If the appraisers find that lands or other property not embraced within the boundaries of the district will be necessary for or affected by, the proposed improvement, or should be included in the district, they shall appraise the benefits and damages to such land and shall file notice, in the court of the appraisal which they have made upon the lands beyond the boundaries of the district, and to land which in their opinion should be included in the district. The appraisers shall report to the court any lands which in their opinion should be eliminated from the district.

Provided, that the court in determining the properties benefited by the proposed project or projects and in assessing the benefits derived therefrom, and in the matter of assessments for the construction and maintenance thereof shall upon proper petition and showing, cause to be brought in and made parties to the action all property owners found to be benefited throughout the course of the stream below the points of diversion of water, or to the crossing of the state line by any such stream, whether such property be originally included in the petition or not, such action to be taken by the court after service of summons as required in civil actions upon such property owners of record.