The whole and its parts: On the coherence theorem of Balinski and Young

A new proof of the Coherence Theorem of Balinski and Young is presented. The theorem elucidates the methods used to apportion parliamentary seats among political parties proportionately to their vote counts, or among geographical districts proportionately to their population figures. A proportional apportionment method is coherent when each seat apportionment among all claimants is such that every part of it is a valid solution for the subset of claimants concerned. The Coherence Theorem states that every coherent apportionment method is compatible with a divisor method.