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