I was glancing through appendix of M Artin algebra in the integers section and here is a proof by using mathematical induction on the basis of Peano's axiom.
Proof for associative law for addition: Definition referred in the proof is given below (keep in mind that $'$ represents succession by one, e.g., $1'= 2$, $3'= 4$)
Addition: $m + 1 = m'$ and $m + n'=(m+n)'$
Multiplication: $m \cdot 1 = m$ and $m \cdot n' = m \cdot n + m$
Now here is the proof:
We are to prove that $(a + b) + n = a + (b + n)$ for all $a, b, n$ belonging to $\mathbb{N}$. We first check the case $n = 1$ for all $a, b$. Three
applications of definition (given above) give
$$(a + b) + 1 = (a + b)' = a + b' = a + (b + 1)$$
Next, assume the associative law true for a particular value of $n$ and for all $a, b$.
Then we verify it for $n'$ as follows:
$(a + b) + n' = (a + b) + (n + 1)$ (definition)
$= ((a + b) + n) + 1$ (case $n = 1$)
$= (a + (b + n)) + 1$ (induction hypothesis)
$= a + ((b + n) + 1)$ (case $n = 1$)
$= a + (b + (n + 1))$ (case $n = 1$)
$= a + (b + n')$ (definition). proved
Now my question is exactly what happened at step $2$, how did it transition from step i.e., $(a+b)+n' = (a+b) + (n+1)$ to $((a+b)+n)+1$, rest all is fine but reference for step $2$ doesn't fit in. Please explain
Although it's bit silly but I'm getting terribly confused.
Reference: M Artin Appendix A2
