I can't get the answer right to this first-order condition derivative

Question

Came across a paper where the authors use LaGrange/shadow variable to solve for an electricity pricing problem.
The objective function: $$\sum_{i\in L} v^i(p_{L},w_L^i)+\sum_{i\in H} v^i(p_{H},w_H^i)$$
*The $v^i$ s are indirect utility functions
The constraint:$$(p_L-c)[n_Hq+\sum_{i\in L} e^i(p_L,w_L^i)]+(p_H-c)[\sum_{i\in H}e^i(p_H,w_H^i)]=F$$
*F is the fixed cost. we treat it as zero here.
The paper went on to generate the first order conditions in regard to $p_H$, $p_L$, and q. $p_H$ stands for the higher block price in an increasing block price model, $p_L$ stands for the lower block price. q is the quantity threshold that separate the blocks.
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I did not get what the paper generated at all. Let me screenshot the three conditions below: I think I did something wrong here, I never get what the paper did for $\partial F/ \partial q$ and the partials for the other two variables.
the constraint equation the budget constraint and more detailed problem setup $ $
Can anyone be so kind and walk me through step by step?the paper's solution