since it implies that $$\begin{aligned}
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R}(X_{1}^{j},\dots ,X_{n}^{j}|y_{1},\dots ,y_{m}[Y_{1}^{j},\dots ,Y_{n}^{j}];k,l,\epsilon /2)\leq  &  & \\
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}};k,l,\epsilon ), &  & \end{aligned}$$ which in turn implies the desired inequality.

\[prop: generation\]Assume that $ {Z_{1},\dots ,Z_{r}}\in W^{*}({Y_{1},\dots ,Y_{m}}) $. Then $$\chi ({X_{1},\dots ,X_{n}}|{Y_{1},\dots ,Y_{m}})\leq \chi ({X_{1},\dots ,X_{n}}|{Z_{1},\dots ,Z_{r}}),$$ and similarly for $ \chi  $ replaced with $ \chi ^{\omega } $.

Fix $ \epsilon >0 $, $ l>0 $, $ R>0 $. Then there exists non-commutative polynomials $ p_{1},\dots ,p_{r} $ in $ m $ variables, such that $$\left( {X_{1},\dots ,X_{n}},p_{1}({Y_{1},\dots ,Y_{m}}),\dots ,p_{r}({Y_{1},\dots ,Y_{m}})\right)$$ approximate $$({X_{1},\dots ,X_{n}},{Z_{1},\dots ,Z_{r}})$$ strongly to any desired accuracy. It follows, that for a suitable choice of such polynomials, there exist $ \epsilon >\epsilon '>0 $, $ l'>l>0 $ and $ R'>R>0 $ such that whenever $$({y_{1},\dots ,y_{m}})\in \Gamma _{R}({Y_{1},\dots ,Y_{m}},{X_{1},\dots ,X_{n}};k,l',\epsilon ')$$ we have the inclusion $$\begin{aligned}
\Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')\subset  &  & \\
\Gamma _{R}({X_{1},\dots ,X_{n}}|p_{1}({y_{1},\dots ,y_{m}}),\dots ,p_{r}({y_{1},\dots ,y_{m}})[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ). &  & \end{aligned}$$ It follows that $$\begin{aligned}
\sup _{{y_{1},\dots ,y_{m}}}\log \lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')\leq  &  & \\
\sup _{z_{1},\dots ,z_{r}}\log \lambda \Gamma _{R}({X_{1},\dots ,X_{n}}|z_{1},\dots ,z_{r}[{Z_{1},\dots ,Z_{r}}];k,l,\epsilon ). &  & \end{aligned}$$ 