This question refers to the following set of lecture notes: http://ocw.mit.edu/courses/physics/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2013/lecture-notes/MIT8_333F13_Lec1.pdf
The notes state the zeroth law of thermodynamics as follows: "If two systems, A and B, are separately in equilibrium with a third system C, then they are also in equilibrium with one another." The notes then assert that despite the simplicity of the zeroth law it implies the existence of an "empirical temperature, such that systems in equilibrium are at the same temperature." To prove this assertion, the following argument is made.
Equilibrium between systems A and C implies a constraint on the thermodynamic coordinates describing A and C. Mathematically, the constraint can be given as follows:
Similarly, equilibrium between B and C can be mathematically stated as follows:
These equations can be solved for $C_1$ and then recast to
where the $F$ equations have a different functional form than the $f$ equations. Since C is simultaneously in equilibrium with A and B we have
The zeroth law implies that A and B are also in equilibrium with each other, therefore we must have
Now we get to the part I didn't really understand. From here, the notes say "Therefore it must be possible to simplify [the second-to-last equation] by cancelling the coordinates of C. Hence, the condition [given in the previous equation] for equilibrium of A and B must be expressible as
$$\Phi_A(A_1, A_2,...)=\Phi_B(B_1, B_2,...).$$
I needed a little help with the last step. I attempted to answer my own question, which is given below, but I'm not sure the answer is correct. Namely, I'm not convinced that the decomposition I've given for my function is the most general decomposition. In fact, I'm fairly convinced that it's not the most general decomposition. However, I don't see what alternative argument can be made, so even if I'm wrong in the particulars, I do believe the general form of my argument is correct. Comments would be appreciated, though.