\begin{definition}[Definition:Proper Subtower in Set]
Let $X$ be a set.
Let $\struct {T, \preccurlyeq}$ be a tower in $X$.
Then $\struct {T, \preccurlyeq}$ is a '''proper subtower in $X$''' {{iff}}:
:$T$ is a proper subset of some set $T' \subseteq X$
and:
:$\struct {T', \preccurlyeq}$ is a tower in $X$.
\end{definition}