Take two affine maps $f,g:\mathbb{R}^n \rightarrow \mathbb{R}^m$ and consider $\Omega=\{p \in \mathbb{R^n}:f(p)=g(p)\}$. If the interior of $\Omega$ is not empty, is it true that $f=g$?
I reason as follows: the set $\Omega$ is the kernel of the affine map $f-g$ and therefore it is an affine subspace. As the only affine subspace with non-empty interior is the total, $\Omega=\mathbb{R^n}$ and therefore $f=g$. Is it correct?