TITLE: Examples of theorems with proofs that have dramatically improved over time
QUESTION [95 upvotes]: I am looking for examples of theorems that may have originally had a clunky, or rather technical, or in some way non-illuminating proof, but that eventually came to have a proof that people consider to be particularly nice.  In other words, I'm looking for examples of theorems for which have some early proof for which you'd say "ok that works but I'm sure this could be improved", and then some later proof for which you'd say "YES! That is exactly how you should do it!"
Thanks in advance.
A sister question: Examples of major theorems with very hard proofs that have NOT dramatically improved over time

REPLY [2 votes]: The four color theorem’s first proof by Appel and Haken, which actually contained mistakes, was proved again much more succinctly and successfully by Robertson, Sanders, Seymour, and Thomas (see for instance here).  As far as I know, there is no better proof as yet, and unlikely there will ever be one, but who knows.