In this paper, we present a formal \eventB model of the Mechanical Lung Ventilator (\MLV), the case study provided by the ABZ'24 conference. This system aims at helping patients maintain good breathing by providing mechanical ventilation. For this purpose, two modes are possible: \textit{Pressure Controlled Ventilation} (\PCV) and \textit{Pressure Support Ventilation} (\PSV). In the former mode, respiratory cycles are completely defined by the patient that is able to start breathing on its own. In the latter mode, the respiratory cycle is constant and controlled by the ventilator. Let us note that it is possible to move from a given mode to the other depending on the breathing capabilities of the patient under ventilation. In this paper, we illustrate the use of a correct-by-construction approach, the \eventB formal method and its refinement process, for the formal modeling and the verification of such a complex and critical system. The development of the formal models has been achieved under the \rodin platform that provides us with automatic and interactive provers used to verify the correctness of the models. We have also validated the built \eventB models using the \ProB animator/model checker.