Proving that force satisfies the laws of vector addition A vector quantity has both magnitude and direction. So, for any physical quantity to be a vector, it should have a direction and a magnitude. Though this is a necessary condition for any quantity to be a vector it is not sufficient. To qualify as a vector a quantity must also follow the laws of vector addition.
Now getting to the point, the most commonly quoted reason to explain why force is a vector is that it has direction and magnitude. Most textbooks say nothing more than this. How can one mathematically prove that force is a vector?
 A: This used to be covered in textbooks.
A fairly recent article about it is "Why do forces add vectorially? A forgotten controversy in the foundations of classical mechanics" by Marc Lange in the American Journal of Physics 79(4) 380-388 (2011); http://dx.doi.org/10.1119/1.3534836
And there are two common answers.
In dynamics you can used Newton's Second Law $$\sum_n\vec F_n=m\vec a$$ and the fact that mass is a scalar. 
In statics you can attempt a symmetry argument where you try to argue that the sum of two forces couldn't point in any direction other than the vector sum.
Of course, if you use the former someone can ask why mass has to be a scalar. And if you use the latter they could ask why vector addition is pairwise, or associative, or so on.
Eventually we say that physics is about understanding the universe so you make theories that have models that can be compared to experiments and observations and when the comparison is good that's a success.
And sometimes we will talk about torques and moments of inertia to challenge mass being a scalar (and torques might more rightly be a bivector or a 2d plane instead of a vector or a 1d line). Or we might say that the sum of the force due to A alone and the force due to B alone is different than the force when both A and B are there if the presence of both A and B causes additional forces beyond what each would produce if it were alone.
But even so you still might want to break the vector down into linear combinations of basis vectors, so you still might end up with the exact same math even in these more complicated situations.
This answer to a related question goes into more details.
A: I don't think that "satisfying the laws of vector addition" is necessary for something to be a vector, depending on what you mean by that.
Take velocities in special relativity. They are vectors; the vector sum of velocities is well defined. But it's rarely useful. More commonly, when you have two velocities and need to combine them somehow, the combination you actually want is given by something like the "velocity addition formula", which isn't the same as vector addition.
The same is true of forces. They don't simply add in general, because no actual force of nature is exactly linear. To prove that they do add, you would have to assume some form of linearity, which is more or less what you are trying to prove.
