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\begin{fcprop}[]
  \label{prop:7.6}
  \nameref{thm:7.5}($\eta$) $\implies$
  \hyperref[thm:7.2]{EPFR}($\eta^{-1}$).
\end{fcprop}

\begin{proof}
  By compactness we can find $U$, $V$ such that
  \[
    \tau_{X, Y}[U; V]
    = \entd UV
    + \eta(\entd UX + \entd VY)
  \]
  is minimised. If $\entd UV \neq 0$ then by
  \nameref{thm:7.5}($\eta$) there exist $Z$, $W$
  such that $\tau_{U, V}[Z; W] < \entd UV$.

  But then
  \begin{align*}
    \tau_{X, Y}[Z; W]
    &= \entd ZW
    + \eta(\entd ZX + \entd WY) \\
    &\le \entd ZW
    + \eta(\entd ZU + \entd WV)
    + \eta(\entd UX + \entd VY) \\
    &\quad \text{(by \nameref{lemma:6.7})}
    \\
    &< \entd UV + \eta(\entd UX + \entd VY) \\
    &= \tau_{X, Y}[U; V]
  \end{align*}
  Contradiction.

  It follows that $\entd UV = 0$. So there exists
  $H$ such that $U$ and $V$ are uniform on cosets
  of $H$, so
  \[
    \eta(\entd{U_H}X + \entd{U_H}Y)
    < \entd XY
  ,\]
  which gives us
  \hyperref[thm:7.2]{EPFR}($\eta^{-1}$).
\end{proof}

\begin{fcdefnstar}[]
  Write $\tau_{X, Y}[U|Z; V|W]$ for
  \divcloze{
  \[
    \sum_{Z, W} \Pbb[Z = z] \Pbb[W = w]
    \tau_{X, Y}[U|Z = z; V|W = w]
  \]
  }
\end{fcdefnstar}

\begin{fcdefnstar}[]
  Write $\tau_{X, Y}[U; V \|Z]$ for
  \divcloze{
  \[
    \sum_z \Pbb[Z = z] \tau_{X, Y}
    [U|z = z; V|Z = z]
  \]
  }
\end{fcdefnstar}

\begin{remark*}
  If we can prove \nameref{thm:7.5} for
  conditional random variables, then by averaging
  we get it for some pair of random variables
  (e.g. of the form $U|Z = z$ and $V|W = w$).
\end{remark*}

\begin{fclemma}[Fibring lemma]
  \label{lemma:7.7}
  % Lemma 5
  Assuming:
  - $G$ and $H$ are abelian groups
  - $\phi : G \to H$ a homomorphism
  - let $X$, $Y$ be $G$-valued random variables.
  Then:
  \[
    \entd XY
    = \entd{\phi(X)}{\phi(Y)}
    + \entd{X|\phi(X)}{Y|\phi(Y)}
    + \muti{X - Y}{\phi(X), \phi(Y) \mid \phi(X) -
    \phi(Y)}
  .\]
\end{fclemma}

\begin{proof}
  \begin{align*}
    \entd XY
    &= \ent{X - Y} - \half \ent X - \half \ent Y
    \\
    &= \ent{\phi(X) - \phi(Y)} + \cent{X -
    Y}{\phi(X) - \phi(Y)} - \half \ent{\phi(X)} \\
    &~~- \half \cent{X}{\phi(X)} - \half \ent{\phi(Y)}
    - \half \cent{Y}{\phi(Y)} \\
    &= \entd{\phi(X)}{\phi(Y)} + \entd{X \mid
    \phi(X)}{Y \mid \phi(Y)}
    + \cent{X - Y}{\phi(X) - \phi(Y)} \\
    &~~- \cent{X -
    Y}{\phi(X), \phi(Y)}
  \end{align*}
  But the last line of this expression equals
  \[
    \cent{X - Y}{\phi(X) - \phi(Y)}
    - \cent{X - Y}{\phi(X), \phi(Y), \phi(X) -
    \phi(Y)}
    = \muti{X - Y}{\phi(X), \phi(Y) \mid \phi(X) -
    \phi(Y)}
    .
    \qedhere
  \]
\end{proof}

We shall be interested in the following special
case.

\begin{fccoro}[]
  \label{coro:7.8}
  % coro 7.6
  Assuming:
  - $G = \Fbb_2^n$ and $X_1, X_2, X_3, X_4$ are
  independent $G$-valued random variables
  Then:
  \begin{align*}
    \entd{(X_1, X_2)}{(X_3, X_4)}
    &= \entd{X_1}{X_3} + \entd{X_2}{X_4} \\
    &= \entd{X_1 + X_2}{X_3 + X_4}
    + \entd{X_1 \mid X_1 + X_2}{X_3 \mid X_3 +
    X_4} \\
    &~~+ \ub{\muti{X_1 + X_3, X_2 + X_4}{X_1 + X_2, X_3 +
    X_4 \mid X_1 + X_2 + X_3 + X_4}}_{(*)}
  \end{align*}
\end{fccoro}

\begin{proof}
  Apply \cref{lemma:7.7} with $X = (X_1, X_2)$, $Y
  = (X_3, X_4)$ and $\phi(x, y) = x + y$.
\end{proof}

We shall now set $W = X_1 + X_2 + X_3 + X_4$.

Recall that \cref{lemma:6.11} says
\[
  \scentd{X}{Y}{X + Y}
  \le 3\muti XY + 2\ent{X + Y} - \ent X - \ent Y
.\]
Equivalently,
\[
  \muti XY \ge \third (\scentd{X}{Y}{X + Y} + \ent
  X + \ent Y - 2\ent{X + Y})
.\]
Applying this to the information term ($*$), we
get that it is at least
\begin{align*}
  &\third(\scentd{X_1 + X_3, X_2 + X_4}{X_1 + X_2,
  X_3 + X_4}{X_2 + X_3, W}
  + \cent{X_1 + X_3, X_2 + X_4}{W} \\
  &~~+ \cent{X_1 + X_2, X_3 + X_4}{W} -
  2\cent{X_2 + X_3, X_2 + X_3}{W})
\end{align*}
which simplifies to
\begin{align*}
  &\third(\scentd{X_1 + X_3, X_2 + X_4}{X_1 + X_2,
  X_3 + X_4}{X_2 + X_3, W}
  + \cent{X_1 + X_3}{W} \\
  &~~+ \cent{X_1 + X_2}{W} - 2\cent{X_2 + X_3}{W})
\end{align*}
