generator for $ B $. Then we have $$\chi ({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}|Y)\leq \chi ({X_{1},\dots ,X_{n}}:Y)=-\infty$$ because of [@dvv:entropy3 Corollary 7.7].

Note that the analogous theorem holds for $ \chi ^{*}(\cdots :B) $, see [@shlyakht:cpentropy].

Let $ p<n $. Then $$\chi ({X_{1},\dots ,X_{n}}|B)\leq \chi (X_{1},\dots ,X_{p}|B)+\chi (X_{m+1},\dots ,X_{p}|B)$$ and a similar inequality holds for $ \chi ^{\omega } $.

We clearly have $$\begin{aligned}
\Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\subset \Gamma _{R}(X_{1},\dots ,X_{p}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\times  &  & \\
\Gamma _{R}(X_{p+1},\dots ,X_{n}|{y_{1},\dots ,y_{m}};k,l,\epsilon ) &  & \end{aligned}$$ so that $$\begin{aligned}
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\leq  &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \left( \Gamma _{R}(X_{1},\dots ,X_{p}|{y_{1},\dots ,y_{m}};k,l,\epsilon )+\right.  &  & \\
\left. \log \lambda \Gamma _{R}(X_{p+1},\dots ,X_{n}|{y_{1},\dots ,y_{m}};k,l,\epsilon \right) \leq  &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \left( \Gamma _{R}(X_{1},\dots ,X_{p}|{y_{1},\dots ,y_{m}};k,l,\epsilon )\right) + &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \left( \Gamma _{R}(X_{p+1},\dots ,X_{nb}|{y_{1},\dots ,y_{m}};k,l,\epsilon \right) . &  & \end{aligned}$$ This implies the proposition.

\[thrm: whenfreefromB\]Let $ {X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}\in (M,\tau ) $ be self-adjoint non-commutative random variables. Assume that the family $ {X_{1},\dots ,X_{n}}$ is free from the von Neumann algebra generated by $ B $ and $ X_{n+1},\dots ,X_{p} $, and assume that $ B $ 