User:IssaRice/Linear algebra/Rank of polynomial matrix is constant everywhere except possibly at finitely many points

This is Corollary 6.2 in Linear Algebra Done Wrong.

I find the proof in the book pretty unclear, so I want to write up a clearer proof.

Corollary statement: Let $A(x)$ be an $m\times n$ polynomial matrix (i.e. a matrix whose entries are polynomials of $x$). Then $x\mapsto \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}A(x)$ is constant everywhere, except possibly at finitely many points, where the rank is smaller.

Proof: