[RedDevil]

No amount of proof will ever convince anyone who does not understand the principle.
And those wo do understand the principle know it should work, and proof is just confirmation.
So please try to understand the principle. It is a real Aha! moment when you see it.

I'll try to explain with a stepwise approach.

Fact - You cannot blow a sail with the wind past wind speed.
But if you take a very long boat and you put the mast on a cart inside the boat and you push that cart towards the stern, then the boat would move faster than if the sail was fixed. The boat would move faster than the wind if you push hard enough.
I think you can agree that would work.

Instead of a sail we use a propeller. Instead of a boat we use a car.
If we motorize the propeller to push against the wind, of course we could move faster than the wind.
With our motorized cart, the force that drives the car forward is the forward pressure on the propeller that transfers through the axle bearings. All obvious, right?

Now comes the tricky part.
We set the motor speed such that the propeller turns against the wind but slightly slower that the wind speed.
The car will speed up beyond the wind speed, as the wind is still pushing against the propeller as its surface moves slower than the wind.
Even though the propeller forces the air backwards compared to the car, it still moves forward compared to the ground - but slower than the wind, while the car moves faster than the wind.
The wind still pushes against the propeller so the car still accelerates.

Because the car still accelerates in those conditions we might tap some of that energy to drive the propeller. The force needed to do so is less than the forward force on the propeller, as long as the propeller pushes the air backwards ar a slower speed than the car is moving.
We don't need the motor after all as long as the overall friction is less than the difference between the forward force on the propeller and the driving force turning it against the wind.

It is the gearing that makes it possible.
Likewise when you have a big sail and a system of pulleys to rile in the boom. If you let go of the rope the wind will catch the sail and the rope end will speed up way past the wind speed.
Here the sail is a propeller, the pulleys are the gearing and the rope end is the car.

As for proof, the wind vane on the car is all you need.
Of course the wind speed is lower close to the ground, but it is enduring quite a strong virtual headwind; much more than you'd expect.
There is a headwind even right in front of the propeller; in fact the headwind there should be even slightly stronger as the propeller slows the wind speed down, so it is a higher headwind relative to the car.