Question: How do I prove the total moments of the horizontal component, $\underline{x}$, and vertical component, $\underline{y}$, of a force, $R$, around a point, $A(\overline{x},\overline{y})$, is independent of their point of application, $B(x,y)$, along the line of action, $y=mx+c$, of said force, $R$?

I guess I have to take an expression for the total moment and reduce it to a constant, independent of $x$ and $y$. I tried but failed.

[![enter image description here](https://i.stack.imgur.com/vqjHr.png)](https://i.stack.imgur.com/vqjHr.png)

Answer: 
Your first two sentences are correct. But then you go astray. A Carnot cycle is indeed less than 100% efficient, yet it is an ideal (reversible) cycle without friction or any other 'dissipative' processes. Its inefficiency is a case of the Second Law of Thermodynamics: no cyclic process can take in heat at one temperature and give out an equal quantity of work; some heat has to be given out at a lower temperature (even if there are no dissipative processes at work). The reason for the Second Law can be understood using the methods of statistical mechanics, but this answer stops here!
No, this is not the way to think about this. In the Carnot cycle some heat is, necessarily deposited in the low temperature reservoir, rather than being used to do work, but this is *not* due to friction and does *not* represent some sort of irreversibly. Your last sentence is true in the sense that it essentially says "a process is reversible if you can reverse it" but I don't think it is a good way to think about it. 

In thermodynamics transferring energy as heat and transferring energy as work are not the same thing. A heat engine takes in energy as *heat* from a reservoir at temperature $T\_1$, does a certain amount of *work*, deposits a certain mount of *heat* in a reservoir at temperature $T\_2$ and returns to its original state. Kelvin's statement of the second law is that not all of the energy you put in can be converted into work, i.e. some heat must be deposited in the second reservoir. In other words the heat being "wasted" is necessary in order to satisfy the second law. It is preventing a net reduction in entropy over the cycle.