All bills, accounts, and claims on account of the Library and its maintenance, except such as are incurred by the trustee commissioners on account of funds under their control, shall be approved by the City Council or, if such authority has been delegated to it, by six or more members of the Board of Library Commissioners and the Mayor's warrant shall be drawn upon the Chief Administrative Officer for the payment of such bills, accounts, and claims. By such date requested annually by the Mayor, the Library staff shall prepare and the Board of Library Commissioners shall approve a recommended Library budget for the ensuing fiscal year. Such proposed budget after being presented to the Mayor may be revised as he or she believes appropriate and ultimately submitted to the City Council for its approval in accordance with the procedures set forth in section 157 of this charter.