In extant literature, deterioration dependence among components can be modelled as inherent dependence and induced dependence. We find that the two types of dependence may co-exist and interact with each other in one multi-component system. We refer to this phenomenon as fault propagation. In practice, a fault induced by the malfunction of a non-critical component may further propagate through the dependence amongst critical components. Such fault propagation scenario happens in industrial assets or systems (bridge deck, and heat exchanging system). In this paper, a multi-layered vector-valued continuous-time Markov chain is developed to capture the characteristics of fault propagation. To obtain the mathematical tractability, we derive a partitioning rule to aggregate states with the same characteristics while keeping the overall aging behaviour of the multi-component system. Although the detailed information of components is masked by aggregated states, lumpability is attainable with the partitioning rule. It means that the aggregated process is stochastically equivalent to the original one and retains the Markov property. We apply this model on a heat exchanging system in oil refinery company. The results show that fault propagation has a more significant impact on the system's lifetime comparing with inherent dependence and induced dependence.