We report the first observation of electron-transfer-mediated decay (ETMD) and interatomic Coulombic decay (ICD) from the triply charged states with an inner-valence vacancy, using the Ar dimer as an example. These ETMD and ICD processes, which lead to fragmentation of Ar(3+)-Ar into Ar(2+)-Ar(2+) and Ar(3+)-Ar+, respectively, are unambiguously identified by electron-ion-ion coincidence spectroscopy in which the kinetic energy of the ETMD or ICD electron and the kinetic energy release between the two fragment ions are measured in coincidence.