If a person was falsely convicted for a crime and served five years before being released as innocent, but then was discovered to have been guilty of another crime committed before their imprisonment, and given five years, would they still have to serve it, or would it cancel out?

For example, lets say that in December 2010 a person commits crime X but is falsely arrested for crime Y. They are tried and convicted for crime Y and maintain innocence but serve five years of their 10 year sentence. After five years new evidence comes about to show that the person is innocent of crime Y but by coincidence, they are proven to be guilty of crime X, and sentenced to 5 years. Do they get a new sentence or is it considered time served?

Extra question: if you are falsely convicted, do you get compensation?