It is well known that between a prime $p$ and $p^2$ always exist a prime , but what is the shortest proof on that(by elementary methods or not) ?

(One can say that we can have it as a collorary of Bertrand's postulate,but it is a stronger result).

I ask it as an example , are there any results that their first proof was large comparing to some proof that someone found later? or

of results that their only known proof/proofs is/are a collorary of the proof of something stronger( maybe the one that i give is not an example for these)