=== Sufficient Condition === Let $\left({g, d}\right)$ be a Galois connection. Thus by definition of Galois connection: :$g$ is an increasing mapping Let $t \in T$. By definition of reflexivity: :$d\left({t}\right) \preceq d\left({t}\right)$ By definition of Galois connection: :$t \precsim g\left({d\left({t}\right)}\right)$ By definition of upper closure: :$g\left({d\left({t}\right)}\right) \in t^\succsim$ By definition of image of set under relation: :$d\left({t}\right) \in g^{-1}\left[{t^\succsim}\right]$ By definition of lower bound: :$\forall s \in S: s$ is lower bound for $g^{-1}\left[{t^\succsim}\right] \implies s \preceq d\left({t}\right)$ We will prove that :$d\left({t}\right)$ is lower bound for $g^{-1}\left[{t^\succsim}\right]$ Let $s \in g^{-1}\left[{t^\succsim}\right]$. By definition of image of set: :$g\left({s}\right) \in t^\succsim$ By definition of upper closure of element: :$t \precsim g\left({s}\right)$ Thus by definition of Galois connection: :$d\left({t}\right) \preceq s$ By definition of infimum: :$g^{-1}\left[{t^\succsim}\right]$ admits an infimum and :$\inf\left({g^{-1}\left[{t^\succsim}\right]}\right) = d\left({t}\right)$ Thus: :$d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$ {{qed|lemma}} === Necessary Condition === Let $g: S \to T$ be an increasing mapping. Let $\forall t \in T: d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$ Thus: :$g$ is increasing mapping. We will prove that :$d$ is increasing mapping. Let $x, y \in T$ such that :$x \precsim y$ By Upper Closure is Decreasing: :$y^\succsim \subseteq x^\succsim$ By Image of Subset under Relation is Subset of Image/Corollary 3: :$g^{-1}\left[{y^\succsim}\right] \subseteq g^{-1}\left[{x^\succsim}\right]$ By assumption :$d\left({x}\right) = \min\left({g^{-1}\left[{x^\succsim}\right]}\right) = \inf\left({g^{-1}\left[{x^\succsim}\right]}\right)$ and :$d\left({y}\right) = \min\left({g^{-1}\left[{y^\succsim}\right]}\right) = \inf\left({g^{-1}\left[{y^\succsim}\right]}\right)$ By Infimum of Subset: :$d\left({x}\right) \preceq d\left({y}\right)$ Thus by definition: :$d$ is an increasing mapping. We will prove that :$\forall s \in S, t \in T:  t \precsim g\left({s}\right) \iff d\left({t}\right) \preceq s$ Let $s \in S, t \in T$. First implication: Let $t \precsim g\left({s}\right)$. By definition of upper closure of element: :$g\left({s}\right) \in t^\succsim$ By definition of image of set: :$s \in g^{-1}\left[{t^\succsim}\right]$ By assumption :$d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right) = \inf\left({g^{-1}\left[{t^\succsim}\right]}\right)$ By definition of infimum: :$d\left({t}\right)$ is lower bound for $g^{-1}\left[{t^\succsim}\right]$ Thus by definition of lower bound: :$d\left({t}\right) \preceq s$ Second implication: Let $d\left({t}\right) \preceq s$ By assumption :$d\left({t}\right) = \min\left({g^{-1}\left[{t^\succsim}\right]}\right)$ By smallest element of set: :$d\left({t}\right) \in g^{-1}\left[{t^\succsim}\right]$ By definition of image of set: :$g\left({d\left({t}\right)}\right) \in t^\succsim$ By definition of upper closure of element: :$t \precsim g\left({d\left({t}\right)}\right)$ By definition of increasing mapping: :$g\left({d\left({t}\right)}\right) \precsim g\left({s}\right)$ Thus by definition of transitivity: :$t \precsim g\left({s}\right)$ Thus by definition: :$\left({g, d}\right)$ is Galois connection. {{qed}}