Definition:Proper Subtower in Set

Definition
Let $X$ be a set.

Let $\struct {T, \preccurlyeq}$ be a tower in $X$.

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

and:


 * $\struct {T', \preccurlyeq}$ is a tower in $X$.