Doubts in the proof of existence of empirical temperature as a result of zeroth law of thermodynamics
I am studying the mathematical proof of existence of empirical temperature online. I have studied the same in the post https://physics.stackexchange.com/q/229794/259524 .
<PERSON> law states that if A and C are in joint (mutual) equilibrium and simultaneously B and C are in joint equilibrium then A and B are also in joint equilibrium.
I have tried to write the proof properly with the help of above references. But I have doubts in various steps of proof.
Consider three systems $A,\;B$ and $C$ described by macroscopic coordinates $(A_1,...,A_n),\;(B_1,...B_k),\;(C_1,...,C_m)$. 749
If A and C are in joint equilibrium then $f_{AC}(A_1,...,A_n,C_1,...,C_m)=0\tag{1}$
Also if B and C are in joint equilibrium then $f_{BC}(B_1,...,B_k,C_1,...,C_m)=0\tag{2}$
$\text{From}\;(1)$, $\implies\;C_1=F_{AC}(A_1,...,A_n,C_2,...,C_m)\tag{3}$ $\text{Also from}\;(2),$ $\implies\;C_1=F_{BC}(B_1,...,B_k,C_2,...,C_m)\tag{4}$
As A and B are simultaneously in equilibrium with C, so from (3) and (4), $F_{AC}(A_1,...,A_n,C_2,...,C_m)=F_{BC}(B_1,...,B_k,C_2,...,C_m)\tag{5}$
As a result of zeroth law, A and B are also in joint equilibrium, so
$f_{AB}(A_1,...,A_n,B_1,...,B_k)=0\tag{6}$
So, equation (6) is independent of $C_i's$, so whatever is the value of $C_i's$, for some particular $A_i's$ and $B_i's$, equation (6) so also (5) continues to hold.
So by fixing $C_i's$ to $C_i^o$ in equation (5) $\forall\;i\in{{2,...,m}}$, define
$\Phi_A(A_1,...,A_n)\triangleq F_{AC}(A_1,...,A_n,C_2^o,...,C_m^o)\;\text{and}\ \Phi_B(B_1,...,B_k)\triangleq F_{BC}(B_1,...,B_k,C_2^o,...,C_m^o)\ C_1^o=\Phi_C(C_2^o,...,C_m^o)\ \text{As system C is also in equilibrium with itself(reflexivity), so $C_1^o$ is determined uniquely}\tag{7}$
So, $\Phi_A(A_1,...,A_n)=\Phi_B(B_1,...,B_k)=\Phi_C(C_2^o,...,C_m^o)\tag{8}$
That empirical temperature is defined as $\Phi(A_1,...,A_n)$ or $\Phi_B(B_1,...,B_k)$ or $\Phi_C(C_2^o,...,C_m^o)$
I have the following doubts while writing the proof - 1) In (3), we are considering $f_{AC}$ and $f_{BC}$ to be an explicit function in $C_1$. So, how we justify it for any general thermodynamic system, like the function $f_{AC}$ and $f_{BC}$ can be implicit in $C_1$ like we can't separate it like the way shown. So, in (3), are we considering only those macroscopic coordinates of C which are explicit or can be separated from $f_{AC}$ and $f_{BC}$? Or there is some different reasoning?
2) The empirical temperature is basically $\Phi(A_1,... 677