$11\mathrm{.}2$ Theorem. Suppose that $V$ is open in $R^2,$ that $(a,b)inV,$ and that $f$:$VR.$ If $f$ is $C^1$ on $V,$ and if one of the mixed second partial derivatives of $f$ exists on $V$ and is continuous at the point $(a,b),$ then the other mixed second partial derivative exists at $(a,b)$ and

$\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{y}\text{d}\text{e}\text{l}\text{x}}(a,b)=\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}}(a,b)$

NOTE: These hypotheses are met if $fin{C}^2(V).$

prove that this theorem