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\import{set/suc.tex}
\import{set.tex}
\section{Natural numbers}
\begin{abbreviation}\label{inductive_set}
$A$ is an inductive set iff $\emptyset\in A$ and for all $a\in A$ we have $\suc{a}\in A$.
\end{abbreviation}
\begin{axiom}\label{naturals_inductive_set}
$\naturals$ is an inductive set.
\end{axiom}
\begin{axiom}\label{naturals_smallest_inductive_set}
Let $A$ be an inductive set.
Then $\naturals\subseteq A$.
\end{axiom}
\begin{abbreviation}\label{naturalnumber}
$n$ is a natural number iff $n\in \naturals$.
\end{abbreviation}
\begin{lemma}\label{emptyset_in_naturals}
$\emptyset\in\naturals$.
\end{lemma}
\begin{signature}\label{addition_is_set}
$x+y$ is a set.
\end{signature}
\begin{axiom}\label{addition_on_naturals}
$x+y$ is a natural number iff $x$ is a natural number and $y$ is a natural number.
\end{axiom}
\begin{abbreviation}\label{zero_is_emptyset}
$\zero = \emptyset$.
\end{abbreviation}
\begin{axiom}\label{addition_axiom_1}
For all $x \in \naturals$ $x + \zero = \zero + x = x$.
\end{axiom}
\begin{axiom}\label{addition_axiom_2}
For all $x, y \in \naturals$ $x + \suc{y} = \suc{x} + y = \suc{x+y}$.
\end{axiom}
\begin{lemma}\label{naturals_is_equal_to_two_times_naturals}
$\{x+y \mid x \in \naturals, y \in \naturals \} = \naturals$.
\end{lemma}
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