I have that $x^{*}(w,z)$ and $y^{*}(w,z)$ is the implicit solution to a the system $F(x^{*}, y^{*},w,z) = 0$ and $H(x^{*}, y^{*},w,z) = 0$. Using the implicit function theorem, I can prove that $\frac{\partial x^{*}(w,z)}{\partial w} > 0$, $\frac{\partial x^{*}(w,z)}{\partial z}<0$, $\frac{\partial y^{*}(w,z)}{\partial w} < 0$ and $\frac{\partial y^{*}(w,z)}{\partial z} > 0$. Can I then claim that $\frac{\partial^2 x^{*}(w,z)}{\partial w \partial z} < 0$? Intuitively I think this must be the case, but I cannot provide a formal proof.
Thanks!