The proof of this follows from the word problem as well.

A second proof of the theorem follows from a more general result published by Reich.

The proof of the theorem now follows: the handle chain complex is exact since .

The proof announced by the four initial authors followed .

This proof follows the same lines as the above proof.

A simpler more illustrative proof is as follows, although it applies only in the discrete case.

The proof follows exactly the same steps, except that the two matrix elements are no longer complex conjugates.

The proof follows by observing that in the Prüfer sequence number appears exactly times.

Formal proofs usually follow a different approach which involve characteristic functions.

The proof of correctness follows from choice of parameters and some probability analysis.