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why formal proofs are not widely used? sometimes non formal proofs are cumbersome. are there any "important" theorems that have been proved formally

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Because are very difficult to write:

To write a correct and complete formal proof one must have a complete command of both maths and logic.

Then a lot of work on all fields of mathematics .

Anyway the only book as far as i know existing which is full of formal proofs is one titled:

FORMAL PROOFS IN MATHS In amazon.com

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