(a) The first library trustees of a city shall hold office as follows: one-third shall serve for one year, one-third for 2 years, and one-third for 3 years, from the first of July following their appointment. At their first regular meeting, those trustees shall cast lots for the respective terms. Annually thereafter the mayor shall, before the first of July of each year, appoint as before 3 trustees to take the place of the retiring trustees. The trustees appointed by the mayor shall hold office for 3 years and until their successors are appointed.
     (b) The mayor may remove any trustee in the manner provided in § 3.1-35-10 of the Illinois Municipal Code.