Mutual information and entropy to prove minimal Relevance Maximum Dependency
For the equations you are reporting, $J(S_m) = I(S_{m-1}; x_m)$, i.e. $J(S_m)$ is the mutual information between $S_{m-1}$ and $x_m$. Since for any two random variables $X$ and $Y$ 

$I(X; Y) = H(X) + H(Y) - H(X; Y)$

the equation follows directly from the definition of mutual information (the chain rule does not apply here, since there is no need to deal with conditional entropy):

$\begin{gathered}
  H({S_{m - 1}},{x_m}) = H({S_m})  \\
   = \sum\limits_{i = 1}^m {H({x_i})}  - J({S_m})  \\
   = \sum\limits_{i = 1}^m {H({x_i})}  - I({S_{m - 1}};{x_m})  \\
   = \sum\limits_{i = 1}^m {H({x_i})}  - (H({S_{m - 1}}) + H({x_i}) - H({S_{m - 1}},{x_m}))  \\
   = \sum\limits_{i = 1}^m {H({x_i})}  - (\sum\limits_{i = 1}^m {H({x_i})}  - H({S_{m - 1}},{x_m}))  \\
   = H({S_{m - 1}},{x_m})  \\ 
\end{gathered}$