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

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



<b>assume </b><a NAME="E1:79"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 &amp; <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> )
 ;<br/>

<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="E2:79"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="pepin.html#T6" target="_self">Th6</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">t</font> = <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#K5">div</a> <font color="Maroon">p</font>;<br/>
<a NAME="E3:79"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">p</font> <a href="nat_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="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</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#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="pepin.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T49">NAT_1:49</a></i>;<br/>
<a NAME="E4:79"/>
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E5:79"/><b>then </b><i><font color="Green">E50</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="E6:79"/>
<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#K2">*</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="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="nat_1.html#K2">*</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="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:79"/><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="pepin.html#K2" target="_self">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</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>
 
;<br/>
<a NAME="E8:79"/><b>then </b><i><font color="Green">E51</font></i>: 
<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="pepin.html#K2" target="_self">^2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E9:79"/><i><font color="Green">E52</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> 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="ref" href="int_1.html#T32">INT_1:32</a></i>;<br/>
<a NAME="E10:79"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_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="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="nat_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="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span>
 
;<br/>
<a NAME="E11:79"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_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="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span>,0 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D3">INT_1:def 3</a></i>;<br/>
<a NAME="E12:79"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_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="nat_1.html#K5">div</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span>,0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> 1</span>)</span> <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#T11" target="_self">Th11</a>, <a class="ref" href="int_1.html#T37">INT_1:37</a></i>;<br/>
<a NAME="E13:79"/><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="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> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="pepin.html#T7" target="_self">Th7</a>, <a class="txt" href="pepin.html#E1"><i><font color="Green">Lemma43</font></i></a></i>;<br/>
<a NAME="E14:79"/><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="pepin.html#K2" target="_self">^2</a> ,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 
;<br/>
<a NAME="E15:79"/><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="pepin.html#K2" target="_self">^2</a> </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#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E16:79"/><b>then </b><i><font color="Green">E98</font></i>: 
<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> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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/>;<br/>
<a NAME="E17:79"/><i><font color="Green">E100</font></i>: 
<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#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:79_1"/>
<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#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<a NAME="E2:79_1"/><b>then </b>
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>,1 <a href="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:79_1"/><b>then </b>
1,2 <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="int_1.html#R1">are_congruent_mod</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T35">INT_1:35</a></i>;<br/>
<a NAME="E4:79_1"/><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="ref" href="pepin.html#T7" target="_self">Th7</a>, , <a class="ref" href="int_1.html#T36">INT_1:36</a></i>;<br/>
<a NAME="E5:79_1"/><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="E6:79_1"/><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="E7:79_1"/>
contradiction
 <b>by </b><i><a class="ref" href="pepin.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:79"/><i><font color="Green">E102</font></i>: 
2,<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#T6" target="_self">Th6</a>, <a class="ref" href="int_2.html#T44">INT_2:44</a>, <a class="ref" href="int_2.html#T47">INT_2:47</a></i>;<br/>
<a NAME="E19:79"/>
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="prepower.html#T13">PREPOWER:13</a></i>;<br/>
<a NAME="E20:79"/><b>then </b><i><font color="Green">E104</font></i>: 
 <a href="pepin.html#K4" target="_self">order</a> 2,<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"><font color="Maroon">n</font> <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#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E3:79"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E21:79"/>
 <a href="pepin.html#K4" target="_self">order</a> 2,<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>, </i>;<br/>
<a NAME="E22:79"/><b>then </b>
 <a href="pepin.html#K4" target="_self">order</a> 2,<font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E23:79"/><b>then </b><i><font color="Green">E124</font></i>: 
(  <a href="pepin.html#K4" target="_self">order</a> 2,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 or  <a href="pepin.html#K4" target="_self">order</a> 2,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 2 = 
<span class="p1">(<span class="default">2 <a href="nat_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="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
2 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>
;<br/>
<a NAME="E25:79"/><b>then </b>
not  <a href="pepin.html#K4" target="_self">order</a> 2,<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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K5">div</a> 2
 
<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#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E5:79"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E26:79"/><b>then </b>
 <a href="pepin.html#K4" target="_self">order</a> 2,<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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="pepin.html#E21"><i><font color="Green">Lemma74</font></i></a>, <a class="txt" href="pepin.html#E22"><i><font color="Green">Lemma75</font></i></a>, , <a class="ref" href="int_2.html#T44">INT_2:44</a>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<a NAME="E27:79"/><b>then </b>
2 <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <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#T6" target="_self">Th6</a>, <a class="txt" href="pepin.html#E20"><i><font color="Green">Lemma73</font></i></a>, <a class="txt" href="pepin.html#E8:79"><i><font color="Green">E51</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">t2</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E29:79"/><i><font color="Green">E125</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">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t2</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">t2</font> = <font color="Maroon">t2</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="E31:79"/>
<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="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E32:79"/><b>then </b>
<font color="Maroon">p</font> <a href="xcmplx_0.html#K6">-</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <font color="Maroon">t2</font>
 
<b>by </b><i><a class="txt" href="pepin.html#E23"><i><font color="Green">Lemma76</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<a NAME="E33:79"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">t2</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E34:79"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">k</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"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/>


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