Every reassessment shall be made upon the property which has been or will be specially benefited by the local improvement and may be made upon property whether or not it abuts upon, is adjacent to, or proximate to the improvement or was included in the original assessment district.

Property not included in the original improvement district when so assessed shall become a part of the improvement district and all payments of assessments shall be paid into and become part of the local improvement fund to pay for the improvement.

Property in the original local improvement district which is excluded in reassessment need not be entered upon the assessment roll.

Every reassessment must be based upon the actual cost of the improvement at the time of its completion.