Extensive rewrite of supporting results (in progress)

- Fix the proofs
- Add results over R+1,...,R'
MAIN PROOFS: restore some results
MAIN PROOFS: unclear amendments
MAIN PROOFS: some work
MAIN PROOFS: Revise and clean up proof of Theorem 1
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