  Recently, the stability of certain topological phases of matter under weak
perturbations was proven. Here, we present a short, alternate proof of the same
result. We consider models of topological quantum order for which the
unperturbed Hamiltonian $H_0$ can be written as a sum of local pairwise
commuting projectors on a $D$-dimensional lattice. We consider a perturbed
Hamiltonian $H=H_0+V$ involving a generic perturbation $V$ that can be written
as a sum of short-range bounded-norm interactions. We prove that if the
strength of $V$ is below a constant threshold value then $H$ has well-defined
spectral bands originating from the low-lying eigenvalues of $H_0$. These bands
are separated from the rest of the spectrum and from each other by a constant
gap. The width of the band originating from the smallest eigenvalue of $H_0$
decays faster than any power of the lattice size.
