The "proof" would consist of the submittal of a reasonably detailed estimate by a qualified person of the cost of all the work to be done to the structure, and a professional appraisal of the current value of the structure.
The estimate must include all work to be done to any addition to the existing house, and any work to be done in the existing house. In other words, all the work covered by your anticipated building permit, except for the cost of permit, license, architect, design, or other fees, and the cost of any improvements not part of the structure, such as pools, landscaping, and detached structures.
The cost estimate must also include minor work such as painting and flooring, built-in appliances, and the fair market value of any free or donated labor or materials.
The appraisal must be by a practicing appraiser (of a stature acceptable to lenders) of the current market value of the structure to be improved. This value cannot include the value of the land or any improvements not a direct part of the structure. For instance, it cannot include the value of pools, guest houses or detached garages (although an attached garage can be included).
The documents will be compared to the submitted building plans and reviewed for reasonableness by Public Works Engineering to determine whether the figures supplied justify the conclusion that the proposed work is not a "substantial improvement". If not, that conclusion will be documented, and the applicant will be able to proceed without needing to meet the flood zone requirements.