Reading a miniature 16 in the book "Thirty-three miniatures" by Jirí Matoušek I can understand the proof only by writing $x_\varnothing = 1$.
In that proof you observe a linear combination $\sum_{I \subseteq \{1,2,...,d\}}\alpha_Ix_I $ of multilinear monomials of the form $x_I = \prod_{i \in I}x_i$, where $I \subseteq \{1,2,...,d\}$, and $d$ is the dimension of a field over $\mathbb{R}$.
Have you ever seen the notation $x_\varnothing = 1$, and if yes - where does it come from?
If $x_I = \prod_{i\in I} x_i$ is a definition, then for $I=\varnothing$ you get the empty product which is defined to be $1$, since $1$ is the unit with respect to multiplication. (Just like the empty sum is $0$)