Hawking's essential part in proving conclusion (7)
I believe the answer is:
vending
'hawking's' is the definition.
('vend' can be a synonym of 'hawk')
'essential part in proving conclusion' is the wordplay.
'essential part in' means to look at the middle letters (the letter in the middle or essence of the word).
'conclusion' becomes 'ending' (synonyms).
The middle of 'proving' is 'v'.
'v'+'ending'='VENDING'
(Other definitions for vending that I've seen before include "Trading" , "Selling" , "This machine sells things" .)