Exercise 21, § 1

Let E be a non-empty totally ordered set, and let $S\{x, y\}$ be the relation "the closed interval with endpoints $x, y$ is scattered" (Exercise 20). Show that S is an equivalence relation which is weakly compatible (Exercise 2) in $x$ and $y$ with the order relation on E, that the equivalence classes with respect to S are scattered sets, and that the quotient ordered set E/S (Exercise 2) is either without gaps or else consists of a single element. Deduce that E is isomorphic to an ordinal sum of scattered sets whose index set is either without gaps or else consists of a single element.