<?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/>

<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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">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">c<sub>1</sub></font> implies  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> )
 <b>by </b><i><a class="ref" href="pepin.html#T44" target="_self">Th44</a>, <a class="ref" href="pepin.html#T55" target="_self">Th55</a></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">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></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">E67</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">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">c<sub>1</sub></font>
 ;<br/><b>assume </b><a NAME="E3:81_1_2"/><i><font color="Green">E68</font></i>: 
not  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2
 <b>by </b><i><a class="ref" href="pepin.html#T60" target="_self">Th60</a></i>;<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="E6:81_1_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E7:81_1_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a>  <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></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">E71</font></i>: 
 <a href="pepin.html#K5" target="_self">Fermat</a> <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#K2">*</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E7:81_1_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E10:81_1_2"/><i><font color="Green">E72</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">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">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E2:81_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="pepin.html#E9:81_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="int_1.html#T41">INT_1:41</a></i>;<br/><a NAME="E11:81_1_2"/>
<font color="Maroon">c<sub>2</sub></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">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 3
 ;<br/>

<a NAME="E2:81_1_2_1"/>
1 <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>3</sub></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#K8">hcf</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="pepin.html#E9:81_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="pepin.html#E1:81_1_2_1"><i><font color="Green">E73</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="txt" href="pepin.html#E1:81_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="pepin.html#T62" target="_self">Th62</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:81_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
3,<font color="Maroon">c<sub>2</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="pepin.html#T44" target="_self">Th44</a>, <a class="ref" href="int_2.html#T47">INT_2:47</a></i>;<br/><a NAME="E13:81_1_2"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="pepin.html#E83"><i><font color="Green">Lemma60</font></i></a></i>;<br/><a NAME="E14:81_1_2"/><b>then </b><i><font color="Green">E75</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="pepin.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E15:81_1_2"/><i><font color="Green">E76</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E16:81_1_2"/><i><font color="Green">E77</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="pepin.html#E82"><i><font color="Green">Lemma59</font></i></a></i>;<br/><a NAME="E17:81_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2
 ;<br/>

<a NAME="E2:81_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>2</sub></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">E79</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font> is <a href="abian.html#V1">even</a>
 
<b>by </b><i><a class="txt" href="pepin.html#E85"><i><font color="Green">Lemma62</font></i></a></i>;<br/>
<a NAME="E5:81_1_2_2"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="pepin.html#E3:81_1_2_2"><i><font color="Green">E79</font></i></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">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E69</font></i></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><a class="txt" href="pepin.html#E7:81_1_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="pepin.html#E5:81_1_2_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="pepin.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:81_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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">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">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E2:81_1_2"><i><font color="Green">E67</font></i></a>, <a class="ref" href="pepin.html#T41" target="_self">Th41</a></i>;<br/><a NAME="E19:81_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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E16:81_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="pepin.html#T28" target="_self">Th28</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>1</sub></font>
 ;<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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E9:81_1_2"><i><font color="Green">E71</font></i></a>, <a class="ref" href="int_1.html#T41">INT_1:41</a></i>;<br/><a NAME="E22:81_1_2"/><b>then </b><i><font color="Green">E79</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E14:81_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="pepin.html#E15:81_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="pepin.html#E57"><i><font color="Green">Lemma42</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E23:81_1_2"/><i><font color="Green">E80</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E12:81_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="pepin.html#E13:81_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="pepin.html#E15:81_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="pepin.html#E22:81_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="pepin.html#T52" target="_self">Th52</a></i>;<br/><a NAME="E24:81_1_2"/><i><font color="Green">E81</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="pepin.html#E12:81_1_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="pepin.html#T54" target="_self">Th54</a></i>;<br/><a NAME="E25:81_1_2"/><i><font color="Green">E82</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="pepin.html#E12:81_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="pepin.html#E15:81_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="pepin.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E26:81_1_2"/><i><font color="Green">E83</font></i>: 
not  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E12:81_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="pepin.html#E15:81_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="pepin.html#T53" target="_self">Th53</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K3">div</a> 2</span>)</span>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pepin.html#E15:81_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="pepin.html#T40" target="_self">Th40</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">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="pepin.html#E10:81_1_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="pepin.html#T43" target="_self">Th43</a></i>;<br/>
<a NAME="E5:81_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> 2
 
