No Order Isomophism Between Distinct Initial Segments of Woset
Let $E$ be a well-ordered set.
Then $S_\alpha = S_\beta$.
Aiming for a contradiction, suppose $S_\alpha \ne S_\beta$.
Then $\alpha \ne \beta$.
By the trichotomy law, $\alpha \prec \beta$ or $\beta \prec \alpha$.
Without loss of generality assume $\alpha \prec \beta$.
Then $S_\alpha \subsetneqq S_\beta$.
That is, $S_\alpha$ is an initial segment of $S_\beta$.
Thus there is an order isomorphism between $S_\beta$ and an initial segment of $S_\beta$.
This contradicts No Isomorphism from Woset to Initial Segment.
Thus $S_\alpha = S_\beta$.