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

<a NAME="E1:57"/>
for <font color="Olive">n</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   holds <br/>(  not <font color="Olive">n</font> <a href="nat_1.html#R1">divides</a> 3 or <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Olive">n</font> <a href="hidden.html#R1">=</a> 3 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> 3
 ;<br/>

<a NAME="E2:57_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="int_2.html#T3">INT_2:3</a></i>;<br/>
<a NAME="E3:57_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 3
 
<b>by </b><i><a class="txt" href="pepin.html#E2"><i><font color="Green">Lemma46</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>
<a NAME="E4:57_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E5:57_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:57_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E7:57_1"/><b>then </b>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:57_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E9:57_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 2
 ;<br/>

3 <a href="nat_1.html#K6">mod</a> 2 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2

<br/>.= 
1 <a href="nat_1.html#K6">mod</a> 2
<b>by </b><i><a class="ref" href="nat_1.html#T73">NAT_1:73</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/>
<b>hence </b><a NAME="E3:57_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E2"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="pepin.html#E8:57_1"><i><font color="Green">E49</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:57_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 3 )
 <b>by </b><i><a class="txt" href="pepin.html#E3"><i><font color="Green">Lemma47</font></i></a>, , <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:57"/>
3 is <a href="int_2.html#V1">prime</a>
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div>
