In the following proof, $\mathbb{Z}_n= \{ 1,2,....n \} $
After $g$ is defined , they defined the composition $gof$ and then they defined $h$ by removing the last ordered pair $(n+1,m)$ from the composition $gof$ and then they applied the induction hypothesis on $h$. There are a couple of thing I don't get in this proof
Why is g defined like that and why should I care about defining it anyway? It seems like unnecessary complication. Wouldn't it be simpler to say: Ok, I have $f$, that goes from the first $ n+1 $natural numbers: $\mathbb{Z}_{n+1}$ to the first $ m $ natural numbers $\mathbb{Z}_{m}$, if I remove the last pair $(n+1,m)$ I get a function from the first $n$ natural numbers : $\mathbb{Z}_{n}$ to the first $m-1$ natural numbers: $\mathbb{Z}_{m-1}$ and I can use the induction over $ n$: $n=m-1$ $\rightarrow$ $n+1=m$
However I've found the same procedure in several books and they all do it in the same way. What am I missing here?