The Secretary of the Interior may, in his discretion, in addition to the extensions authorized by sections 333 to 335 of this title or other law existing prior to February 25, 1925, grant to any entryman under the desert-land laws of the United States a further extension of time of not to exceed three years within which to make final proof: Provided, That such entryman shall, by his corroborated affidavit, filed in the land office of the district where such land is located, show to the satisfaction of the Secretary that because of unavoidable delay in the construction of the irrigation works intended to convey water to the land embraced in his entry, he is, without fault on his part, unable to make proof of the reclamation and cultivation of said lands as required by law within the time limited therefor: And provided further, That the entryman, his heirs, or his duly qualified assignee, has in good faith complied with the requirements of law as to yearly expenditures and proof thereof, and shall show, under rules and regulations to be prescribed by the Secretary of the Interior, that there is a reasonable prospect that if the extension is granted he will be able to make the final proof of reclamation, irrigation, and cultivation required by law.