Any desert-land entryman whose desert-land entry has been embraced within the exterior limits of any land withdrawal or irrigation project under the Act of June 17, 1902, known as the reclamation Act, and who may have obtained a water supply for the land embraced in any such desert-land entry from the reclamation project by the purchase of a water-right certificate, may at any time after having complied with the provisions of the law applicable to such lands and upon proof of the cultivation and reclamation of the land to the extent required by the reclamation Act for homestead entrymen, submit proof of such compliance, which proof, if found regular and satisfactory, shall entitle the entryman to a patent and a final water-right certificate under the same terms and conditions as required of homestead entrymen under the Act entitled “An Act providing for patents on reclamation entries, and for other purposes, approved August ninth, nineteen hundred and twelve [43 U.S.C. 541 et seq.].” 1