<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:53"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:53"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/>
<a NAME="E4:53"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<a NAME="E1:53_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="newton.html#K12" target="_self">!</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E2:53_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="newton.html#K12" target="_self">!</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:53"/>
<font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font> <a href="newton.html#K12" target="_self">!</a> 
 <b>by </b><i><a class="txt" href="newton.html#E3:53"><i><font color="Green">E33</font></i></a></i>;<br/>


</div>
