This paper deals with the question of whether a proof of Clausius inequality can be given on the basis of a finite number of processes involving the union of an arbitrary system and a finite number of copies of a perfect gas, but avoiding the pass to the limit, which is neither mathematically nor physically admissible. This requires to put on a firm basis consisting of two conceptual elements: the notion of ``union'' of systems and processes, and the way to render mathematically the notion of ``heat exchanged at a set of hotnesses''. These problems have been analysed in the context of measure theory by \textit{M. Šilhavý} [Arch. Ration. Mech. Anal. 81, 221-243 (1983; Zbl 0517.73002)] and \textit{M. Feinberg} and \textit{R. Lavine} [Arch. Ration. Mech. Anal. 82, 203-293 (1983; Zbl 0587.73008)], and without direct use of measure-theoretic concepts by \textit{J. Serrin} [Arch. Ration. Mech. Anal. 70, 355 (1979)]. This paper follows the latter way, but in contrast to Serrin, which required a stronger version of the Kelvin formulation, it establishes a framework in which Kelvin-like formulation of the second law suffices for proving the general Clausius inequality, but changes the set of ideal processes from those that are piecewise smooth to those that are absolutely continuous. Four theorems are proved: the first one is able to show some results concerning thermodynamic processes which were considered as axioms in previous formulations; the second one states and proves which is the fundamental thermodynamical property of a perfect gas necessary and sufficient to prove the general Clausius inequality from the Kelvin formulation of the second law, and thus provides a rigorous abstract characterization of ``thermometric substances'', conceptually more general than the perfect gas. The third and fourth theorems show that Kelvin's second law is equivalent to the accumulation inequality by Serrin, and thus Kelvin's and Serrin's versions of the second law are shown to be equivalent.