*3.8. Theorem* ¬(*C I* |→ *C*0)

**Proof.** Based on (13) and (15) we can see that the model family for the *C I* principle is larger than the uniform set of processes (stable model) for the *C*0 principle. These are additional models that contain engine processes with an efficiency greater than that of the Carnot cycle (*η<sup>C</sup>* < *η<sup>m</sup>* < 1). In addition, there is even a {*C I*}<sup>1</sup> model containing *perpetuum mobile* type II, which is contrary to the Carnot principle.

First of all, it should be shown that processes with the maximum efficiency in the range *η<sup>C</sup>* < *η<sup>m</sup>* < 1 are allowed by the *C I* principle. Although this principle distinguishes between higher and lower temperatures and the direction of heat flow, it does not impose quantitative restrictions on efficiency (even a model with *η<sup>m</sup>* = 1 is possible).

Consider the Carnot engine cycle process ↓→↓ (*Q*1, *W*, *Q*2). We assume that this process complies with the Clausius I principle. Let us consider whether there is a possible model of the *C I* principle, in which there is a process with even greater efficiency, i.e., process ↓→↓ (*Q*<sup>1</sup> − *q*, *W*, *Q*<sup>2</sup> − *q*), where *q* > 0. The inclusion of such a process will not lead to a contradiction, if only some refrigeration processes exist in the model. For example, ↑←↑ (−*Q*<sup>1</sup> + *q* + *p*, −*W*, −*Q*<sup>2</sup> + *q* + *p*) refrigeration processes will be acceptable for *p* ≥ 0. Their addition to the Carnot cycle process and their addition to the more efficient process under consideration leads to downward heat flow processes ↓ 0 ↓ or to a null process 000, which are in accordance with the *C I* principle. In the considered model of the *C I* principle, however, the refrigeration process type ↑←↑ (−*Q*1, −*W*, −*Q*2) cannot occur because the higher efficiency engine added to the process would generate a heat flow upwards ↑ 0 ↑ contrary to the *C I* principle. However, the absence of (−*Q*1, −*W*, −*Q*2) in the model does not make it contradictory. Thus, the considered model of the *C I* rule is acceptable. At the same time, this model of the *C I* principle is not compatible with the model of the *C*0 principle, as it contains engine processes with efficiency greater than the Carnot cycle.
