*3.7. Theorem C*0 |→ *CII*

**Proof.** Based on (13) and (25) we can see that {*C*0} = {*CII*}, so in particular also {*C*0} ⊂ {*CII*}. Thus, on the basis of the models (sets of processes), there is implication (or even equivalence).

Consider the Carnot engine process ↓→↓ (*Q*1, *W*, *Q*2) and the natural heat flow downward process ↓ 0 ↓ (*q*, 0, *q*). On the basis of (28) within the Carnot principle we know that the process inverse to the second process cannot take place (so *q* ≥ 0). The completeness condition implies the existence of a summary process (*Q*<sup>1</sup> + *q*, *W*, *Q*<sup>2</sup> + *q*). Let us calculate the entropy change of the heat reservoirs in this process:

$$
\Delta S = \frac{-Q\_1 - q}{T\_1} + \frac{Q\_2 + q}{T\_2} = 0 + \frac{T\_1 - T\_2}{T\_1 T\_2} q \ge 0. \tag{30}
$$

which is in line with the Claussius II principle.

Now consider the Carnot refrigeration process ↑←↑ (−*Q*1, −*W*, −*Q*2). By adding the ↓ 0 ↓ (*q*, 0, *q*) process to it, we get the allowed (−*Q*<sup>1</sup> + *q*, −*W*, −*Q*<sup>2</sup> + *q*) process. The change in the entropy of the heater and cooler in this refrigeration process is:

$$
\Delta S = \frac{Q\_1 - q}{T\_1} + \frac{-Q\_2 + q}{T\_2} = 0 + \frac{T\_1 - T\_2}{T\_1 T\_2} q \ge 0. \tag{31}
$$

which, again, is in line with the Clausius II principle. The change of entropy would be negative for the (−*Q*<sup>1</sup> − *q*, −*W*, −*Q*<sup>2</sup> − *q*) refrigeration process—however, such a process is not compatible with the Carnot principle, as the convexity condition allows for addition, not subtraction, of process diagrams.

Equality of process sets of the *C*0 and *CII* principles and the fulfillment of the *CII* principle for processes permitted by the *C*0 principle ends the proof.

Now we turn to the implications of Clausius I principle. The first implication, however, does not hold true (as do the other two).
