Tensor equations in General Relativity In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent.
Question: why is this true?
I'm looking for a mathematical argument/proof about this fact.
 A: 
In the context of general relativity it is often stated that one of the main purposes of tensors is that of making equations frame-independent. Question: why is this true?

Actually this isn't quite true. General relativity doesn't have frames of reference (except locally, which is trivially true because GR is the same as SR locally). A better way of saying this would be:
The purpose of tensors is to make equations coordinate-independent.
The idea is that when we assign coordinates to something, that's just a name. The laws of nature should be expressible in a manner such that the names don't matter.

I'm looking for a mathematical argument/proof about this fact.

A tensor is defined as something that transforms in a certain way under a change of coordinates. Since the transformation of tensors is well-defined, it follows that a tensorial equation retains the same form under a change of coordinates.
A: What we want of a law of nature is that is has the same form for every equivalent observer.
Therefore, these laws should be construct with geometrical objects which transform into themselves up to multiplicative factors. This is also known as an homogeneous transformation under certain group (typically Lorentz or diffeomorphism).
The geometrical object which satisfy this homogeneous transformation rule are tensors (there are also spinors). Thus physical theories are described (so far) successfully by these objects (or fields).
