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The paper addresses the problem of reasoning with multiple ontologies interconnected by semantic mappings. This problem is becoming more and more relevant due to the necessity of building the interoperable Semantic Web. In contrast to the so called global reasoning approach, in this paper we propose a distributed reasoning technique that accomplishes reasoning through a combination of local reasoning chunks, internally executed in each separate ontology. Using Distributed Description Logics as a formal framework for representation of multiple semantically connected ontologies, we define a sound and complete distributed tableau-based reasoning procedure which is built as an extension to standard Description Logic tableau. Finally, the paper describes the design and implementation principles of a distributed reasoning system, called DRAGO (Distributed Reasoning Architecture for a Galaxy of Ontologies), that implements such distributed decision procedure.