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The process of designing and modelling an ontology can be difficult, especially if the user finds the syntax to be relatively inaccessible. Providing users with graphical syntax with which they can model and visualise their ontology has the potential to be helpful. Previously, we informally introduced concept diagrams for ontology visualisation and modelling. We present a case study comprising: (a) a set of axioms for an ontology, and (b) a set of theorems that follow from the axioms, together with their proofs. The proofs have been constructed so that they are, in our opinion, of an intuitive style. From these proofs, we derive a set of sound inference rules that can be used to formally reason about ontologies following the same intuitive style. This approach to designing inference rules differs from previous efforts where the primary focus has been on obtaining a set of sound and complete inference rules, rather than on intuitiveness.