On the Jordan-Hölder decomposition of proof nets

Having defined a notion of homology for paired graphs, Métayer ([Ma]) proves a homological correctness criterion for proof nets, and states that for any proof net there exists a Jordan-Hölder decomposition of H0(G). This decomposition is determined by a certain enumeration of the pairs in G. We correct his proof of this fact and show that there exists a 1-1 correspondence between these Jordan-Hölder decompositions of H0(G) and the possible ‘construction-orders’ of the par-net underlying G.