
 * $d \left({t}\right)$ is an infimum of $g^{-1} \left[{\left\{ {t}\right\} }\right]$

Thus by Lower Bound is Lower Bound for Subset:
 * $d \left({t}\right)$ is lower bound for $g^{-1} \left[{\left\{ {t}\right\} }\right]$

Thus by definition:
 * $\forall s \in S: s$ is lower bound for $g^{-1} \left[{\left\{ {t}\right\} }\right] \implies s \preceq d \left({t}\right)$

Thus by definition of minimum:
 * $d \left({t}\right) = \min \left({g^{-1} \left[{\left\{ {t}\right\} }\right]}\right)$