Carnot's theorem: no engine beats a Carnot engine between the same reservoirs

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, let $k<h$, and let $e$ be a Carnot engine between $h$ and $k$. Then every realizable cycle $c$ operating between $h$ and $k$ and absorbing $c(h)>0$ from the hot reservoir satisfies

This is the quantitative content of Carnot's theorem: the efficiency of any heat engine operating between two given reservoirs is at most that of a reversible (Carnot) engine operating between the same two reservoirs.