Why does thermal equilibrium between two systems $A$ and $C$ imply the constraint $f_{AC}(A1, A2, · · · ; C1, C2, · · ·) = 0$?

I am trying to understand the mathematical formulation of the zeroth law of thermodynamics. I get that if a system A is in equilibrium with a system C and a system B is in equilibrium with the same system C, then A and B are in equilibrium with each other, but i can't understand this conditional statement in terms of coordinates of a state function.

I have 3 problems with this "proof":

1. Why does thermal equilibrium between two systems A and C imply the constraint fAC(A1, A2, · · · ; C1, C2, · · ·) = 0.? Wouldn't another constraint different that 0 also imply that a change in A1 must be accompanied by some changes in {A2, · · · ; C1, C2, · · ·} to maintain equilibrium?
2. How do we know the equations can be solved for C1, as stated in 1.3 and what is the relationship between F and f?
3. Why does cancelling the coordinates of C in 1.4 makes us introduce another function, Θ, different from f and F?
• the title should be changed to something better describing the body of the question. Mar 12, 2023 at 15:42
• 1. Any constraint of the form $f(\vec{x}) = a\neq0$ can be recast as $\tilde{f}(\vec{x}) = f(\vec{x}) - a = 0$. Mar 13, 2023 at 4:22