In this, the last chapter of our axiomatic-algebraic approach, the Kelvin–Planck version of the Second Law for a heat engine and the Clausius version for a heat pump are presented and shown to be equivalent. The extent to which these versions of the Second Law can be said to be equivalent to either of Carathéodory’s versions is examined. Both the Kelvin–Planck and Clausius versions of the Second Law involve a system performing a cycle consisting of two isothermal process effected by thermal contact with heat reservoirs of different temperatures separated by adiabatic processes. We consider all six possibilities for this setup, corresponding to each combination of signs of temperatures for the two reservoirs and whether the system has increasing or decreasing entropy in an adiabatic process (positive or negative heat capacity, respectively). We show that the Kelvin–Planck and Clausius versions of the Second Law follow from the analysis of such a system performing a cycle, when both reservoirs are at a positive temperature. Proposed modifications to the laws for negative temperatures and negative heat capacities are presented, and the possibility of processes between states with positive and negative temperatures is considered.