Corollary 3

a) Let $\alpha \in \mathbf{R}$. Then $Y(\alpha)$ is a connected subset of $B(C)$ (Chap. IV, Appendix).

b) Let $Y$ be a non-empty connected subset of $B(C)$. Then $\sum_{\beta \in Y} \beta$ belongs to $\mathbf{R}$.

To prove a), we can assume that $\alpha$ is positive. We argue by induction on $\mathrm{Card}(Y(\alpha))$, the assertion being trivial if $\mathrm{Card}(Y(\alpha)) = 1$. By Prop. 19, there exists $\beta \in B(C)$ such that $\alpha - \beta \in \mathbf{R}$. Let $p$ be the largest integer $\geqslant 0$ such that $\gamma = \alpha - p\beta \in \mathbf{R}$. Since $\gamma - \beta \notin \mathbf{R}$ and $\gamma + p\beta \in \mathbf{R}$, $(\gamma|\beta) \neq 0$ (Prop. 9); thus $\beta$ is linked to at least one element of $Y(\gamma)$. But $Y(\alpha) = Y(\gamma) \cup \{\beta\}$, and $Y(\gamma)$ is connected by the induction hypothesis. Thus $Y(\alpha)$ is connected, which proves a).

Now let $Y$ be a non-empty connected subset of $B(C)$; we show by induction on $\mathrm{Card}(Y)$ that $\sum_{\beta \in Y} \beta$ is a root. The case in which $\mathrm{Card}(Y) \leqslant 1$ is trivial. Assume that $\mathrm{Card}(Y) \geqslant 2$. Since $X$ is a forest (Chap. V, § 4, no. 8, Prop. 8), $Y$ is a tree and has a terminal vertex $\beta$ (Chap. IV, Appendix). The set $Y - \{\beta\}$ is connected, and one of its element is linked to $\beta$. By the induction hypothesis, $\alpha = \sum_{\gamma \in Y - \{\beta\}} \gamma \in \mathbf{R}$, and since $(\alpha|\beta) < 0$, it follows that $\alpha + \beta \in \mathbf{R}$ (Th. 1). Q.E.D.