Understanding proofs Lemma 11 & 10 in this specific Computer Science Scheduling Paper

I’m having a hard time wrapping my head around the proofs Lemma 11 and Lemma 10 (Pages 10 and 11) in this paper called: Preemptive and Non-Preemptive Real-Time UniProcessor Scheduling.

Generally the proofs have very few steps in between and I can’t seem to understand how the author continues from one step to another. Therefore, I would be very gratefull in a more step by step detailed explanation of the proofs mentioned.

Thank you in advance!