In reading the appendix of Lectures on Chevalley Groups by Steinberg, I'm having trouble understanding the uniqueness aspect of Proposition 9 (in both parts).
Here is the setup. Let $V$ be an inner product space over $\mathbb{R}$, and let $\Sigma$ be a root system in $V$. We fix a subset $V^+ \subset V$ which is the "positive part," i.e. $V^+$ is closed under addition and multiplication by positive scalars, and satisfies "trichotomy," which means $\forall v,w \in V^+$, exactly one of $v<w, w< v, w=v$ is true.
A positive system $P$ is a subset of $\Sigma \cap V^+$. A simple system $\Pi$ is a linearly independent subset of $\Sigma$, such that every element of $\Sigma$ is a linear combination of elements of $\Pi$ where all nonzero coefficients have the same sign.
Proposition 9 (a) Each simple system is contained in a unique positive system. (b) Each positive system contains a unique simple system.
For (a), Steinberg just says that the uniqueness is clear, but I don't see it. In part (b), he writes, "From the definition of a simple system any simple system contained in $P$ consists of those elements of $P$ which are not positive combinations of others, hence is uniquely determined by $P$." I do not understand how this is immediate from the definition.