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I am working with the securit yof CRYSTAL's Dilithium signature in the QROM. I am working with Kiltz et al.'s approach through lossy ID-schemes and looking at the proof of minimal entropy for the $DFS[ID, H, PRF]$ derived Dilithium-QROM signature. Unfortunately, I am not able to see how the proof of the lemma flows, nor understand what is beeing done and why it is "ok" to make the claims that are made... I was hoping someone here would be able to help me.

The definition of minimal entropy I am working with:

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and the lemma with proof I can't seem to understand:

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Any and all help would be greatly appreciated :) Thank you!

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