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Within the Schengen Area, there are no border checks between countries. This is actually by law; a Schengen country can't introduce systematic passport checks at the border with another Schengen country even if they wanted to.
So you should have no trouble moving around within the Schengen area. If you choose to leave the Schengen area of course, you would likely need a new visa to get back in (the documentation you have might work to get you back in, but I wouldn't count on it). If an official such as a police officer asks you for your papers, then the documentation you have should be fine.