To give a conclusive demonstration, runs over doctor in Leith? (5,4)

I believe the answer is:

prove true

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'to give a conclusive demonstration' is the definition.
I don't know anything about this answer so I cannot judge whether this works.

'runs over doctor in leith?' is the wordplay.
'runs over' becomes 'ro' (this could be a standard abbreviation which I don't know about).
'doctor' becomes 'vet' (vetting is a kind of doctoring).
'in' is an insertion indicator.
'leith?' becomes 'prue' (I've seen this before).
'ro'+'vet'='rovet'
'rovet' placed into 'prue' is 'PROVE TRUE'.

Can you help me to learn more?

(Another definition for prove true that I've seen is " Establish".)

I've seen this clue in The Independent.

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