How do you come up with such proofs?

[MhYao2014]

This awesome work inspired me a lot, and I believe it deserves more exploration to further release its potential.

However, the last step of proofs in the appendix (the proof of why F_p1(P)-F_p1(P*) >= 0) is confusing. Would you please update an example to illustrate how to manipulate the index sets? Some pictures regarding the example would be awesome.