*3.6. Theorem C*0 |→ *K*

**Proof.** Based on (13), (19) and (21) we can see that {*C*0}⊂{*K*}<sup>↓</sup> *<sup>η</sup><sup>m</sup>* for *η<sup>m</sup>* = *ηC*. Since essentially {*K*} = {*K*} <sup>0</sup>, {*K*}↓, {*K*}↑, it can be assumed that {*C*0}⊂{*K*}. Thus, the set of Carnot principle processes follows the Kelvin principle. These are, of course, processes with the natural direction of heat flow downwards (as in principle of Clausius I).

Consider a process involving an ↓→? heat engine in which we do not know if the cooler absorbs heat (*Q*<sup>2</sup> =?). Let us describe this process with three parameters (*Q*1, *W*, *Q*2), where *Q*<sup>1</sup> > 0, *W* > 0, *Q*<sup>2</sup> = *Q*<sup>1</sup> − *W*. The Carnot principle implies the condition for *Q*2:

$$
\eta = 1 - \frac{Q\_2}{Q\_1} \le \eta\_\mathbb{C} < 1 \quad \Rightarrow \quad Q\_2 \ge (1 - \eta\_\mathbb{C}) Q\_1 > 0. \tag{29}
$$

This means that heat must be transferred to the cooler, so it is a ↓→↓ process with less than 1 efficiency, which is in line with the Kelvin principle.
