From 29f32e2031eafa087323d79d812a1b38ac78f977 Mon Sep 17 00:00:00 2001
From: Simon-Kor <52245124+Simon-Kor@users.noreply.github.com>
Date: Mon, 23 Sep 2024 01:20:05 +0200
Subject: working commit

---
 library/nat.tex | 22 +++++++++++-----------
 1 file changed, 11 insertions(+), 11 deletions(-)

(limited to 'library/nat.tex')

diff --git a/library/nat.tex b/library/nat.tex
index ac9a141..841ac36 100644
--- a/library/nat.tex
+++ b/library/nat.tex
@@ -3,48 +3,48 @@
 
 \section{Natural numbers}
 
-\begin{abbreviation}\label{inductive_set}
+\begin{abbreviation}\label{num_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}
+\begin{axiom}\label{num_naturals_inductive_set}
     $\naturals$ is an inductive set.
 \end{axiom}
 
-\begin{axiom}\label{naturals_smallest_inductive_set}
+\begin{axiom}\label{num_naturals_smallest_inductive_set}
     Let $A$ be an inductive set.
     Then $\naturals\subseteq A$.
 \end{axiom}
 
-\begin{abbreviation}\label{naturalnumber}
+\begin{abbreviation}\label{num_naturalnumber}
     $n$ is a natural number iff $n\in \naturals$.
 \end{abbreviation}
 
-\begin{lemma}\label{emptyset_in_naturals}
+\begin{lemma}\label{num_emptyset_in_naturals}
     $\emptyset\in\naturals$.    
 \end{lemma}
 
-\begin{signature}\label{addition_is_set}
+\begin{signature}\label{num_addition_is_set}
     $x+y$ is a set.
 \end{signature}
 
-\begin{axiom}\label{addition_on_naturals}
+\begin{axiom}\label{num_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}
+\begin{abbreviation}\label{num_zero_is_emptyset}
     $\zero = \emptyset$.
 \end{abbreviation}
 
-\begin{axiom}\label{addition_axiom_1}
+\begin{axiom}\label{num_addition_axiom_1}
     For all $x \in \naturals$ $x + \zero = \zero + x = x$.
 \end{axiom}
 
-\begin{axiom}\label{addition_axiom_2}
+\begin{axiom}\label{num_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}
+\begin{lemma}\label{num_naturals_is_equal_to_two_times_naturals}
     $\{x+y \mid x \in \naturals, y \in \naturals \} = \naturals$.
 \end{lemma}
 
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