Outdo, transcend (7)
I believe the answer is:
surpass
I believe this is a double definition.
'outdo' is the first definition.
(I know that outdo can be written as surpass)
'transcend' is the second definition.
(I know that transcend is a more specific form of surpass)
(Other definitions for surpass that I've seen before include "eclipse visible" , "Outdo" , "supply best" , "Transcend" , "Go one better" .)