I have a convex polytope $P\in\mathbb{R}^{n+m}$. I denote coordinates of an element $p\in P$ by $p(k)$.
I need to compute the intersection of this polytope with a subspace given by some number of equality constraints acting only on the first $n$ dimensions
$\{E^i\equiv \sum_{k=1}^n\beta^i_k P(k)=c^i\}_i$.
Afterwards, I need to project the resulting polytope onto a subspace given by equality constraints acting only on the second $m$ dimensions, specifically
$P(k)=0\quad\forall k\in\{n+1\ldots n+m\}$.
I want to know if I can perform the projection first and the cut after in this particular scenario. Intuitively, it seems to me that since the constraints for the cut are on acting on separate dimensions than the ones of the projection, this might be possible, but I am neither sure if this is true nor do I know how I could potentially prove this.
For some context: I only know the V-representation of $P$, and computing the intersection first and the projection second would be quite computationally difficult. Also is is actually a parametrized family of polytopes, and if I was able to perform to the two operations in the reverse order I might be able to get at least some valid parametrized inequalities of the final polytope.
I would be happy for any pointers on how to approach this. It does not seem to be a problem that is encountered very often, at least my search did not yield any results.