For the proof of the above equation, I understand the first step which has been obtained from the definition but in the second step I don't understand why they are summing over $j$ first (shouldn't the summation still be in $i$?) and then multiplying by $i$ in the second line the terms in the bracket. Also don't understand how they got each of the terms in the second line from the terms in the first line. Also, how did they get for the second term in the second line, $g(\partial f/\partial q_j)(\partial h/\partial p_j)$? when the other three terms have $q$'s and $p$'s in terms of $i$ so shouldn't the second term also have $q_i$ and $p_i$ in the denominator instead of $j$'s?
Any help would be much appreciated.