That proof of your admission to the United States for permanent residency means you need to show the Green Card or some other similar proof (for example foreign passport with permanent resident visa stamp). I understand that it is tough to rebuild your identity when you lose the originating documents (like passport, naturalization certificate). In particular USCIS should be much more efficient and fast in providing a replacement certificate. However, it is not an easy problem for them to solve, as I am sure many people try to find vulnerabilities in the system to obtain or provide others with ill gotten proofs of citizenship. Whole thing stresses the importance of safeguarding important documents and spreading out proof of citizenship (passport book/card/certificate) at several locations.