The current proof of the probabilistically checkable proofs (PCP) theorem (i.e., NP = PCP(log, O(1))) is very complicated. One source of difficulty is the technically involved analysis of low-degree tests. Here, we refer to the difficulty of obtaining strong results regarding low-degree tests; namely, results of the type obtained and used by Arora and Safra [J. ACM, 45 (1998), pp. 70-122] and Arora et al. [J. ACM, 45 (1998), pp. 501-555]. In this paper, we eliminate the need to obtain such strong results on low-degree tests when proving the PCP theorem. Although we do not remove the need for low-degree tests altogether, using our results it is now possible to prove the PCP theorem using a simpler analysis of low-degree tests (which yields weaker bounds). In other words, we replace the strong algebraic analysis of low-degree tests presented by Arora and Safra and Arora et al. by a combinatorial lemma (which does not refer to low-degree tests or polynomials.