Puffed up old French president’s losing papers (7)

I believe the answer is:

pompous

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'puffed up' is the definition.
(I've seen this in another clue)

I don't understand how the rest of the clue works.

(Other definitions for pompous that I've seen before include "Proud" , "Grandiloquent" , "Puffed up with vanity" , "Affectedly grand" , "Bombastic" .)

I've seen this clue in The Times.

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