*3.5. Theorem C*0 |→ *C I*

**Proof.** Based on (13) and (15), (17) we can see that {*C*0}⊂{*C I*}*η<sup>m</sup>* for *η<sup>m</sup>* = *ηC*. Consider a Carnot heat engine denoted by the diagram (↓→↓)*η<sup>C</sup>* , in which heat input, work done and output heat can be written as (*Q*1, *W*, *Q*2). Similarly, we will write the process of heat flow (*q*, 0, *q*), which is a flow down (towards a lower temperature) ↓ 0 ↓ for *q* > 0 or a flow up (towards a higher temperature) ↑ 0 ↑ for *q* < 0. Now let us combine these two processes:

$$(Q\_1, W, Q\_2) + (q, 0, q) = (Q\_1 + q, W, Q\_2 + q). \tag{27}$$

Let us only consider the resultant processes which follow the Carnot principle:

$$
\eta = \frac{\mathcal{W}}{Q\_1 + q} \le \frac{\mathcal{W}}{Q\_1} = \eta\_{\mathcal{C}} \quad \Rightarrow \quad q \ge 0. \tag{28}
$$

Therefore, on the basis of the completeness condition, we can conclude that the Carnot principle is compatible with the processes of heat flow downwards ↓ 0 ↓ (*q* > 0), and the processes of heat flow upwards ↑ 0 ↑ (*q* < 0) do not occur. Thus, we obtained the thesis of Clausius I principle.
