Equivalence of two thermostatistical formalisms based on the Havrda&Charvat-Daroczy-Tsallis entropies

We show that the latest thermostatistical formalism based on the Havrda & Charvat-Dar'oczy-Tsallis entropy $S_q[\rho ]=(q-1{)}^{-1}(1-tr({\rho }^q))$ proposed by Tsallis, Mendes and Plastino is {\em equivalent} to the first one proposed by Tsallis in 1988. Here, equivalent means: {\em the ``equilibrium’’ state predicted by either formalism using $q$ leads to the same expectation values for all observables as that predicted by the other formalism using $1\mathrm{/}q$’'}. We also point out once again that the basic property of {\em transitivity of equilibrium} (e.g., the $0^{th}$ Law of Thermodynamics) fails in these formalisms.