Add notation and clarification to beginning of appendix

- Defines a few terms that had been accidentally deleted at the beginning
  of the appendix
- Add text specifying that the proofs are for scalars.
Rephrase parts of proof of Theorem 1
Remove redundant theorem from appendix
In-progress edits to theory
TEXT: some edits
TEXT: unclear edits
TEXT: remove some of the material added earlier
TEXT: minor changes
TEXT: Intermediate commit