The part of the proof that shows that you can actually reach all the
configurations in a particular equivalence class is not particularly
elegant. Basically, you have to appeal to the details of a particular
cube solving algorithm.

For example, I have a tool that "flips" two edge cubies in place,
without desturbing anything else. This tool shows that I can orient
the edge cubies in ANY even permutation of the edge cubie faces. The
fact that I cannot obtain any odd permutations is a result of the fact
that a quarter twist is itself an even permutation of the edge cubie
faces.

Most people can examine their own cube solving tools and see that in
fact, they are capable of obtaining all the configurations not
forbidden by the familiar constraints.