The Kelvin-Planck statement is one way of saying the second law of thermodynamics. A thought experiment can prove it. Take a power plant just like we drew before, but assume the heat rejected to the river (lost!) has been reduced to 0. Great, right? Well, no: that results in what is called a perpetual motion machine, which cannot exist. Because now only one reservoir is involved!
To complete the thought experiment, say we take all the energy (A) we make with the power plant and use it to power a refrigerator between the hot and cold reservoirs. That is drawn above on the left side. You see that the energy A isn’t really doing anything but leaving the hot reservoir and then returning to it, so we can just subtract it out, giving us the picture on the right side.
This proves it, because now we have energy (heat) flowing from cold to hot all by itself! Which is not allowed! We still have the refrigerator drawn there, but nothing is powering it. This is the equivalent of heat flowing from ice water to a warm room. We all know heat flows the other way: from hot things to cold.
