Apart from the group theory technicalities this proof is interesting because it becomes clear how to generalize the Bloch theorem for groups that are not only translations. This is typically done for space groups which are a combination of a translation and a point group and it is used for computing the band structure, spectrum and specific heats of crystals given a specific crystal group symmetry like FCC or BCC and eventually an extra basis. In this proof it is also possible to notice how it is key that the extra point group is driven by a symmetry in the effective potential but it shall commute with the Hamiltonian.