*3.10. Theorem* ¬(*C I* |→ *CII)*

**Proof.** Based on (15) and (25) we can see that {*C I*} ⊂ {*CII*}. From the whole family of *η<sup>m</sup>* ∈ [0, 1] models for the *C I* principle, only the *η<sup>m</sup>* = *η<sup>C</sup>* model coincides with the *CII* principle model. Even the zero-efficiency *η<sup>m</sup>* = 0 model is not included in the {*CII*} model because it contains too many refrigeration processes. Thus, there is not formal implication from the *C I* principle to the *CII* principle.

Consider two counterexamples of implications in the form of processes, that are allowed in the *C I* principle models but not in the *CII* principle model. The first counterexample would be *perpetuum mobile* type II ↓→ 0, which was already given in the previous section in the context of the *C I* rule. The second counterexample should be the efficient refrigeration process (−*Q*<sup>1</sup> − *q*, −*W*, −*Q*<sup>2</sup> − *q*) related to the Carnot cycle (*Q*1, *W*, *Q*2). It is easy to verify that for *q* > 0 the refrigeration process would increase the entropy of the heat reservoirs (Δ*S* > 0), so it does not follow the *CII* rule. However, this refrigeration process will follow the *C I* principle model, which does not include the Carnot cycle, but the less efficient cycle (*Q*<sup>1</sup> + *q*, *W*, *Q*<sup>2</sup> + *q*). Then adding these processes does not lead to contradiction with the model for the *C I* principle. The indicated counterexamples disproves the implication *C I* |→ *CII*.

Errors in the alleged proof of the disproved implication will now be shown. Consider the virtual engine process ↓→↓ (*Q*<sup>1</sup> − *q*, *W*, *Q*<sup>2</sup> − *q*) with efficiency greater than that of a Carnot engine ↓→↓ (*Q*1, *W*, *Q*2) for *q* > 0. Such a process obviously increases the entropy of the system and is against the *CII* principle. Additionally, consider the Carnot refrigeration cycle ↑←↑ (−*Q*1, −*W*, −*Q*2). The superposition of the virtual engine cycle with the Carnot refrigeration cycle gives:

$$(Q\_1 - q, \mathcal{W}, Q\_2 - q) + (-Q\_1, -\mathcal{W}, -Q\_2 - q) = (-q, 0, -q), \tag{32}$$

which is contrary to the *C I* principle of the upward heat flow process ↑ 0 ↑. One might think that this is proof by contradiction for the implication under consideration. However, nothing could be more wrong. Firstly, the simultaneous use of the process contrary to the *CII* principle and the second process consistent with the principles of *C I* and *CII* raises doubts as to whether the *CII* principle was actually negated (in the proof by contradiction). Secondly, the proof should apply to all processes, not just one process (material implication vs formal implication).

Thus, based on the lack of inclusion of models, two counterexamples, and an error indication in the supposed typical proof, the implication *C I* |→ *CII* is rebutted.

It will now be shown that the other principles do not formally follow from the Kelvin principle.
