According Einstein the gravity is caused by curvation of Spacetime. I understood that the formula of this curvation is the Riemann curvation tensor. So should this formula with the proper approximations result in Newton formula of gravity?

Or how is it mathematical proven that gravity is caused bij curvation?

It is possible to derive Newtonian gravity from GR, for example:

http://www.astro.ex.ac.uk/people/sing/Notes/12.EinsteinFieldEquation_NewtonianLimit.pdf

But I certainly don't understand that!

 

You can't prove that gravity is space-time curvature. All you can do is test the predictions of GR and see that they give better results than other theories.