Finiteness of basic solutions

Let $\iota$ be a finite type and $C : \iota \to$ linear constraints on $\mathbb{R}^n$. The theorem asserts the conjunction of two finiteness claims: (a) the set of all $x \in \mathbb{R}^n$ that are basic solutions of $C$ (every "$=$"-tagged constraint active at $x$, and some $n$-element index set $s$ with all constraints active at $x$ and linearly independent coefficient vectors $(a_i)_{i\in s}$) is a finite set; and (b) the set of all basic feasible solutions (basic solutions that moreover satisfy every constraint) is a finite set. No nonemptiness or feasibility hypotheses are made; when $n = 0$ the ambient space $\mathbb{R}^0$ is itself a single point, and when $\#\iota < n$ both sets are empty — in all such cases the claims hold trivially.