Section 13. Upon the entry of the appeal, the supposed creditor shall file a written statement of his claim, setting forth briefly and distinctly all the material facts necessary in a declaration for the same cause of action; and like proceedings shall thereupon be had in the pleadings, trial and determination of the cause as in an action at law; but no execution shall be awarded against the executor or administrator for a debt found due to the claimant. The final judgment shall be conclusive, and the list of debts allowed shall be altered by the probate court, if necessary, to conform thereto.