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Subtyping relations for object-oriented formalisms describe relationships between classes which satisfy the substitutability requirement imposed on types and their subtypes. Subtyping relations have been proposed both for state-based and for behaviour-oriented specification formalisms. In this paper we state and proof correspondence results between these two forms of subtyping: starting from a state-based specification language we show that via a behaviour-oriented semantics state-based subtyping induces behaviour-oriented subtyping. In the comparison we study three different kinds of subtyping achieving substitutivity to a different degree. For all three kinds of definitions a correspondence result can be shown.