Hybrid systems are complex systems where a software controller interacts with a physical environment, usually named a plant, through sensors and actuators. The specification and design of such systems usually rely on the description of both continuous and discrete behaviours. From complex embedded systems to autonomous vehicles, these systems became quite common, including in safety critical domains. However, their formal verification and validation as a whole is still a challenge. To address this challenge, this article contributes to the definition of a reusable and tool supported formal framework handling the design and verification of hybrid system models that integrate both discrete (the controller part) and continuous (the plant part) behaviours. This framework includes the development of a process for defining a class of basic theories and developing domain theories and then the use of these theories to develop a generic model and system-specific models. To realise this framework, we present a formal proof tool chain, based on the Event-B correct-by-construction method and its integrated development environment Rodin, to develop a set of theories, a generic model, proof processes, and the required properties for designing hybrid systems in Event-B. Our approach relies on hybrid automata as basic models for such systems. Discrete and continuous variables model system states and behaviours are given using discrete state changes and continuous evolution following a differential equation. The proposed approach is based on refinement and proof using the Event-B method and the Rodin toolset. Two case studies borrowed from the literature are used to illustrate our approach. An assessment of the proposed approach is provided for evaluating its extensibility, effectiveness, scalability, and usability.