Theorem 1

Let $\alpha, \beta$ be two roots.

(i) If $n(\alpha, \beta) > 0$, $\alpha - \beta$ is a root unless $\alpha = \beta$. (ii) If $n(\alpha, \beta) < 0$, $\alpha + \beta$ is a root unless $\alpha = -\beta$.

If $n(\alpha, \beta) > 0$, the possibilities, by the list above, are the following:

1) $n(\alpha, \beta) = 1$; then $\alpha - \beta - s_{\beta}(\alpha) \in \mathbf{R}$; 2) $n(\beta, \alpha) = 1$; then $\beta - \alpha = s_{\alpha}(\beta) \in \mathbf{R}$, so $\alpha - \beta \in \mathbf{R}$; 3) $\beta = \alpha$.

This proves (i), and (ii) follows by changing $\beta$ to $-\beta$.