This article builds upon the formalization of the deterministic part of the original Kyber algorithms (see the CRYSTALS-Kyber entry). The correctness proof is expanded to cover both the deterministic part from the previous formalization and the probabilistic part of randomly chosen inputs. Indeed, the probabilistic version of the delta-correctness in the original paper was flawed and could only be formalized for a modified delta'. The authors of Kyber also remarked, that the security proof against indistinguishability under chosen plaintext attack (IND-CPA) does not hold for the original version of Kyber. Thus, the newer version (round 2 and 3 of NIST standarisation process) was formalized as well, including the adapted deterministic and probabilistic correctness theorems. Moreover, the IND-CPA security proof against the new version of Kyber has been verified using the CryptHOL library. Since the new version also included a change of parameters, the Kyber algorithms have been instantiated with the new parameter set as well.