Now we come back to the proof of the theorem.

The proof of the theorem, as it stood around 1985 or so, can be called first generation.

It is also the basis for the proof of Taylor's theorem.

You could give it a human proof of the theorem.

A proof of Beck's theorem can be given as follows.

Since then, multiple proofs of the theorem have been discovered.

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

This method can be used for a very compact proof of the theorem under discussion.

This outline should not be considered a rigorous proof of the theorem.

Later, the paper offers a proof of the theorem.