Need some help with this proof?

I am currently trying to implement this paper : Reinforcement Learning for Uplift Modeling

I have skimmed through the paper have intuitive idea of the process they are describing.

but am struggling with the 2.2 Uplift Modeling General Metric part. could someone have a look at it and help me understand the thought process?

I am struggling to understand the Lemma 1. would greatly appreciate some help over there.

just wanted to understand the maths behind the proof in detail: