Proper Subtower is Initial Segment

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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:

\(\displaystyle S_x \left({T_1}\right)\) \(=\) \(\displaystyle \left\{ {b \in T_1, x \in T_1: b \prec x}\right\}\) Definition of Initial Segment
\(\displaystyle \) \(=\) \(\displaystyle \left\{ {b \in T_2, x \in T_2: b \prec x}\right\}\) Definition of Proper Subtower in Set, as $T_1 \subseteq T_2$
\(\displaystyle \) \(=\) \(\displaystyle S_x \left({T_2}\right)\) Definition of Initial Segment

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

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

$\blacksquare$


Sources