Put lips thus to separate (5,2)

I believe the answer is:

split up

I'm a little stuck... Click here to teach me more about this clue!

'separate' is the definition.
(I know that separate can be written as split up)

'put lips thus' is the wordplay.
'put lips' anagrams to 'SPLIT UP'.
But, I am unsure how this is indicated.

'to' acts as a link.

Can you help me to learn more?

(Other definitions for split up that I've seen before include "Take apart" , "End a relationship" , "Divorce" , "Parted company" , "Separate" .)

I've seen this clue in The Irish Independent.

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