\begin{definition}[Definition:Infimum of Mapping]
Let $S$ be a set.
Let $\struct {T, \preceq}$ be an ordered set.
Let $f: S \to T$ be a mapping from $S$ to $T$.
Let $f \sqbrk S$, the image of $f$, admit an infimum.
Then the '''infimum''' of $f$ (on $S$) is defined by:
:$\ds \inf_{x \mathop \in S} \map f x = \inf f \sqbrk S$
\end{definition}