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?

I am not an expert on publishing.
It of course makes good sense to publish articles and then later collect all the research and publish a book covering the material. Often articles covering the same topic can change in notation and so collecting the articles and editing them into a book form can make things a lot more readable.
Could you do the reverse? You might. But from what I understand you would probably need something new. I don't think that you will get something published as an article if the main results of the article are already contained in a book. However, you might be able to do so if you add extra research. 
As a sidenote: You might be able to write a survey article. But from what I have seen, these tend to be trying to collect material from a lot of articles and try to make things easier to read. They can function as a "gentle" introduction to a topic so that a new researcher doesn't have  to jump in the deep end starting with the original articles. But again, this doesn't (from what I understand) cover taking well presented material from a book and making an article out of it.
