Proposition 3

Let $(\mathrm{X}_\iota)_{\iota \in \mathbf{I}}$ be a family of well-ordered sets such that for each pair of indices $(\iota, \varkappa)$ one of the sets $\mathrm{X}_\iota$, $\mathrm{X}_\varkappa$ is a segment of the other. Then there exists a unique ordering on the set $\mathrm{E} = \bigcup_{\iota \in \mathbf{I}} \mathrm{X}_\iota$ which induces the given ordering on each of the $\mathrm{X}_\iota$. Endowed with this ordering, E is a well-ordered set. Every segment of $\mathrm{X}_\iota$ is a segment of E; for each $x \in \mathrm{X}_\iota$, the segment with endpoint $x$ in $\mathrm{X}_\iota$ is equal to the segment with endpoint $x$ in E; and each segment of E is either E itself or a segment of one of the $\mathrm{X}_\iota$.

The first assertion is a consequence of the following general lemma :