I would also suggest changing the last sentence of the proof to "since otherwise one of the plans wouldn't be optimal". (the next natural step my internal autocomplete expected in the proof was substituting back to the first equation, and I kept wondering for quite a while what the unspecific "one of them wouldn't appear" means in that context)
And if you'll be editing: there's probably also a typo in the second paragraph of the Technical Appendix: a confusing "N" instead of an "R" .