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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_1"><b>suppose </b></a><a NAME="E1:81_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_1"/>
( 3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4">-</a> 1 <a href="int_1.html#R1">are_congruent_mod</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> implies  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2"><b>suppose </b></a><a NAME="E1:81_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>assume </b><a NAME="E2:81_1_2"/><i><font color="Green">E45</font></i>: 
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4">-</a> 1 <a href="int_1.html#R1">are_congruent_mod</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font>
 ;<br/><b>assume </b><a NAME="E3:81_1_2"/><i><font color="Green">E49</font></i>: 
not  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a>
 ;<br/><a NAME="E4:81_1_2"/>
 <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 <b>by </b><i><a class="txt" href="pepin.html#E6"><i><font color="Green">Lemma57</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:81_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E7:81_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:81_1_2"/><i><font color="Green">E52</font></i>: 
 <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E11:81_1_2"/><i><font color="Green">E53</font></i>: 
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4">-</a> 1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="pepin.html#T12" target="_self">Th12</a>, <a class="ref" href="int_1.html#T41">INT_1:41</a></i>;<br/><a NAME="E12:81_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1_2_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 3
 ;<br/>

<a NAME="E2:81_1_2_1"/>
1 <a href="nat_1.html#K10">hcf</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T64">NEWTON:64</a></i>;<br/>
<a NAME="E3:81_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> 1</span>)</span> <a href="nat_1.html#K10">hcf</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 3
 
<b>by </b><i><a class="ref" href="euler_1.html#T6">EULER_1:6</a></i>;<br/>
<a NAME="E4:81_1_2_1"/><b>then </b>
not 3, <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="pepin.html#T12" target="_self">Th12</a>, <a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>hence </b><a NAME="E5:81_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="pepin.html#T7" target="_self">Th7</a>, <a class="ref" href="pepin.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:81_1_2"/><b>then </b><i><font color="Green">E100</font></i>: 
3,<font color="Maroon">p</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="pepin.html#T11" target="_self">Th11</a>, <a class="txt" href="pepin.html#E2:81_1_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="int_2.html#T47">INT_2:47</a></i>;<br/><a NAME="E14:81_1_2"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="pepin.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E15:81_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i/>;<br/><a NAME="E16:81_1_2"/><i><font color="Green">E124</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="pepin.html#T11" target="_self">Th11</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E17:81_1_2"/><i><font color="Green">E125</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="pepin.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E18:81_1_2"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1_2_2"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 ;<br/>

<a NAME="E2:81_1_2_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E3:81_1_2_2"/><b>then </b><i><font color="Green">E127</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:81_1_2_2"/>
not  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> is <a href="abian.html#V1">even</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:81_1_2_2"/><b>then </b><i><font color="Green">E128</font></i>: 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="pepin.html#T11" target="_self">Th11</a>, , <a class="ref" href="int_2.html#D5">INT_2:def 5</a>, <a class="ref" href="nat_2.html#T24">NAT_2:24</a></i>;<br/>
<a NAME="E6:81_1_2_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="pepin.html#T11" target="_self">Th11</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:81_1_2_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><a NAME="E19:81_1_2"/>
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="pepin.html#K2" target="_self">^2</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="pepin.html#K2" target="_self">^2</a>  <a href="int_1.html#R1">are_congruent_mod</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="pepin.html#E1"><i><font color="Green">Lemma43</font></i></a></i>;<br/><a NAME="E20:81_1_2"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <a href="pepin.html#K2" target="_self">^2</a>  <a href="int_1.html#R1">are_congruent_mod</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="pepin.html#E24"><i><font color="Green">Lemma77</font></i></a>, </i>;<br/><a NAME="E21:81_1_2"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 <a href="int_1.html#R1">are_congruent_mod</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font>
 ;<br/><a NAME="E22:81_1_2"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="pepin.html#T12" target="_self">Th12</a>, <a class="ref" href="int_1.html#T41">INT_1:41</a></i>;<br/><a NAME="E23:81_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E22"><i><font color="Green">Lemma75</font></i></a>, <a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, </i>;<br/><b>set </b><font color="Maroon">d</font> =  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font>;<br/><a NAME="E24:81_1_2"/><i><font color="Green">E130</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E21"><i><font color="Green">Lemma74</font></i></a>, <a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, , <a class="txt" href="pepin.html#E3:81_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E25:81_1_2"/><i><font color="Green">E131</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T11" target="_self">Th11</a>, <a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E7:81_1_2"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E26:81_1_2"/><i><font color="Green">E132</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, </i>;<br/><a NAME="E27:81_1_2"/><i><font color="Green">E133</font></i>: 
not  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1_2_3"/>
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2
 ;<br/>

