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Perhaps you should use their work as a guide.
(From your subsequent posts, your question almost certainly does not belong in the Pre-Algebra and Algebra subforum). It is impossible to know what level of proof you require. Unless you can make this crystal clear, I don't see the point in posting help with your question.
In Peano's axioms for the natural numbers we have the "successor function", s(n). Addition is defined by
a) n+ 1= s(n) and
b) if then there exist p such that m= s(p) and then n+ m= s(n+ p).
While 2 is defined as s(1) so it follows that 1+ 1= s(1)= 2.
If you don't like that then:
1) How are you defining "1"?
2) How are you defining "2"?
3) How are you defining "+"?