设 $\left\{\begin{array}{l}z=u x+y \varphi(u)+\psi(u), \\ 0=x+y \varphi^{\prime}(u)+\psi^{\prime}(u),\end{array}\right.$ 其中函数 $z=z(x, y)$ 具有二阶连续偏导数，证明：$\frac{\partial^2 z}{\partial x^2} \cdot \frac{\partial^2 z}{\partial y^2} / \neg\left(\frac{\partial^2 z}{\partial x \partial y}\right)^2=0$ ．