exact differential proof

Question

I was looking at this proof https://proofwiki.org/wiki/Solution_to_Exact_Differential_Equation which is the same proof in my differential equations book. The part I do not understand is how they were allowed to commute the partial derivatives in this step $$\frac{\delta N}{\delta x}-\frac{\delta ^2}{\delta x \delta y}\int M(x,y)dx$$ $$ \frac{\delta N}{\delta x}-\frac{\delta ^2}{\delta y \delta x}\int M(x,y)dx$$ $$\frac{\delta N}{\delta x}-\frac{\delta M}{\delta y}=0$$

Answer 1

Clairaut's theorem is used in the proof so that you may interchange partial differential operators $\dfrac{\partial}{\partial x}$ and $\dfrac{\partial}{\partial y}$.

Clairaut's theorem says the following: let $f$ be a function defined on a domain $D\subseteq \mathbb{R}^2$ and let $\mathbf{a}\in D$. If $f_{xy}$ and $f_{yx}$ are continuous on $D$, then $f_{xy}(\mathbf{a})=f_{yx}(\mathbf{a})$.