Crossword Genius

Having had one’s fill at tea, is lively (7)

Ross

I believe the answer is:

satiate

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'having had one's fill' is the definition.
I can't judge whether this defines the answer.

'at tea is lively' is the wordplay.
'lively' is an anagram indicator (letters get livened up or moved around).
'at'+'tea'+'is'='atteais'
'atteais' is an anagram of 'SATIATE'.

Can you help me to learn more?

(Other definitions for satiate that I've seen before include "content" , "stuff" , "Have more than enough" , "Fill to satisfaction" , "Satisfy fully, cloy with excess" .)

I've seen this clue in The Times.
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