Why is change in potential energy is defined as PE1 - PE0 = -W I mean I could see it for example for gravity if we took PE0 to be zero at ground and we integerated -mgy(y^) we get -mg(y0 - y1) -> -mgh,but is their a proof somewhere where it shows it will be always negative work ? Thank you.
It won't always be negative work. If the potential energy increases, PE1 > PE0, so W < 0. That says that rather than doing work the system absorbed work. If the potential energy reduces, PE1 < PE0, so W > 0.
W, is work, defined as energy lost from the system. If energy is gained, the system has lost negative work. It's just a convention of direction of energy flow between environment and system.
We define it this way so that work can be interpreted as a transfer of energy from one object to another: the "giver" of the work decreases its energy, and the "recipient" of the work increases its energy. Consider lifting an object at constant velocity against gravity (so its kinetic energy doesn't change). You do positive work mgΔh (the force you exert is in the same direction as the motion), and your own internal energy decreases in the process. The object's gravitational potential energy increases, therefore its PE_{final} - PE_{initial} = W_{done by you}. The force that you exert on the object as you lift it is equal in magnitude and opposite in direction to the gravitational force on the object. Therefore the gravitational force does work on the object that is equal in magnitude but with opposite sign to the work that you do. Therefore we can also write PE_{final} - PE_{initial} = -W_{gravity}