*3.13. Theorem* ¬(*K* |→ *CII*)

**Proof.** Based on (19) oraz (25) we can see that {*K*} ⊂ {*CII*}. The family of models for the *K* principle is too large to be a subset of the *CII* principle model. The *K* principle models include too efficient engine processes *η* > *ηC*, as well as refrigeration processes with too low efficiency *η*˜ < *ηC*. Such processes, on the other hand, are contrary to the *CII* principle.

The processes ↓→↓ (*Q*<sup>1</sup> − *q*, *W*, *Q*<sup>2</sup> − *q*), ↑←↑ (−*Q*<sup>1</sup> − *q*, −*W*, −*Q*<sup>2</sup> − *q*) are counterexamples for the implication under consideration. They were referenced to the ↓→↓ (*Q*1, *W*, *Q*2) process with Carnot efficiency *ηC*. For *q* > 0 these two counterexamples contradict the *CII* principle, since for them Δ*S* < 0. However, they do not contradict the *K* principle, which does not allow 100% efficiency, but does not give a stronger limitation with Carnot efficiency.

However, one might get the impression that the processes (*Q*<sup>1</sup> − *q*, *W*, *Q*<sup>2</sup> − *q*) and (−*Q*<sup>1</sup> − *q*, −*W*, −*Q*<sup>2</sup> − *q*) do not follow the *K* principle because, when added to the inverse (−*Q*1, −*W*, −*Q*2) or the regular (*Q*1, *W*, *Q*2) of the Carot process, they give respectively in both cases the process ↑ 0 ↑ (−*q*, 0, −*q*), which is impossible in the considered models (disregarding the existence of a separate class of models of the type {*K*}↑). The point is, however, that in the model suitable for the (*Q*<sup>1</sup> − *q*, *W*, *Q*<sup>2</sup> − *q*) process, there is no inverse Carnot cycle with Carnot efficiency (*η*˜ = *ηC*). For the model suitable for the (−*Q*<sup>1</sup> − *q*, −*W*, −*Q*<sup>2</sup> − *q*) process, there is no Carnot engine cycle with Carnot efficiency (*η* = *ηC*). Moreover, both counterexamples of processes belong to different models, so these processes should not be added, either.

It is also worth pointing to the errors in the alleged proof by the contradiction of the implication under consideration. It is known that the ↑ 0 ↑ (−*q*, 0, −*q*) process for *q* > 0 does not comply with the *CII* rule because in this case Δ*S* < 0. By combining the above process with the ↓→↓ (*q* + *W*, *W*, *q*) engine process, we get *perpetuum mobile* type II, i.e., the process ↓→ 0(*W*, *W*, 0). However, this does not prove anything, since the proof of formal implication cannot be based on one example of a process impossible for a given principle (and consequently impossible for a second principle). Moreover, considering both the impossible and the possible process (for the *CII* principle) is unclear. However, consider another impossible process ↓→↓ (*Q*<sup>1</sup> − *q*, *W*, *Q*<sup>2</sup> − *q*) with an efficiency greater than that of the Carnot process. Combining it with the Carnot inverse cycle ↑←↑ (−*Q*1, −*W*, −*Q*2) we get the process ↑ 0 ↑ (−*q*, 0, −*q*) impossible for the *CII* principle, but not excluded by the *<sup>K</sup>* rule. Such a process ↑ <sup>0</sup> ↑ occurs, for example, in the model {*K*} <sup>0</sup>, which includes any refrigeration process {*K*} <sup>0</sup>, but does not contain any engine process ↓→↓.

On the basis of the structure of the {*K*}, {*CII*} models, two counterexamples and pseudo-proof errors, the formal implication of *K* |→ *CII* has been rebutted.

Three formal implications of the *CII* principle remain to be proved.
