Beukers relation basis has full rank at every complex specialization

Every finite family of complex formal power series admits a polynomial basis for all its polynomial relations whose rows remain linearly independent at every complex point. Explicitly, there exist $r$ polynomial rows $C_j$ such that each $\sum_iC_{ji}f_i=0$, every polynomial relation is a polynomial linear combination of these rows, and $(C_j(\xi))_{j=1}^r$ is linearly independent over $\mathbb C$ for every $\xi\in\mathbb C$. This is the full-rank specialization assertion of Beukers' relation-basis lemma. It follows by evaluating the polynomial left inverse constructed in the proved relation-basis theorem.