Help me to prove that driving an automible requires work.?

Work is the vector dot product of force and distance. Therefore, if you apply a force on a car over a given distance you are doing work. If the car accelerates, by Newton's second law, a force is being applied, and as the car moves a given distance work is being done. Here's another way to look at it, the net work done on a car is equal to its change in kinetic energy, so if the car ever changes speed (like going from rest to any given speed), work must have been done.

Work being an expenditure of energy, you have already performed "work" by unlocking the car, by putting the key in the ignition, etc, etc.

you need work to overcome friction when you press on the accelerator. The fuel/air mixture undergoes combustion and this does work against the piston