<a NAME="E2:81_1_2_3"/><b>then </b>
<span class="p1">(<span class="default">3 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, <a class="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E3:81_1_2_3"/><b>then </b>
3 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K5">div</a> 2</span>)</span>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, </i>;<br/>
<a NAME="E4:81_1_2_3"/><b>then </b>
1, <a href="xcmplx_0.html#K4">-</a> 1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="pepin.html#E19"><i><font color="Green">Lemma72</font></i></a>, <a class="txt" href="pepin.html#E1:81_1_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E5:81_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="int_1.html#R2">divides</a> 1 <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="int_2.html#T19">INT_2:19</a></i>;<br/>
<a NAME="E6:81_1_2_3"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 2
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:81_1_2_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">2N</font> = 2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>;<br/><a NAME="E28:81_1_2"/><i><font color="Green">E134</font></i>: 
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i/>;<br/><a NAME="E29:81_1_2"/><i><font color="Green">E155</font></i>: 
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i/>;<br/><a NAME="E30:81_1_2"/><i><font color="Green">E156</font></i>: 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><a NAME="E31:81_1_2"/><b>then </b><i><font color="Green">E157</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E32:81_1_2"/><i><font color="Green">E158</font></i>: 
not  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 2
 <b>by </b><i>, <a class="ref" href="pepin.html#T21" target="_self">Th21</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E33:81_1_2"/>
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1_2_4"/><i><font color="Green">E159</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:81_1_2_4_1_1"/>
(  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 or  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="pepin.html#E40"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:81_1_2_4_1_1_1"/>
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E31"><i><font color="Green">Lemma82</font></i></a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="81_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:81_1_2_4_1_1_2"/>
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:81_1_2_4_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:81_1_2_4"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>then </b> <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> = 
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="pepin.html#T24" target="_self">Th24</a>, <a class="ref" href="pepin.html#T26" target="_self">Th26</a>, , <a class="ref" href="int_2.html#T44">INT_2:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i><a class="ref" href="pepin.html#T21" target="_self">Th21</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E35:81_1_2"/><b>then </b><i><font color="Green">E160</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="pepin.html#T21" target="_self">Th21</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E37:81_1_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">p</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="pepin.html#T16" target="_self">Th16</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E38:81_1_2"/>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E39:81_1_2"/><b>then </b><i><font color="Green">E162</font></i>: 
<font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="pepin.html#T27" target="_self">Th27</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E40:81_1_2"/><b>then </b><i><font color="Green">E163</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E40"><i><font color="Green">Lemma89</font></i></a></i>;<br/><b>then </b><i><font color="Green">E164</font></i>: <font color="Maroon">p</font> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> = 
1 <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<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/><a NAME="E42:81_1_2"/><i><font color="Green">E165</font></i>: 
<font color="Maroon">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1_2_7"/>
<font color="Maroon">x</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/>

<a NAME="E2:81_1_2_7"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="pepin.html#T29" target="_self">Th29</a>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>hence </b><a NAME="E3:81_1_2_7"/>
contradiction
 <b>by </b><i><a class="ref" href="pepin.html#T11" target="_self">Th11</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E43:81_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T12" target="_self">Th12</a>, </i>;<br/><a NAME="E44:81_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K6">mod</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T31" target="_self">Th31</a>, <a class="ref" href="euler_2.html#T10">EULER_2:10</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E46:81_1_2"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 &amp; 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/> <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="pepin.html#T12" target="_self">Th12</a>, <a class="txt" href="pepin.html#E40"><i><font color="Green">Lemma89</font></i></a>, <a class="ref" href="pepin.html#T28" target="_self">Th28</a>, <a class="txt" href="pepin.html#E47"><i><font color="Green">Lemma95</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1

;<br/><a NAME="E49:81_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E50:81_1_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p0">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><b>then </b>1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p5">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pepin.html#T19" target="_self">Th19</a>, <a class="txt" href="pepin.html#E3"><i><font color="Green">Lemma47</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="pepin.html#T19" target="_self">Th19</a>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>
;<br/><b>then </b><font color="Maroon">p</font> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i>, <a class="ref" href="pepin.html#T29" target="_self">Th29</a>, <a class="txt" href="pepin.html#E46"><i><font color="Green">Lemma94</font></i></a>, </i>
<br/>.= 
 <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">n</font>

;<br/><b>hence </b><a NAME="E53:81_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="pepin.html#T10" target="_self">Th10</a>, <a class="ref" href="pepin.html#T11" target="_self">Th11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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