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**particlejohn** Prove for any natural numbers $\displaystyle a,b,c $, we have $\displaystyle (a+b)+c = a+(b+c) $. So fix $\displaystyle a $ and $\displaystyle b $ and induct on $\displaystyle c $. For $\displaystyle c = 0 $, $\displaystyle a+b = a+b $. Suppose inductively that $\displaystyle (a+b)+c = a+(b+c) $. We have to prove that $\displaystyle (a+b) + c++ = a+(b+c++) $. So $\displaystyle (a+b)+c++ = (a+b+c)++ $. Also $\displaystyle a+(b+c++) = (a+b+c)++ $. $\displaystyle \blacksquare $