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It's clear that

\begin{align}
d_j-1&=\sum_{t=2}^{l}\Big({\sum_{j_2,\cdots ,j_t\in e_*}{\beta_{jj_2\cdots j_t}^{(t)}}}\Big)+\beta_j^{(1)}\label{beta1},\\
&=\sum_{t=1}^{l}{\beta_j^{(t)}},\qquad\qquad \forall j\in e_*\label{beta2}.
\end{align}
Equations (\ref{beta1}) and (\ref{beta2}) result from not considering
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