Solemnly declare as almost cut-rate (10)

I believe the answer is:

asseverate

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'solemnly declare' is the definition.
(I've seen this before)

'as almost cut-rate' is the wordplay.
'almost cut' becomes 'seve' (I am not sure about this - if you are sure you should believe this answer much more).
'as'+'seve'+'rate'='ASSEVERATE'

Can you help me to learn more?

(Other definitions for asseverate that I've seen before include "Swear" , "State categorically" , "Solemnly declare" , "Solemnly say" , "Claim" .)

I've seen this clue in The Irish Independent.

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