That in respect to all cases, proceedings, and matters now pending in the Supreme or district Courts of either of the territories mentioned in this act at the time of the admission into the union of either of the states mentioned in this act, and arising within the limits of any such state, whereof the circuit or district courts by this act established might have had jurisdiction under the laws of the United States had such courts existed at the time of the commencement of such cases, the said circuit and district courts, respectively, shall be the successors of said Supreme and district Courts of said territory; and in respect to all other cases, proceedings and matters pending in the Supreme or district Courts of any of the territories mentioned in this act at the time of the admission of such territory into the union, arising within the limits of said proposed state, the courts established by such state shall, respectively, be the successors of said Supreme and district territorial Courts; and all the files, records, indictments, and proceedings relating to any such cases, shall be transferred to such circuit, district, and state courts, respectively, and the same shall be proceeded with therein in due courts of law; but no writ, action, indictment, cause, or proceeding now pending, or that prior to the admission of any of the states mentioned in this act, shall be pending in any territorial court in any of the territories mentioned in this act, shall abate by the admission of any such state into the union, but the same shall be transferred and proceeded with, in the proper United States circuit, district, or state court, as the case may be; provided, however, that in all civil actions, causes, and proceedings, in which the United States is not a party, transfers shall not be made to the circuit and district courts of the United States except upon written request of one of the parties to such action or proceeding filed in the proper court; and in the absence of such request, such cases shall be proceeded with in the proper state courts.