*3.12. Theorem* ¬(*K* |→ *C I*)

**Proof.** Based on (19) and (15) we can see that {*K*} ⊂ {*C I*}. Both principles have large families of models, but the Kelvin family of models is larger with models with opposite heat flow directions.

Counterexamples for the considered implication may be the process of heat flow upward ↑ 0 ↑ or a process of finite efficiency ↑→↑ of heat conversion from the lower temperature reservoir to work. Both processes are simultaneously included in the two models {*K*}<sup>↑</sup> <sup>0</sup><*η*2<*η<sup>m</sup>* , {*K*}<sup>↑</sup> <sup>1</sup><sup>−</sup> of Kelvin principle. Within these models, these processes cannot be excluded by adding another process from a given model, because as one can check the operation of addition, it is an internal operation in these models. At the same time, the indicated two counter-examples are directly contrary to the *C I* principle.

Now the errors in the supposed typical proof of *K* |→ *C I* implication will be pointed out. This typical pseudo-proof is proof by contradiction. It relies on the ↑ 0 ↑ (−*Q*2, 0, −*Q*2) process, which contradicts the *C I* principle (*Q*<sup>2</sup> > 0). This process is then combined with the engine process ↓→↓ (*Q*1, *W*, *Q*2):

$$(-Q\_2, 0, -Q\_2) + (Q\_1, W, Q\_2) = (Q\_1 - Q\_2, W, 0). \tag{33}$$

The resulting process ↓→ is a *perpetuum mobile* type II, which contradicts the *K* principle. Then it is claimed that the *K* |→ *C I* implication has been proved, but this is not true. First, it is not clear whether it is allowed to add a process ↓→↓ that complies with the *C I* principle, since we want to contradict this principle in the proof by contradiction. The result of adding this process is a process that formally no longer contradicts the *C I* principle. Secondly, the formal implication should be checked for all processes, not just for one type of processes ↑ 0 ↑ in the proof scheme by contradiction. There are more processes that contradict the *C I* principle, e.g., ↑→↑, 0 →↑.

Thus, the structure of the *K* and *C I* models and the two counterexamples, as well as the alleged proof errors, disprove the implication *K* |→ *C I*.
