is contained in $$\Gamma _{R}(f(X_{1}),\dots ,X_{n},{Y_{1},\dots ,Y_{m}};k,l,\epsilon ).$$ Choose $ {y_{1},\dots ,y_{m}}$ such that $$\begin{aligned}
\log \sup _{z_{1},\dots ,z_{m}}\lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|z_{1},\dots ,z_{m}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')- &  & \\
\log \lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l;,\epsilon ')<\delta . &  & \end{aligned}$$ Then we have that $$\begin{aligned}
\sup _{z_{1},\dots ,z_{m}}\log \lambda \Gamma _{R'}(f(X_{1}),X_{2},\dots ,X_{n}|z_{1},\dots ,z_{m}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )\geq  &  & \\
\log \lambda \Gamma _{R}(f(X_{1}),\dots X_{n}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l,\epsilon )\geq  &  & \\
\log \lambda F\left( \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ')\right) \geq  &  & \\
\log \lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ') &  & \\
+\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t)-\delta k^{2}\geq  &  & \\
\log \sup _{z_{1},\dots ,z_{m}}\lambda \Gamma _{R'}({X_{1},\dots ,X_{n}}|z_{1},\dots ,z_{m}[{Y_{1},\dots ,Y_{m}}];k,l',\epsilon ') &  & \\
+\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t)-2\delta k^{2}. &  & \end{aligned}$$ Taking $ \limsup _{k\to \infty } $ gives us that $$\begin{aligned}
\chi _{R}(f(X_{1}),\dots ,X_{n}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l,\epsilon )\geq  &  & \\
\chi _{R'}({X_{1},\dots ,X_{n}}|{y_{1},\dots ,y_{m}}[{Y_{1},\dots ,Y_{m}}];l',\epsilon ') &  & \\
+\int \int \frac{\log |f(s)-f(t)|}{\log |s-t|}d\mu _{1}(s)d\mu _{1}(t), &  & \end{aligned}$$ which implies the theorem. The proof for $ \chi ^{\omega } $ is exactly the same.

If $ \Vert X_{i}\Vert <R $ for all $ i $, then $$\chi _{R}({X_{1},\dots ,X_{n}}|B)=\chi ({X_{1},\dots ,X_{n}}|B),$$ 