How to justify the shell theory by using Gauss Law?

[Negi Magi]

"As the ball gets deeper under the surface of the Earth, the layers above it stop exerting gravitational force, and all that matters is the mass underneath."

How to justify this sentence is true?

 

[Nabeshin]

"As the ball gets deeper under the surface of the Earth, the layers above it stop exerting gravitational force, and all that matters is the mass underneath."

How to justify this sentence is true?

Well it was proven in general by Newton a few hundred years ago: http://en.wikipedia.org/wiki/Shell_theorem

Using Gauss' law, it's manifest in that you only consider the mass inside the Gaussian sphere. In a sense, Gauss' law embodies the shell theorem.