Proper Subtower is Initial Segment

Theorem
Let $\struct {T_1, \preccurlyeq}$ be a proper subtower of $\struct {T_2, \preccurlyeq}$.

Then $\struct {T_1, \preccurlyeq}$ is an initial segment of $\struct {T_2, \preccurlyeq}$.

Proof
Define the set:


 * $Y = \set {y \in T_1: S_y \text { is an initial segment of } \struct {T_2, \preccurlyeq} }$.

Then:

By Induction on Well-Ordered Set, $Y = T_1$.

That is, $\struct {T_1, \preccurlyeq}$ is an initial segment in $\struct {T_2, \preccurlyeq}$.