For example, leaving East Ridge on a horse (7)
I believe the answer is:
astride
'on a horse' is the definition.
(I've seen this in another clue)
'for example leaving east ridge' is the wordplay.
'for example' becomes 'e.g.' (short for 'exempli gratia', 'for example' in Latin).
'leaving' is a deletion indicator.
'east'+'ridge'='eastridge'
'eastridge' with 'eg' taken away is 'ASTRIDE'.
(Other definitions for astride that I've seen before include "Straddling (horse)" , "On the back of" , "With feet apart" , "Passing across" , "Not side saddle" .)