I have the following proof for the chain rule for entropy of random variables: We write:
\begin{eqnarray*} H(X_1,X_2,...,X_n)&=&-\sum\limits_{x_1,x_2,...,x_n}p(x_1,x_2,...,x_n)logp(x_1,x_2,...,x_n) \\&=&-\sum\limits_{x_1,x_2,...,x_n}p(x_1,x_2,...,x_n)log\prod\limits_{i=1}^{n}p(x_i|x_{i-1},...,x_1) \\&=&-\sum\limits_{x_1,x_2,...,x_n}\sum\limits_{i=1}^n p(x_1,x_2,...,x_n)logp(x_i|x_{i-1},...,x_1) \\&=&-\sum\limits_{x_1,x_2,...,x_i}\sum\limits_{i=1}^n p(x_1,x_2,...,x_n)logp(x_i|x_{i-1},...,x_1) \\&=&\sum\limits_{i=1}^n H(X_i|X_{i-1},...,X_1) \end{eqnarray*}
Basically, I have 2 questions: first how we get from line 1 to line 2. Is it some kind of Markov property? And second, why the summation lower boundary changes in line 4?