One prince lying in state, not breathing out (10)

I believe the answer is:

inhalation

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'breathing out' is the definition.
'inhalation' can be an answer for 'breathing' (thesaurus). I am not certain of the 'out' bit.

'one prince lying in state not' is the wordplay.
'one' becomes 'i' (Roman numeral).
'prince' becomes 'hal' (Shakespearean nickname of Henry V when a prince).
'lying in' means one lot of letters goes inside another.
'state not' becomes 'nation' (both can mean a country. I am not sure about the 'not' bit.).
'hal' inserted within 'nation' is 'nhalation'.
'i'+'nhalation'='INHALATION'

Can you help me to learn more?

I've seen this clue in The Times.

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