Let $X$ be a quasi-projective scheme over $\mathbb{F}_q$. If its base change to $\overline{\mathbb{F}}_q$ is an affine space $\mathbb{A}^n$, then it is known that $|X(\mathbb{F}_q)|=q^n$.

The only proof I have seen, for example, 10.11 in Digne--Michel's book *Representations of Finite Groups of Lie type*, uses Grothendieck's Lefschetz trace formula on \'etale cohomology.

As it seems this result should be known before the development of \'etale cohomology (please correct me if I am wrong), I would like to know if there is any more elementary proof?