I am studying N. [Katz's paper][1] where I found a fairly good account on $p$-curvatures.

I don't understand the following proof:
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[![enter image description here][2]][2]
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Let : 

$\Psi: \operatorname{Der}(S|T) \to \operatorname{End}_T(\mathcal E)$   

$D \to (\nabla(D))^p -\nabla(D^p)$

Where $\nabla: \operatorname{Der}(S|T) \to \operatorname{End}_T(\mathcal E)$ such that: $\nabla(D)(ge) = D(g)e+g\nabla(D)(e)$, $g$ and $D$ sections of $\mathcal O_S$ and $\operatorname{Der}(S|T)$ respectively.

 1. To prove $(5.4.4)$, we have by additivity of the p-curvature:

 $$\Psi(D)=\sum_i a_i^p \Psi\big( \frac{\partial}{\partial s_i}\big) = \sum_i a_i^p \Big(\nabla\big(\frac{\partial}{\partial^p s_i}\big)\Big)^p -\sum_i a_i^p \nabla\big(\frac{\partial^p}{\partial s_i^p}\big)$$ but the term $\sum_i a_i^p \nabla\big(\frac{\partial^p}{\partial s_i^p}\big)$ disappears in the proof and I don't see why?

 2. At the end of the proof, it looks like we use the fact that $\frac{\partial}{\partial s_i}$ and $\frac{\partial}{\partial s_j}$ commute, but why is that true?

Thank you for your help.

  [1]: http://www.numdam.org/article/PMIHES_1970__39__175_0.pdf
  [2]: https://i.sstatic.net/Z0xcn.png