The proof for the definition of energy is based on hard mathematics, the undisputed characteristics of constant powered rockets and the Leibniz analysis of falling objects. In other words, it corrects the Leibniz analysis. There are two major points to be made about this. One, it is not a law of nature that says energy must do something other than momentum does. In fact, the absolute conservation of energy is an assumption in conflict with evidence.
Potential energy is an example.
It is not energy. If a person robs a bank and leaves potential money behind, he gets thrown in jail. The same should be true of energy. The assumptions with observations are very tricky in not being recognized as assumptions at all. Logic over Perception.