Say we take $\pmb X \pmb \beta$ and partition the columns of $\pmb X$ so that we have $\pmb X \pmb \beta_1$ and $\pmb X_2 \pmb \beta_2$. Why does:
$\pmb y = \pmb X \pmb \beta + \pmb e = \pmb X_1 \pmb \beta_1+ \pmb X_2 \pmb \beta_2 + \pmb e$
I can reason through it geometrically. We're finding the the vector that gets as close to $\pmb y$ as possible in the column space of $\pmb X$. Therefore splitting up the matrix isn't going to change $\hat{ \pmb y}$. Furthermore, the coeficients in $\pmb \beta$ will correspond to the the same columns in the split regression because the solution to the original problem is unique.
So I feel comfortable with why the result is true but I don't know how to prove it beyond that... Is it possible to do this reasonably?