Proper Subtower is Initial Segment

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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:

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

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

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

$\blacksquare$


Sources