(f_{1}(X_{1}),\dots ,f_{n}(X_{n})|B)=\chi ({X_{1},\dots ,X_{n}}|B)+\sum ^{n}_{i=1}\int \int \frac{\log |f_{i}(s)-f_{i}(t)|}{\log |s-t|}d\mu _{i}(s)d\mu _{i}(t),$$ and the same formula holds for $ \chi ^{\omega } $ in place of $ \chi  $.

It is sufficient to prove the statement assuming further that $ f_{i} $ are identity diffeomorphisms for $ i>1 $; we write $ f=f_{1} $. It is moreover sufficient to show that given $ {Y_{1},\dots ,Y_{m}}\in B $ $$\begin{aligned}
\chi (f_{1}(X_{1}),\dots ,f_{n}(X_{n})|{Y_{1},\dots ,Y_{m}})\geq \chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})+ &  & \\
\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t), &  & \end{aligned}$$ since the reverse inequality follows by replacing $ f $ with its inverse. It is shown in [@dvv:entropy4 Proposition 3.1] that given $ \delta >0 $, $ \epsilon >0 $, $ l>0 $, $ R>0 $, there exist $ k_{0}>0 $, $ \epsilon >\epsilon _{0}>0 $, $ l_{0}>l>0 $, such that for all $ k>k_{0} $, $ 0<\epsilon '<\epsilon _{0} $ and $ l'>l_{0} $, the determinant of the map $$F:({x_{1},\dots ,x_{n}},{y_{1},\dots ,y_{m}})\mapsto (f(x_{1}),x_{2},\dots ,x_{n},{y_{1},\dots ,y_{m}})$$ is bounded below by $$\exp \left( k^{2}\left[ \int \int \log \frac{|f(s)-f(t)|}{|s-t|}d\mu _{1}(s)d\mu _{1}(t)-\delta \right] \right)$$ for $$({x_{1},\dots ,x_{n}},{y_{1},\dots ,y_{m}})\in \Gamma _{R'}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};k,l',\epsilon ').$$ Moreover, the image of $$\Gamma _{R'}({X_{1},\dots ,X_{n}},{Y_{1},\dots ,Y_{m}};k,l',\epsilon ')$$ under this map 