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I am trying to understand the following proof: enter image description here

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enter image description here enter image description here

Main Confusion: In particular, I am having a very hard time understanding the chain of inequalities in the proof for (3)': I think the first equality is due to $X$ being symmetric. However, how did we obtain the rest of the inequalities? Could someone expand and explain the details behind these inequalities?

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    $\begingroup$ I think the last inequality requires some additional information. Perhaps, it is part of the proof of (1) implies (2). $\endgroup$
    – geetha290krm
    Commented Jun 26 at 6:32
  • $\begingroup$ @geetha290krm I have updated the post for the previous proofs. $\endgroup$
    – Partial T
    Commented Jun 26 at 6:34

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