Proves smarter than fools at Brasenose? (7)
I believe the answer is:
outwits
'proves smarter than' is the definition.
(to outwit someone is to be smarter than them)
I don't understand how the rest of the clue works.
(Other definitions for outwits that I've seen before include "kids" , "Gulls" , "Is too smart for" .)