That all cases of appeal or writ of error heretofore prosecuted and now pending in the Supreme Court of the United States upon any record from the Supreme Court of either of the territories mentioned in this act, or that may hereafter lawfully be prosecuted upon any record from either of said courts, may be heard and determined by said Supreme Court of the United States. And the mandate of execution or of further proceedings shall be directed by the Supreme Court of the United States to the circuit or district court hereby established within the state succeeding the territory from which such record is or may be pending, or to the Supreme Court of such state, as the nature of the case may require; provided, that the mandate of execution or of further proceedings shall, in cases arising in the territory of Dakota, be directed by the Supreme Court of the United States to the circuit or district court of the district of South Dakota, or to the Supreme Court of the state of South Dakota, or to the circuit or district court of the district of North Dakota, or to the Supreme Court of the state of North Dakota, or to the Supreme Court of the territory of North Dakota, as the nature of the case may require. And each of the circuit, district, and state courts, herein named, shall, respectively, be the successor of the Supreme Court of the territory, as to all such cases arising within the limits embraced within the jurisdiction of such courts respectively, with full power to proceed with the same, and award mesne or final process therein; and that from all judgments and decrees of the Supreme Court of either of the territories mentioned in this act, in any case arising within the limits of any of the proposed states prior to admission, the parties to such judgment shall have the same right to prosecute appeals and writs of error to the Supreme Court of the United States as they shall have had by law prior to the admission of said state into the union.