Composition law $f(T_1,T_3)=f(T_1,T_2)f(T_2,T_3)$ for the Carnot heat ratio

Throughout, $\Theta$ is a type of empirical temperatures carrying a linear order ($s<t$ means the reservoir at $s$ is colder than the one at $t$), and a cycle is a finitely supported function $c:\Theta\to\mathbb R$: for each reservoir temperature $t$, the number $c(t)$ is the net heat absorbed by the device from the reservoir at $t$ in one complete cycle (negative when heat is rejected to it), and only finitely many reservoirs are touched. The work delivered in the cycle is

which is the first law for a device returning to its initial state. A set $\mathcal C$ of cycles stands for the physically realizable ones; it is assumed to be a convex cone: $0\in\mathcal C$, $c+d\in\mathcal C$ (run two devices side by side), and $r\cdot c\in\mathcal C$ for $r\ge 0$ (scale a device). A cycle is reversible when $c\in\mathcal C$ and $-c\in\mathcal C$. A cycle operates between $h$ and $k$ when $c(t)=0$ for every $t\notin\{h,k\}$. A Carnot engine between a hot reservoir $h$ and a cold reservoir $k$ is a reversible cycle operating between $h$ and $k$ with $c(h)>0$, $c(k)<0$ and $W(c)>0$. The efficiency relative to the hot reservoir is $\eta=W(c)/c(h)$.

Statement. Let $\mathcal C$ be a convex cone of realizable cycles satisfying the Clausius statement and let $T_3<T_2<T_1$ be three reservoir temperatures. Let $e_1$ be a Carnot engine between $T_1$ and $T_3$, $e_2$ one between $T_1$ and $T_2$, and $e_3$ one between $T_2$ and $T_3$. Then the heat ratios compose:

Writing $f(T_{\mathrm H},T_{\mathrm C})=|q_{\mathrm C}|/|q_{\mathrm H}|$ for the (substance-independent) ratio of a reversible engine, this is the article's identity $f(T_1,T_3)=f(T_1,T_2)\,f(T_2,T_3)$: a reversible engine between $T_1$ and $T_3$ has the same efficiency as the two-cycle engine passing through an intermediate reservoir at $T_2$, the heat $q_2$ passing through the intermediate reservoir without loss.