Is there a simple proof that shows:

1. Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron.
2. Minkowski sum of two $\mathcal{H}$-polyhedrons is a $\mathcal{H}$-polyhedron.

I know a proof of (1.) based on Fourier-Motzkin elimination. and, I know (2.) is a simple consequence of (1.).

Every different approach is appreciated.