Let $\mathbb{N}=\{1,2,3,\dots\}$, and let $s: \mathbb{N} \to \mathbb{N}$ be the successor map $s(n)=n+1$. Let addition and multiplication on $\mathbb{N}$ be defined recursively by
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\begin{align*}
a+1=s(a)
\end{align*}
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and
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\begin{align*}
a+s(b)=s(a+b)
\end{align*}
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for addition, and by
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\begin{align*}
a1=a
\end{align*}
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and
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\begin{align*}
a\,s(b)=ab+a
\end{align*}
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for multiplication, for all $a,b \in \mathbb{N}$. Then for all $a,b,c \in \mathbb{N}$,