Mechanical power is not a state function because it changes with the change of the path taken from a specific initial point to a specific final point.
The same for thermal power. It is not a state function.
But the sum of mechanical and thermal power is the variation of total energy of the system, and the total energy is considered a state function.
I am not convinced since if the mechanical power decreased by some amount, the thermal power doesn't increase by the same amount..
So, why is the variation of the total energy considered a state function?