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Sometimes a mathematical result is initially proved with a very long and convoluted argument, but later after the proof and truth is well-understood, a better, simpler proof of the same result will become apparent.

Are there any examples of problems which had huge first-proofs, but later insight yielded extremely short and elegant proofs?

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    It would make more sense to ask for proofs that have not been simplified over the years. – 2017-01-05

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Irrationality of $\pi$

The proof given by Lambert is long but Niven's proof is just a paragraph long.