You would use the natural number axiom(basic assumptions we hold to be true/that have been proven to be true) which allows for one to logically induce that all natural numbers follow in an order.(The variable n represents all natural numbers in this case.) Then you would use addition property, then the property of equality and so on and so on.
You are representing the fact that 1 +1 = 2 by using the symbols(language)/rules/laws of Mathematics.
It is not unlike--Bonjour means hello regardless if it's in English or French it's just a different representation of the same thing. It still holds true that both are ways(facts)of saying hello.
The two types of proof are the same, but one is more accessible to everyone than the other. Mathematics and language are both human constructs representing physical truths or realities. MAthematics indeed IS a form of language.
PS i did pre university double maths in a time when there were no calculators or computers. We learned first as young children how to do mental arithmetic, such as our times tables and simple and long division and mulitlication, in our heads, and second how to use slide rules and logarithmic tables for calculations. Thus we had to have the knowledge and discipline to understand the very basic and more complex building blocks of mathemeatics. We had to memorise, and hold in our heads, all the mathematical and geometric formulae/processes used in maths, and work out complex questions in our heads, or on paper, in a few minutes under test conditions. At the end of our final year we had two, state set and assessed, three hour exams, one on each maths subject, in order to assess our state ranking. Maths wasn't my strong suite but I achieved in the top 25% of the state's students in both exams.