,x_{p}) $ be an open ball of radius $ \epsilon ' $ for the operator norm on $ M_{k}^{n-p} $, centered at $ x_{n+1},\dots ,x_{p} $. Then for sufficiently small $ \epsilon ' $ (depending only on $ k $ and $ l $), we have $$\Gamma _{k,x_{n+1},\dots ,x_{p}}\times O\subset \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ).$$

Let $$\Gamma _{k}=\bigcup _{x_{n+1},\dots ,x_{n}}\Gamma _{k,x_{n+1},\dots \, x_{p}}\times O(x_{n+1},\dots ,x_{n}).$$ Then $ \Gamma _{k}\subset \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ), $ so that $$\begin{aligned}
\lambda \Gamma _{R}({X_{1},\dots ,X_{n}},X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )\geq \lambda \Gamma _{k} &  & \\
\geq \inf _{x_{n+1,\dots \, x_{p}}}\lambda \Gamma _{k,x_{n+1},\dots ,x_{p}}\times \lambda \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ) &  & \\
\geq (1-\delta )\cdot \lambda \Gamma _{R}({X_{1},\dots ,X_{n}};k,l,\epsilon )\times \lambda \Gamma _{R}(X_{n+1},\dots ,X_{p}|y_{1},\dots ,y_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon ). &  & \end{aligned}$$ The statement of the theorem follows. The proof for $ \chi ^{\omega } $ is identical.

We note that the preceding theorem implies that $ \chi ({X_{1},\dots ,X_{n}}|B) $ is not always $ -\infty  $. For example, if $ S_{1},\dots ,S_{n} $ is a free semicircular family free from a unital von Neumann algebra $ B $, which can be embedded into the ultrapower of the hyperfinite II$ 