Proper Subtower is Initial Segment

Theorem
Let $\left({T_1,\preccurlyeq}\right)$ be a proper subtower of $\left({T_2,\preccurlyeq}\right)$.

Then $\left({T_1,\preccurlyeq}\right)$ is an initial segment of $\left({T_2,\preccurlyeq}\right)$.

Proof
Define the set:


 * $Y = \left\{ { y \in T_1: S_y \text{ is an initial segment of } \left({T_2,\preccurlyeq}\right) } \right\}$.

Then:

By induction on well-ordered sets, $Y = T_1$.

That is, $\left({T_1,\preccurlyeq}\right)$ is an initial segment in $\left({T_2,\preccurlyeq}\right)$.