+ 1} w_{j_0 + 1} 
+ \cdots + x_{d_0} w_{d_0} > -  x_{j_0} \lambda_1 \ \Big) \nonumber\\
&\fallingdotseq& 
P \Big(x_1 w_1 + \cdots + x_{j_0 - 1} w_{j_0 -1} + x_{j_0 + 1} w_{j_0 + 1} 
+ \cdots + x_{d_0} w_{d_0} > 0 \ \Big) \nonumber\\
& & (\mbox{because $\vert-  x_{j_0} \lambda_1 \vert = \vert x_{j_0} \vert 
\vert \lambda_1 \vert \leq \alpha \cdot (1/d_0)$ and 
the number of input neurons $d_0$ is} \nonumber\\
& & \mbox{sufficiently large}) \nonumber\\
&=& \frac{1}{2}.  \ \ (\mbox{for the same reason that the last equality 
of Eq. (\ref{eqn4-3}) holds true.}) 
$$

\[For $H$\] This case corresponding to a deep nonlinear neural network with $H$ hidden layers, we show that if the case of $H-1$ holds, then case $H$ also holds. Assuming that the case of $H-1$ holds, then $$\begin{aligned}
\label{eqn4-8}
& & P \left( {\mbox{\boldmath $X$}}_i^T {\mbox{\boldmath $w$}}_{j_1} > 0, \left[ \varphi( U_1^{(1)} ) \cdots \varphi(U_{d_1}^{(1)} ) \right] {\mbox{\boldmath $w$}}_{j_2} > 0, 
\cdots, \left[ \varphi( U_1^{(H-1)} ) \cdots \varphi(U_{d_{H-1}}^{(H-1)}) \right] 
{\mbox{\boldmath $w$}}_{j_H} > 0 \right. \nonumber\\
& & \hspace*{8.0cm} \left. \Big\vert \ w_{j_1 j_0} = \lambda_1, 
\cdots, w_{j_H j_{H-1} } = \lambda_H \right) \nonumber\\
&=& P \left( \left[ \varphi( U_1^{(1)} ) \cdots \varphi(U_{d_1}^{(1)} ) \right] 
