Prove that hot can never be colder than cold?
In your example heat transfer is proportional to minus the temperature gradient, so heat never flows into hot. To prove it?... Isn't it the second law in some statements? Wiki says, the simplest formulation of the second law, the heat formulation or Clausius statement: Heat generally cannot spontaneously flow from a material at lower temperature to a material at higher temperature. EDIT: Classical Thermodynamics assumes relatively slow processes, so that a system changes state smoothly.
Assuming that heat flow lengthwise in the conductor is negligible, if at any time Th < Tc, then at some point between the entrance and where the temperatures were reversed there has to be a balance Th = Tc. But once fluids have reached that balance, there wouldn't be any further heat flow, since that only occurs when there's an temperature imbalance.
