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% 18/03/2025 09AM

\begin{fccoro}[]
  \label{coro:7.15}
  % Corollary 7.13
  Assuming:
  - $(U, V)$ is \gls{Crel} to $(X, Y)$
  Then:
  $(U + V, U + V)$ is \glsref[Crel]{$2(C +
  1)$-relevant} to $(X, Y)$.
\end{fccoro}

\begin{proof}
  \begin{align*}
    \entd{U + V}X
    &\le \half (\entd UX + \entd VX + \entd UV) \\
    &\le \half (\entd UX + \entd VY + \entd XY +
    \entd UX + \entd XY + \entd VY) \\
    &= \entd UX + \entd VY + \entd XY
  \end{align*}
  Similarly for $\entd{U + V}Y$.
\end{proof}

\begin{fclemma}[]
  \label{lemma:7.16}
  % Lemma 7.14
  Assuming:
  - $U, V, X$ are independent $\Fbb_2^n$-valued
  random variables
  Then:
  \[
    \entd{U \mid U + V}X
    \le \half (\entd UX + \entd VX + \entd UV)
  .\]
\end{fclemma}

\begin{proof}
  \begin{align*}
    \entd{U \mid U + V}X
    &\le \cent{U + X}{U + V} - \half \cent{U}{U +
    V} - \half \ent X \\
    &\le \ent{U + X} - \half \ent U - \half \ent V
    + \half \ent{U + V} - \half \ent X
  \end{align*}
  But $\entd{U \mid U + V}X = \entd{V \mid U +
  V}X$, so it's also
  \[
    \le \ent{V + X}
    - \half \ent U - \half \ent V + \half \ent{U +
    V} - \half \ent X
  .\]
  Averaging the two inequalities gives the result
  (as earlier).
\end{proof}

\begin{fccoro}[]
  \label{coro:7.17}
  % Corollary 7.15
  Assuming:
  - $U, V$ are independent random variables
  - $(U, V)$ is \gls{Crel} to $(X, Y)$
  Thens:[(i)]
  - $(U_1 \mid U_1 + U_2, V_1 \mid V_1 + V_2)$ is
  \glsref[Crel]{$2C$-relevant} to $(X, Y)$.
  - $(U_1 \mid U_1 + V_1, U_2 \mid U_2 + V_2)$ is
  \glsref[Crel]{$2(C + 1)$-relevant} to $(X, Y)$.
\end{fccoro}

\begin{proof}
  Use \cref{lemma:7.16}. Then as soon as it is
  used, we are in exactly the situation we were in
  when bounding the relevance of $(U_1 + U_2, V_1
  + V_2)$ and $(U_1 + V_1, U_2 + V_2)$.
\end{proof}

It remains to tackle the last two terms in
\cref{lemma:7.10}. For the fifth term we need to
bound
\[
  \entd{X_1 + X_2 \mid X_2 + Y_1, X_1 + Y_2}X
  + \entd{X_1 + Y_1 \mid X_2 + Y_1, X_1 + Y_2}Y
.\]
But first term of this is at most (by
\cref{lemma:7.12})
\[
  \half(
    \entd{X_1}{X_2 + Y_1, X_1 + Y_2}{X}
    + \entd{X_2 \mid X_1 + Y_1, X_1 + Y_2}{X}
    + \scentd{X_1}{X_2}{X_2 + Y_1, X_1 + Y_2}
  )
.\]
By \nameref{lemma:6.7} and independence, this is at
most
\begin{align*}
  &\le \entd{X_1 \mid X_1 + Y_2}{X} + \entd{X_2
  \mid X_2 + Y_1}{X} \\
  &= 2\entd{X \mid X + Y}{X}
\end{align*}
Now we can use \cref{lemma:7.16}, and similarly
for the other terms.

In this way, we get that the fifth and sixth terms
have relevances bounded above by $\lambda C$ for
an absolute constant $\lambda$.
