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:
 * $t \precsim g\left({d\left({t}\right)}\right)$