<b>by </b><i><a class="txt" href="pepin.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:81_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E17:81_1_2"><i><font color="Green">E78</font></i></a>, <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">c<sub>5</sub></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">c<sub>1</sub></font></span>)</span>;<br/><a NAME="E27:81_1_2"/><i><font color="Green">E84</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">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/><a NAME="E28:81_1_2"/><i><font color="Green">E85</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">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma4</font></i></a></i>;<br/><a NAME="E29:81_1_2"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></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="E30:81_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pepin.html#E23:81_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E31:81_1_2"/><i><font color="Green">E88</font></i>: 
not  <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">div</a> 2
 <b>by </b><i><a class="txt" href="pepin.html#E26:81_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="pepin.html#E29:81_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E32:81_1_2"/>
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></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">E89</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="pepin.html#E1:81_1_2_4"><i><font color="Green">E89</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">c<sub>2</sub></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#E25:81_1_2"><i><font color="Green">E82</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">c<sub>2</sub></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="txt" href="pepin.html#E26:81_1_2"><i><font color="Green">E83</font></i></a>, <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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E30:81_1_2"><i><font color="Green">E87</font></i></a>, <a class="txt" href="pepin.html#E31:81_1_2"><i><font color="Green">E88</font></i></a>, <a class="ref" href="pepin.html#T39" target="_self">Th39</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">c<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
<b>by </b><i><a class="txt" href="pepin.html#E29:81_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E34:81_1_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pepin.html#E29:81_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E36:81_1_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K5">-'</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">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="pepin.html#E24:81_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E37:81_1_2"/>
<font color="Maroon">c<sub>2</sub></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#E15:81_1_2"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E38:81_1_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" target="_self">order</a> 3,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pepin.html#E36:81_1_2"><i><font color="Green">E90</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E39:81_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E34:81_1_2"><i><font color="Green">E89</font></i></a></i>;<br/><b>then </b><i><font color="Green">E93</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K4">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">c<sub>1</sub></font></span>)</span></span>)</span> = 
1 <a href="nat_1.html#K4">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">c<sub>1</sub></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="txt" href="pepin.html#E27:81_1_2"><i><font color="Green">E84</font></i></a>, <a class="ref" href="nat_1.html#T76">NAT_1:76</a></i>
;<br/><a NAME="E41:81_1_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E39:81_1_2"><i><font color="Green">E92</font></i></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="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E42:81_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K4">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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E9:81_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="pepin.html#E84"><i><font color="Green">Lemma61</font></i></a></i>;<br/><a NAME="E43:81_1_2"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K4">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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="pepin.html#E40:81_1_2"><i><font color="Green">E93</font></i></a>, <a class="ref" href="euler_2.html#T10">EULER_2:10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E45:81_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K1">+</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">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/> <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E9:81_1_2"><i><font color="Green">E71</font></i></a>, <a class="txt" href="pepin.html#E34:81_1_2"><i><font color="Green">E89</font></i></a>, <a class="txt" href="pepin.html#E38:81_1_2"><i><font color="Green">E91</font></i></a>, <a class="txt" href="pepin.html#E45:81_1_2"><i><font color="Green">E95</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1

;<br/><a NAME="E47:81_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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 ;<br/><a NAME="E48:81_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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="nat_1.html#K2">*</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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K3">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">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="pepin.html#E28:81_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="pepin.html#T48" target="_self">Th48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K2">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="pepin.html#E28:81_1_2"><i><font color="Green">E85</font></i></a>, <a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>
;<br/><b>then </b><font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="txt" href="pepin.html#E27:81_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="pepin.html#E39:81_1_2"><i><font color="Green">E92</font></i></a>, <a class="txt" href="pepin.html#E41:81_1_2"><i><font color="Green">E94</font></i></a>, <a class="txt" href="pepin.html#E3"><i><font color="Green">Lemma3</font></i></a></i>
<br/>.= 
 <a href="pepin.html#K5" target="_self">Fermat</a> <font color="Maroon">c<sub>1</sub></font>

;<br/><b>hence </b><a NAME="E51:81_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pepin.html#E3:81_1_2"><i><font color="Green">E68</font></i></a>, <a class="txt" href="pepin.html#E6:81_1_2"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
