The robustness and correctness of SAT solvers are receiving more and more attention. In recent SAT competitions, a proof of unsatisfiability emitted by SAT solvers must be checked. So far, no proof checker has been efficient for every case. In the SAT competition 2016, some proofs were not verified within 20000 seconds. For this reason, we decided to develop a more efficient proof checker called TreeRat. This new checker uses a window shifting technique to improve the level of efficiency at which it verifies proofs of unsatisfiability. At the same time, we suggest that tree-search-based SAT solvers should use an equivalent relation encoding to emit proofs of subproblems. In our experiments, TreeRat was able to verify almost all proofs within 20000 seconds. On this point, TreeRat is shown to be superior to gratgen, which is an improved version of DRAT-trim. Also, in most cases, TreeRat is faster than gratgen. Like DRAT-trim, TreeRat can output also trace dependency graphs. Its output format is LRAT. The correctness of TreeRat can be ensured by checking its LRAT output.