how does one prove the associative rule for addition of positive real numbers in elementary terms I want to know how do I prove the associative rule for addition in elementary terms. I searched about the proof on Google but I was not able to figure head or tail about it. So how does one prove the associative rule?

In case we have axiomatized:

1. $\sum_{i=1}^n x_i=\sum_{i^1}^n x_{\sigma(i)}$ for a permutation $\sigma\in S_n$ (addition is commutative)
2. $x+y+z=(x+y)+z$ (you evaluate addition from left to right - reading direction)



Then we can prove: $$ (x+y)+z=x+y+z=y+z+x=(y+z)+x=x+(y+z) $$ But it has to be based on axioms, after all.