[proofplan] The associated base $\Delta$ is chosen so that every positive root is expressible as a nonnegative integral linear combination of elements of $\Delta$, which gives existence. To prove uniqueness, compare two such expansions of the same positive root and subtract them. The difference is a real linear relation among the simple roots, and the defining [linear independence](/page/Linear%20Independence) of a base forces every coefficient in that relation to vanish. [/proofplan]

[step:Use the defining spanning property of the associated base to obtain an expansion] Let $\gamma \in \Phi^+$ be fixed. Since $\Delta$ is the base associated to the positive system $\Phi^+$, every positive root is a nonnegative integral linear combination of elements of $\Delta$. Therefore there exists a family $(n_\alpha)_{\alpha \in \Delta}$ with $n_\alpha \in \mathbb{Z}_{\ge 0}$ for each $\alpha \in \Delta$ such that \begin{align*} \gamma = \sum_{\alpha \in \Delta} n_\alpha \alpha. \end{align*} This proves existence of at least one expansion. [/step]

[step:Subtract two expansions to obtain a linear relation among simple roots] Suppose that $(n_\alpha)_{\alpha \in \Delta}$ and $(m_\alpha)_{\alpha \in \Delta}$ are two families of nonnegative integers satisfying \begin{align*} \gamma = \sum_{\alpha \in \Delta} n_\alpha \alpha \qquad \text{and} \qquad \gamma = \sum_{\alpha \in \Delta} m_\alpha \alpha. \end{align*} Subtracting the second equality from the first equality in the real [vector space](/page/Vector%20Space) $V$ gives \begin{align*} 0 = \sum_{\alpha \in \Delta} n_\alpha \alpha - \sum_{\alpha \in \Delta} m_\alpha \alpha = \sum_{\alpha \in \Delta} (n_\alpha - m_\alpha)\alpha. \end{align*} For each $\alpha \in \Delta$, the coefficient $n_\alpha - m_\alpha$ is an integer, hence also a real number. [/step]

[step:Apply linear independence of the base to force equality of coefficients] By definition of a base of a root system, the set $\Delta$ is linearly independent over $\mathbb{R}$. Applying this linear independence to the relation \begin{align*} \sum_{\alpha \in \Delta} (n_\alpha - m_\alpha)\alpha = 0 \end{align*} shows that \begin{align*} n_\alpha - m_\alpha = 0 \end{align*} for every $\alpha \in \Delta$. Hence $n_\alpha = m_\alpha$ for every $\alpha \in \Delta$. Thus any two nonnegative integral expansions of $\gamma$ in the simple roots agree coefficient by coefficient. Since $\gamma \in \Phi^+$ was arbitrary, every positive root has a unique expression as a nonnegative integral linear combination of the elements of $\Delta$. [/step]