Michigan Compiled Laws, Chapter 397. Libraries § 397.173c

(a) The district library shall continue to be recognized for all purposes as a lawfully established district library that may continue to exercise all powers, duties, functions, and responsibilities, including the levy of taxes authorized by the electors of the district, as provided in the agreement, this act, and any other applicable law.

(b) If 2 or more participating municipalities remain in the district library, those municipalities shall amend the agreement to reflect the disincorporation, annexation, consolidation, or merger.  If only 1 participating municipality remains in the district library after the disincorporation, annexation, consolidation, or merger, or if all participating municipalities are merged or consolidated into 1 municipality, that municipality shall assume the powers, duties, functions, and responsibilities of the former participating municipality or municipalities without amendment of the agreement.

(c) The terms of the district library board members shall continue as provided in the agreement and this act, except that if the district library board members are appointed, the legislative body of a single remaining participating municipality or a consolidated or merged municipality shall appoint district library board members as the terms of district library board members expire.