A very specialized form of a theorem begins with the statement â€śThe following are equivalentâ€¦,â€ť which is then followed by a list of statements. Informally, this lead-in sometimes gets abbreviated by â€śTFAE.â€ť This formulation means that any two of the statements on the list can be connected with an â€śif and only ifâ€ť to form a theorem. So if the list has $$n$$ statements then, there are $$\tfrac{n(n-1)}{2}$$ possible equivalences that can be constructed (and are claimed to be true).
Suppose a theorem of this form has statements denoted as $$A\text{,}$$ $$B\text{,}$$ $$C\text{,}$$ â€¦, $$Z\text{.}$$ To prove the entire theorem, we can prove $$A\Rightarrow B\text{,}$$ $$B\Rightarrow C\text{,}$$ $$C\Rightarrow D\text{,}$$ â€¦, $$Y\Rightarrow Z$$ and finally, $$Z\Rightarrow A\text{.}$$ This circular chain of $$n$$ equivalences would allow us, logically, if not practically, to form any one of the $$\tfrac{n(n-1)}{2}$$ possible equivalences by chasing the equivalences around the circle as far as required.