transformation of independent random variables

Here's how I would do it:

If $X_1$ and $X_2$ are independent then: $f_{X_1,X_2}(x_1,x_1)=f_{X_1}(x_1)f_{X_2}(x_2)$ let $Y_1=g(x_1)$ and $Y_2=h(x_2)$ be invertible functions then: $X_1=g^{-1}(y_1)$ and $X_2=h^{-1}(y_2)$ then we have: $\frac{\partial X_1}{\partial y_1}=\frac{dg^{-1}(y_1)}{dy_1}$ , $\frac{\partial X_1}{\partial y_2}=0$ , $\frac{\partial X_2}{\partial y_1}=0$ , $\frac{\partial X_2}{\partial y_2}=\frac{dh^{-1}(y_2)}{dy_2}$ . Such that the Jacobian $|J|=|\left(\frac{dg^{-1}(y_1)}{dy_1}\right)\left(\frac{dh^{-1}(y_2)}{dy_2}\right)|$ So then $f_{Y_1,Y_2}(y_1,y_2)=f_{X_1,X_1}(g^{-1}(y_1),h^{-1}(y_2))|J|=\left(f_{X_1}(g^{-1}(y_1))(|\frac{dg^{-1}(y_1)}{dy_1}|)\right)\left(f_{X_2}(h^{-1}(y_2))(|\frac{dh^{-1}(y_2)}{dy_2}|)\right)$ Which is quite clearly separable and therefore $h(x_1)$ and $g(x_2)$ are independent.

The only problem I could see would be assuming that the transformations are invertible.