claimed in the lemma.
\end{proof}
++<<<<<<< HEAD
+We can now prove the Theorem~\ref{t:chaos des general}.
++=======
+ We can now prove Theorem~\ref{t:chaos des general}...
++>>>>>>> e55d237aba022a66cc2d7650d295b29169878f45
\begin{proof}[Theorem~\ref{t:chaos des general}]
Firstly, strong transitivity implies transitivity.
by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
one has
++$\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]$ and,
++therefore,
\begin{equation}\label{PCH-2}
--\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1].
++\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(U_{kN})=1].
\end{equation}
Now, using (\ref{PCH-1}) again, one has for every $x$,