Lemma 2

Let H be an isolated subgroup of a totally ordered commutative group G. If H is a direct factor of G, the ordered group G is isomorphic to the group $(G/H) \times H$ ordered lexicographically.

Let $j$ be an isomorphism of $(G/H) \times H$ onto G such that $j(0, x) = x$ for all $x \in H$ and $j(y, x)$ belongs to the coset of H. As $(G/H) \times H$ is totally ordered, it amounts to showing that $j$ is increasing (Set Theory, Chapter III, § 1, no. 12, Proposition 11). Let $(y, x)$ be an element $\geq 0$ of $(G/H) \times H$ ordered lexicographically. If $y > 0$, the coset of H containing $j(y, x)$ is an element $> 0$, whence $j(y, x) > 0$, for, otherwise, $y \leq 0$ (§ 4, no. 2, Proposition 3). If $y = 0$ and $x \geq 0$, then $j(y, x) = x \geq 0$. Hence $j$ is certainly increasing.