I would say it's stylistically questionable to include the proof that it's in NP under the proof of NP-hardness. They could have written it in a paragraph befor

I would say it's stylistically questionable to include the proof that it's in NP under the proof of NP-hardness. They could have written it in a paragraph before. However it could still be true that this result was known beforehand (and judging by the length of the proof, it doesnt seem too difficult). So their main result would still be the NP-hardness, which justifies the name of the paper.