Upon a petition and bond being filed by one or more landowners, either within or without the boundaries of a district, and like proceedings being had as in the case of the original establishment and construction of a system of improvement, the county commissioners may declare any system of improvement or any part thereof, abandoned or may strike from the district lands no longer benefited or served thereby, or they may cause any system of improvement to be altered, reduced, enlarged, added to or in any other manner bettered or improved, either within or without the district, and to effect such subsequent improvements, may exercise any of the powers which are in this chapter, or may be hereafter conferred upon such districts. But the striking of any lands from a district shall not in any way affect any assessment theretofore levied against such lands. When such improvements shall have been completed the costs thereof shall be apportioned and assessed against the lands benefited thereby in the manner hereinbefore provided for such apportionment and assessment in the case of original proceedings. New lands assessed for any such improvement shall become a part of such district. The construction and maintenance of any such new improvement, unless let by contract by the board of county commissioners, shall be under the direction of the board of supervisors of the district in which they are made or to which said improvement is added. The lands assessed for such new improvements, of less than the entire district, shall be designated, alphabetically, “subdistrict . . . . . . of . . . . . . improvement district No. . . . . .”