Exercise 12, § 2

Let I be a totally ordered set and let $(\mathbf{E}_\iota)_{\iota \in \mathbf{I}}$ be a family of ordered sets indexed by I. Let $\mathbf{R}\{x, y\}$ denote the following relation on $\mathbf{E} = \prod_{\iota \in \mathbf{I}} \mathbf{E}_\iota$ : "the set of indices $\iota \in \mathbf{I}$ such that $\mathrm{pr}_\iota x \neq \mathrm{pr}_\iota y$ is well-ordered, and if $\varkappa$ is the least element of this subset of I, we have $\mathrm{pr}_\varkappa x < \mathrm{pr}_\varkappa y$". Show that $\mathbf{R}\{x, y\}$ is an order relation between $x$ and $y$ on E. If the $\mathbf{E}_\iota$ are totally ordered, show that the connected components of E with respect to the relation "$x$ and $y$ are comparable" (Chapter II, § 6, Exercise 10) are totally ordered sets. Suppose that each $\mathbf{E}_\iota$ has at least two elements. Then E is totally ordered if and only if I is well-ordered and each $\mathbf{E}_\iota$ is totally ordered (use Exercise 3); and E is then the lexicographic product of the $\mathbf{E}_\iota$.