Proposition 10

Let $\alpha, \beta$ be two non-proportional roots such that $\beta + \alpha$ is a root. Let $p, q$ be the integers in Prop. 9. Then

Let S be the $\alpha$-chain defined by $\beta, \gamma$ its origin; its length $l$ is $\geq 1$ since $\beta + \alpha$ is a root. The following cases are possible:

1) $l = 1$; then $\beta = \gamma, q = 0, p = 1, (\beta + \alpha|\beta + \alpha) = (\beta|\beta)$. 2) $l = 2, \beta = \gamma$; then $q = 0, p = 2, (\beta + \alpha|\beta + \alpha) = \frac{1}{2}(\beta|\beta)$. 3) $l = 2, \beta = \gamma + \alpha$; then $q = 1, p = 1, (\beta + \alpha|\beta + \alpha) = 2(\beta|\beta)$.

4) $l = 3, \beta = \gamma$; then $q = 0, p = 3, (\beta + \alpha|\beta + \alpha) = \frac{1}{3}(\beta|\beta)$. 5) $l = 3, \beta = \gamma + \alpha$; then $q = 1, p = 2, (\beta + \alpha|\beta + \alpha) = (\beta|\beta)$. 6) $l = 3, \beta = \gamma + 2\alpha$; then $q = 2, p = 1, (\beta + \alpha|\beta + \alpha) = 3(\beta|\beta)$.

In each case, the formula to be proved is satisfied.