Proof: By Induction

(related to Proposition: Addition Of Natural Numbers Is Associative)

We shall show the associativity of "\( + \)" by induction.

Base case

For arbitrary \(n,m\in\mathbb N\), it follows from the definition of addition that \[n+(m+0)=n+m=(n+m)+0.\]

Induction step

Now, let assume that \(n+(m+p_0)=(n+m)+p_0\) has been proven for all \(p_0\le p\), where we use "\(\le\)" as the order relation of natural numbers. Then it follows again from the definition of addition that \[n+(m+p^+)=n+(m+p)^+=(n+(m+p))^+,\] which according to our assumption equals \[(n+(m+p))^+=((n+m)+p)^+=(n+m)+p^+.\] This proves the associativity of "\( + \)" for all \(n,m,p\in\mathbb N\).

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  1. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013