Proposition: Finite injective evaluation exists

If $U\subseteq {\mathbb{R}}^X$ has dimension $d<\infty$, some family of at most $d$ labels has injective evaluation on $U$.

Proof. If the common kernel of the labels chosen so far is nonzero, choose a nonzero function in it and a label where it does not vanish. That evaluation decreases the kernel dimension. Repeat at most $d$ times.