Galois Connection with Upper Adjoint Surjective implies Scond Ordered Set and Image of Lower Adjoint are Isomorphic

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/

Let $N = \left({d\left[{T}\right], \preceq'}\right)$ be an ordered subset of $L$.

Then $R$ and $N$ are order isomorphic.

Proof
By Galois Connection implies Upper Adjoint is Surjection iff Lower Adjoint is Injection:
 * $d$ is an injection.

Define $d' = d:T \to g\left[{T}\right]$

By definition:
 * $d'$ is an injection.

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

We will prove that
 * $d$ is order embedding.

Let $x, y \in T$.

By definition of increasing mapping:
 * $x \precsim y \implies d\left({x}\right) \preceq d\left({y}\right)$

Thus by definition of $d'$ and ordered subset:
 * $x \precsim y \implies d'\left({x}\right) \preceq' d'\left({y}\right)$

Assume that
 * $d'\left({x}\right) \preceq' d'\left({y}\right)$