Publishing an article after a book? If I first publish an article, afterward I may publish a book containing materials from the article.
What's about the reverse: If I first publish a book does it make sense to publish its fragment as an article AFTERWARD?

"If I first publish a book does it make sense to publish its fragment as an article AFTERWARD?"
Sure, why not? You might write a book for one audience, and very usefully re-publish a fragment in the form of a journal article for another audience. I have done this with some stuff buried near the end of a long textbook book aimed at beginning grad students, which colleagues are unlikely to read, but which I thought (and the journal editor thought) was interesting enough for stand-alone publication in one of the journals. And I've done things the other way around too, taken something from a book aimed at a professional audience and reworked it as an article in a publication with a more general readership. [This was techie philosophy rather than straight maths, but I'd have thought the same principles would apply.]
So I guess you'd need to think about whether you would or would not be reaching significantly different audiences. (Of course, re-publication for the sake of padding out a CV is not a good idea!) 
