Exercise 10, § 2

Let I be an ordered set and let $(\mathbf{E}_\iota)_{\iota \in \mathbf{I}}$ be a family of ordered sets, all equal to the same ordered set E. Show that the ordinal sum $\sum_{\iota \in \mathbf{I}} \mathbf{E}_\iota$ (§ 1, Exercise 3) is isomorphic to the lexicographical product of the sequence $(\mathbf{F}_\lambda)_{\lambda \in \{\alpha, \beta\}}$, where the set $\{\alpha, \beta\}$ of two distinct elements is well-ordered by the relation whose graph is $\{(\alpha, \alpha), (\alpha, \beta), (\beta, \beta)\}$, and where $\mathbf{F}_\alpha = \mathbf{I}$ and $\mathbf{F}_\beta = \mathbf{E}$. This product is called the lexicographic product of E by I and is written E.I.