Positive bases with respect to another fixed base

Question

Let $\Delta$ be a base of a root system, i.e. a subset of a root system $ E =span \phi$ with $E$ Euclidean space, such that $\Delta$ is a basis of $E$ and every root can be written as a linear combination of elements of $\Delta$ with coefficients either all positive or negative. Suppose $\Delta_1$ and $\Delta_2$ are other two bases, positive with respect to $\Delta$. Is is true that $\Delta_1 = \Delta_2$? If so, why?