This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below.

[![Claim 6.2][1]][1]


  [1]: https://i.sstatic.net/0INnb.png

In [the remark](https://i.sstatic.net/RQYbQ.png), he does this:
$$
\begin{align}
\sum_{k=1}^N \frac{\partial }{\partial q_k} \left( \frac{\partial x_i}{\partial q_m} \right) \dot{q_k} = \frac{\partial }{\partial q_m} \sum_{k=1}^N \frac{\partial x_i}{\partial q_k} \dot{q_k}
\end{align}
$$
He has essentially changed the order of differentiation from $\frac{\partial }{\partial q_k} \frac{\partial x_i}{\partial q_m}$ to $\frac{\partial }{\partial q_m} \frac{\partial x_i}{\partial q_k}$. 

However, isn't this only valid if all the $q_k$ are independent? If he is assuming that all the $q_k$ are independent, isn't this a very limited proof? <del>It wouldn't show that the E-L equations hold in a coordinate system with a non-orthogonal basis, for instance.</del>