Are you asking how one proves that "addition" works, or that the integer 2 can be represented by 1+1?
I would check out this entry on Addition; it is one of those "basic" ideas you just accept I suppose: when we "add" something we are combining one set of objects with another (I also like the "length" argument).
If your intent is to prove that 2 is equal to 1+1, you can just note that 2 is an even number, and all even numbers can be represented as 2k, where k is some integer. In this case your integer is 1 (1+1 => 2(1)).
Hopefully that answers, a little bit, your question - assuming I understood the question.