Basic Math.

Let $A,B,C \in \{0,1,\infty,\alpha\}$, be such that $g(\{A,B,C\})=\{0,1,\infty\}$. Define
\[
h(x)=h_{A,B,C}(x)=\frac{(x-A)(B-C)}{(x-C)(B-A)}.
\]
Then $h(\{A,B,C\})=\frac{1}{h}(\{A,B,C\})=\{0,1,\infty\}$.

My question is as follows: In any mapping, at all, if you have three elements mapping to the same thing those three things map to in the inverse, then all those three elements are the same, and then $\alpha$ cannot be apart of $A$, $B$, or $C$. amirite?

I'm still sick something awful. but this little bit tripped me up in the paper I'm reading, and I can't follow the proof without understanding this point.