Let \(Z_1,Z_2\) 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 \(Z_i\) determines \(\pi (Z_i)\)). 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 \(A := Z_1 - Z_2\), \(B := \pi (Z_1 - Z_2)\) and \(C := (\pi (Z_1),\pi (Z_{2}))\), and noting that in this case \(\mathbb {H}[A | B,C] = \mathbb {H}[A | C]\) since \(C\) uniquely determines \(B\) (this may require another helper lemma about entropy). This completes the proof.

By Proposition 5.1 and the nonnegativity of conditional Ruzsa distance (from Lemma 3.15) we have

The inequality follows from \(d[X\mid \pi (X);Y\mid \pi (Y)]\geq 0\) (Lemma 3.15).

We apply Proposition 5.1 with \(H := G \times G\), \(H' := G\), \(\pi \) the addition homomorphism \(\pi (x,y) := x+y\), and with the random variables \(Z_1 := (Y_1,Y_3)\) and \(Z_2 := (Y_2,Y_4)\). Then by independence (Corollary 2.24)

while by definition

Furthermore,

since \(Z_1=(Y_1,Y_3)\) and \(Y_1\) are linked by an invertible affine transformation once \(\pi (Z_1)=Y_1+Y_3\) is fixed, and similarly for \(Z_2\) and \(Y_2\). (This has to do with Lemma 2.12) Finally, we have

where in the last line we used the fact that \((Y_1+Y_2, Y_1+Y_2+Y_3+Y_4)\) uniquely determine \(Y_3+Y_4\) and similarly \((Y_2+Y_4, Y_1+Y_2+Y_3+Y_4)\) uniquely determine \(Y_1+Y_3\). (This requires another helper lemma about entropy.)