*3.9. Theorem* ¬(*C I* |→ *K*)

**Proof.** Based on (15), (18) and (19), (22) we can see that {*C I*} ⊂ {*K*}. The absence of inclusions here is quite subtle and is due to the lack of inclusions of special models {*C I*}<sup>1</sup> ⊂ {*K*}<sup>↓</sup> <sup>1</sup><sup>−</sup> . This is due to the existence of a *perpetuum mobile* process of the II type ↓→ 0 ∈ {*C I*}1, which is not allowed by the Kelvin principle ↓→ 0 ∈ {*K*}<sup>↓</sup> 1− .

Thus, the rebuttal of the implication in this case is based on the existence of one counterexample: the ↓→ 0 process in one model of Clausius I principle. It is about the {*C I*}<sup>1</sup> model the processes of which are listed in (18). The ↓→ 0 process in this model does not lead to a contradiction because there are no refrigeration processes in this model that could lead to a contradiction. Refrigeration processes may exist in other models, however, where there is no process ↓→ 0. The existence of the {*C I*}<sup>1</sup> model removes the formal implication between the *C I* principle and the *K* principle.

It is also necessary to point out the error in the alleged contemporary proof by the contradiction of the considered implication. Consider two processes that contradict Kelvin principle: ↓→ 0(*W*, *W*, 0) and 0 →↑ (0, *W*, −*W*), whose efficiency is 100%. Let us assume a non-zero refrigeration process ↑←↑ (−*Q*1, −*W*, −*Q*2). Combining such a refrigeration process with the above 100% efficiency processes leads to upward heat flow processes, respectively: ↑ 0 ↑ (−*Q*2, 0, −*Q*2) and ↑ 0 ↑ (−*Q*1, 0, −*Q*1). Thus, it might seem that breaking Kelvin principle entails breaking Clausius I principle (alleged proof by contradiction). However, it is not so. First, the consideration of the ↑←↑ refrigeration process is consistent with, and not in contradiction to, the Kelvin Principle—upon which proof by contradiction should be based. Not surprisingly, considering contradicting the Kelvin principle and adopting the Kelvin principle leads to a contradiction. Second, there is nothing to prove that the combination of a virtual process (outside the model) with a process within the model that leads to a process outside the model (one or the other principle). The condition for the internality of adding processes in the completeness condition applies to the situation when both the processes being added belong to the model. Third, the alleged proof by contradiction relies on a material implication for two cases, while it should be based on a formal implication based on all possible processes.
