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\begin{align*}
&& \displaystyle \prod_{i=1}^{n} (s_i)-1\overset{\text{I.V.}}{=}& \displaystyle \sum_{y\in \Omega_n}(\prod_{i=1}^{n} {(s_i-1)}^{y_i}) &&+1\\
&\Rightarrow \,& \displaystyle \prod_{i=1}^{n} (s_i)=&\displaystyle \sum_{y\in \Omega_n}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+1 && \cdot (s_{n+1}-1) \\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i)=&(s_{n+1} -1) (\displaystyle \sum_{y\in \Omega_n}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+1) && \\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i)=&(s_{n+1} -1) \displaystyle \sum_{y\in \Omega_n}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+(s_{n+1}-1) && \\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i)=&(s_{n+1} -1) \displaystyle \sum_{y\in \Omega_n}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+(\prod_{\substack {i=1\\y=0\dots01}}^{n+1} (s_i-1)^{y_i}) && \\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i)=& \displaystyle \sum_{y\in \Omega_n}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})(s_{n+1} -1)^1+(\prod_{\substack {i=1\\y=0\dots01}}^{n+1} (s_i-1)^{y_i}) && \\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i)=& \displaystyle \sum_{\substack {y\in \Omega_{n+1}\setminus y=0\dots 01\\y_{n+1}=1}}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+(\prod_{\substack {i=1\\y=0\dots01}}^{n+1} (s_i-1)^{y_i}) && \\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i)=& \displaystyle \sum_{\substack {y\in \Omega_{n+1} \\y_{n+1}=1}}(\prod_{i=1}^{n} {(s_i-1)}^{y_i}) && +\displaystyle \prod_{i=1}^{n} (s_i)-1
\end{align*}
\begin{align*}
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i) +\displaystyle \prod_{i=1}^{n} (s_i)-1 =& \displaystyle \sum_{\substack {y\in \Omega_{n+1} \\y_{n+1}=1}}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+\displaystyle \prod_{i=1}^{n} (s_i)-1 &&\\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i) +\displaystyle \prod_{i=1}^{n} (s_i)-1 \overset{\text {I.V.}}{=}& \displaystyle \sum_{\substack {y\in \Omega_{n+1} \\y_{n+1}=1}}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+ \displaystyle \sum_{y\in \Omega_n}(\prod_{i=1}^{n} {(s_i-1)}^{y_i}) &&\\
&\Rightarrow \,& (s_{n+1}-1) \displaystyle \prod_{i=1}^{n} (s_i) +\displaystyle \prod_{i=1}^{n} (s_i)-1 =& \displaystyle \sum_{\substack {y\in \Omega_{n+1} \\y_{n+1}=1}}(\prod_{i=1}^{n} {(s_i-1)}^{y_i})+ \displaystyle \sum_{\substack {y\in \Omega_{n+1} \\y_{n+1}=0}}(\prod_{i=1}^{n+1} {(s_i-1)}^{y_i}) &&\\
&\Rightarrow \,& (s_{n+1}-1+1) \displaystyle \prod_{i=1}^{n} (s_i) )-1 =& \displaystyle \sum_{y\in \Omega_{n+1}}(\prod_{i=1}^{n} {(s_i-1)}^{y_i}) &&\\
&\Rightarrow \,& \displaystyle \prod_{i=1}^{n+1} (s_i) )-1 =& \displaystyle \sum_{y\in \Omega_{n+1}}(\prod_{i=1}^{n} {(s_i-1)}^{y_i}) &&\hfill \Box
\end{align*}