Let be independent throughout (this is possible by Lemma 3.10 and Lemma 3.7). By Lemma 3.20, We have
In the middle step, we used Corollary 2.20, and in the last step we used the fact that
(thanks to Lemma 2.13 and Lemma 2.2) and that
(since determines ). This gives the claimed inequality. The difference between the two sides is precisely
To rewrite this in terms of (conditional) mutual information, we use the identity
(which follows Lemma 2.26) taking , and , and noting that in this case since uniquely determines (this may require another helper lemma about entropy). This completes the proof.