Hi I am trying to learn how the number systems was created, and there are two very basic thing I don't get. first question: My book describes and proves that addition is well-defined for integers Z. that is if, z2=z3, then z1+z2 = z1+z3 It also does the same for rational number, it gives a proof that if q2 = q3, then q1+q2=q1+q3 However one thing that puzzles me is that I can not find a proof that it holds for naturlar numbers. That is if: n2= n3 then: n1+n2 = n1+n3 I know this is probably very basic, can I assume it is correct, or should it also be proved? They define the natural numbers as cardinal numbers of sets. And proves many laws like m+n=n+m etc. for natural numbers, but not the one I asked above. second question: This question is probably very stupid, but since it seems like everything should be proved at this basic level, why can I assume that if a = b, then b = a, is this how = is defined?