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Because of the 0th Principle of Thermodynamics it is possible to say that being in thermal equilibrium is an equivalence relation. Therefore there exists a function $F(x_1,x_2)$, where $x_1,x_2$ are the thermodynamical coordinates, such that two systems are in thermal equilibrium if and only if their values of $F$ are the same. Later on, for a given $x$ a new function (the empirical temperature) $\theta(x_2)=F(x,x_2)$ is defined and it is assumed to be of the form $ax_2 +b$.

I don't see why, I mean most functions are not affine. And by defining $\theta$ in such a way it is already assumed that $F(x,x_2) \neq F(x,y_2)$ whenever $x_2 \neq y_2$, that is, if two identical systems share the value of one of their coordinates they can only be in equilibrium when the second one is also equal.

So why is it done so? Isn't generality lost in this way?

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