Let $\Phi$ be a finite root system in a real Euclidean space $E$ with inner product $(\cdot,\cdot)$, let $\Delta \subset \Phi$ be a simple system, and let $\Phi^+$ be the corresponding set of positive roots. If $\gamma \in \Phi^+$ and $\gamma \notin \Delta$, then there exists a simple root $\alpha \in \Delta$ such that $\gamma - \alpha \in \Phi^+$.