All Infima Preserving Mapping is Upper Adjoint of Galois Connection/Lemma 1

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:
 * $g$ is an increasing mapping.

Proof
Define a mapping $d: T \to S$:
 * $\forall t \in T: \map d t := \map \inf {g^{-1} \sqbrk {t^\succsim} }$

Let $x, y \in S$ such that
 * $x \preceq y$

By Upper Closure is Decreasing:
 * $y^\succeq \subseteq x^\succeq$

By Infimum of Upper Closure of Element:
 * $\map \inf {x^\succeq} = x$ and $\map \inf {y^\succeq} = y$

By definition of all infima preserving mapping:
 * $g$ preserves the infimum on $x^\succeq$

and:
 * $g$ preserves the infimum on $y^\succeq$

By definition of infimum on subset preserving mapping:
 * $\map \inf {\map {g^\to} {x^\succeq} } = \map g x$ and $\map \inf {\map {g^\to} {y^\succeq} } = \map g y$

By Image of Subset under Mapping is Subset of Image:
 * $\map {g^\to} {y^\succeq} \subseteq \map {g^\to} {x^\succeq}$

Thus by Infimum of Subset:
 * $\map g x \precsim \map g y$