Eliminating arguments from a function

[thegreenlaser]

My issue comes from thermodynamics (https://www.physicsforums.com/threads/zeroth-law-of-thermodynamics-and-empirical-temperature.774255/), but I guess this is really a math problem rather than a physics problem:

Let $\boldsymbol{x} \in \mathbb{R}^m$, $\boldsymbol{y} \in \mathbb{R}^n$, $\boldsymbol{z} \in \mathbb{R}^p$.

Given
$$f_1 (\boldsymbol{x}, \boldsymbol{z} ) = f_2 (\boldsymbol{y}, \boldsymbol{z})$$
and
$$f_3(\boldsymbol{x}, \boldsymbol{y}) = 0,$$
how can we show that there exist functions $g_1, g_2$ such that
$$g_1(\boldsymbol{x}) = g_2(\boldsymbol{y})$$

(i.e., $f_3 = 0$ means that $f_1$ and $f_2$ can be reduced so they're independent of $\boldsymbol{z}$)

In my physics class, we were given a sort of "hand-waving" argument, but I'm having trouble finding a more rigorous explanation. Can anyone help me out?

 

[mathman]

I have trouble understanding the question, but I'll take a stab at it. The last comment says f1 and f2 can be reduced to be independent of z. In that case wouldn't g1 and g2 simply be these reduced versions of f1 and f2?