Mate finishes this chapter - he's succeeded (5)

I believe the answer is:

chess

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'mate finishes this' is the definition.

'he's succeeded' is the wordplay.
'he's' becomes 'ches'.
'succeeded' becomes 's' (genealogical abbreviation).
'ches'+'s'='CHESS'

'chapter' is the link.
I am not very happy about this link. It may belong to another bit of the clue.

Can you help me to learn more?

(Other definitions for chess that I've seen before include "that has over thirty men on board" , "Game with kings and queens" , "Pure-skill board game" , "form of battle" , "Musical - board game" .)

I've seen this clue in The Telegraph.

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