We have a two-dimensional smooth function $f(x_1,x_2)$. $x,y,z$ are two dimensional vectors. We know that:
1) $x_1=y_1$ and $f(x)=f(y)$ implies that $\exists v\in\mathbb R^2$, $\forall a\in\mathbb R,$ $f(x+av)=f(y+av)$.
2) $x_2=z_2$ and $f(x)=f(z)$ implies that $\exists u\in\mathbb R^2$, $\forall a\in\mathbb R,$ $f(x+au)=f(z+au)$.
Taking the two conditions as given, we are asked to show that whether there exists a one-to-one change of coordinates ($w=h(x)$),
such that $g(w)=g(h(x))=f(x)$ and $\forall w_1,w'_1,w_2,w'_2\in\mathbb R$ we have $g(w_1,w_2)=g(w_1,w'_2)$ and $g(w_1,w_2)=g(w'_1,w_2)$?
The new condition asked is basically saying that in the new system, the new $v$ and $u$ are perpendicular to each other and they are parallel to the axis. To be more specific, for $g$, $v=(1,0)$ and $u=(0,1)$.
At first, I think it is very simple: we just need to make $v$ and $u$ perpendicular to each other. However, now I feel that it is impossible to do so, because changing the coordinate would also change the direction of $(x_1,x_2)$ and $(x_1,y_2)$, and the line passing those two points would no longer parallel to the axis after changing the coordinates. For example, if we directly use $(u,v)$ as the basis vectors, the required conditions won't be satisfied.
I am not sure if my explanation is clear enough.