Definition:Proper Subtower in Set

Definition
Let $X$ be a set.

Let $\left({T, \preccurlyeq}\right)$ be a tower in $X$.

Then $\left({T, \preccurlyeq}\right)$ is a proper subtower in $X$ :
 * $T$ is a proper subset of some set $T' \subseteq X$

and:


 * $\left({T', \preccurlyeq}\right)$ is a tower in $X$.