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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> implies  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> implies  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:80_1_1"/>
( 3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> implies  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T44" target="_self" title="PEPIN:th.44">Th44</a>, <a class="ref" href="pepin.html#T55" target="_self" title="PEPIN:th.55">Th55</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_2"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> implies  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  )</span><br/><div class="add"><a NAME="E2:80_1_2"/>
 <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pepin.html#T60" target="_self" title="PEPIN:th.60">Th60</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:80_1_2"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
<font color="Maroon" title="c2">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E5:80_1_2"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T48" title="INT_2:th.48">INT_2:48</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c3">d</font> =  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:80_1_2"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
 <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_1_2"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E8:80_1_2"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:80_1_2_1"/>
<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:80_1_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E3:80_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A6</font></span>: 
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E4:80_1_2_1"/><span class="lab"><font color="Green" title="E63">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/>
<a NAME="E5:80_1_2_1"/>
not  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> is  <a href="abian.html#V1" title="ABIAN:attr.1">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="pepin.html#E79"><span class="lab"><font color="Green" title="E53">Lm16</font></span></a></span>;<br/>
<a NAME="E6:80_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E3:80_1_2_1"><span class="lab"><font color="Green" title="E62">A6</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a>, <a class="ref" href="nat_2.html#T24" title="NAT_2:th.24">NAT_2:24</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:80_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:80_1_2"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="txt" href="#E4:80_1_2_1"><span class="lab"><font color="Green" title="E63">A7</font></span></a>, <a class="ref" href="pepin.html#T6" target="_self" title="PEPIN:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E9:80_1_2"/><span class="lab"><font color="Green" title="E62">A8</font></span>: 
<span class="p1">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><span class="kw">assume </span><a NAME="E10:80_1_2"/><span class="lab"><font color="Green" title="E63">A9</font></span>: 
3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> </span><br/><a NAME="E11:80_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A10</font></span>: 
<span class="p1">(<span class="default">3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> ,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T39" title="INT_1:th.39">INT_1:39</a></span>;<br/><a NAME="E12:80_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="pepin.html#E76"><span class="lab"><font color="Green" title="E50">Lm13</font></span></a></span>;<br/><a NAME="E13:80_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A11</font></span>: 
3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a>  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:80_1_2"><span class="lab"><font color="Green" title="E64">A10</font></span></a>, <a class="ref" href="pepin.html#T28" target="_self" title="PEPIN:th.28">Th28</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c5">2N</font> = 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><span class="kw">assume </span><a NAME="E14:80_1_2"/><span class="lab"><font color="Green" title="E66">A12</font></span>: 
not  <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c4">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/><a NAME="E16:80_1_2"/><span class="lab"><font color="Green" title="E67">A13</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>;<br/><a NAME="E17:80_1_2"/><span class="lab"><font color="Green" title="E68">A14</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:80_1_2_2"/>
1 <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T64" title="NEWTON:th.64">NEWTON:64</a></span>;<br/>
<a NAME="E2:80_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A15</font></span>: 
<span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 1</span>)</span> <a href="nat_d.html#K6" title="NAT_D:func.6">gcd</a> <span class="p1">(<span class="default">3 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_1.html#T6" title="EULER_1:th.6">EULER_1:6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:80_1_2_2"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:80_1_2_2"/><span class="kw">then </span>
not 3, <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_2"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E2:80_1_2_2"><span class="lab"><font color="Green" title="E69">A15</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:80_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_2"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="ref" href="pepin.html#T62" target="_self" title="PEPIN:th.62">Th62</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:80_1_2"/><span class="kw">then </span>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="pepin.html#T44" target="_self" title="PEPIN:th.44">Th44</a>, <a class="ref" href="pepin.html#T54" target="_self" title="PEPIN:th.54">Th54</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E20:80_1_2"/><span class="lab"><font color="Green" title="E69">A16</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><a NAME="E21:80_1_2"/><span class="lab"><font color="Green" title="E70">A17</font></span>: 
3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:80_1_2"><span class="lab"><font color="Green" title="E63">A9</font></span></a>, <a class="txt" href="#E7:80_1_2"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="ref" href="int_1.html#T41" title="INT_1:th.41">INT_1:41</a></span>;<br/><a NAME="E22:80_1_2"/><span class="lab"><font color="Green" title="E71">A18</font></span>: 
not  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:80_1_2_3"/>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:80_1_2_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E17:80_1_2"><span class="lab"><font color="Green" title="E68">A14</font></span></a>, <a class="txt" href="#E16:80_1_2"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="pepin.html#T44" target="_self" title="PEPIN:th.44">Th44</a>, <a class="ref" href="pepin.html#T53" target="_self" title="PEPIN:th.53">Th53</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></span>;<br/>
<a NAME="E3:80_1_2_3"/><span class="kw">then </span>
3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2</span>)</span>,1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80_1_2"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="pepin.html#T40" target="_self" title="PEPIN:th.40">Th40</a></span>;<br/>
<a NAME="E4:80_1_2_3"/><span class="kw">then </span>
1, <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:80_1_2"><span class="lab"><font color="Green" title="E70">A17</font></span></a>, <a class="ref" href="pepin.html#T43" target="_self" title="PEPIN:th.43">Th43</a></span>;<br/>
<a NAME="E5:80_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="int_1.html#R2" title="INT_1:pred.2">divides</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:80_1_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:80_1_2"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E23:80_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A19</font></span>: 
not  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_1_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E24:80_1_2"/><span class="lab"><font color="Green" title="E73">A20</font></span>: 
3,<font color="Maroon" title="c2">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E17:80_1_2"><span class="lab"><font color="Green" title="E68">A14</font></span></a>, <a class="ref" href="pepin.html#T44" target="_self" title="PEPIN:th.44">Th44</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></span>;<br/><a NAME="E25:80_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A21</font></span>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80_1_2"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="pepin.html#D2" target="_self" title="PEPIN:def.2">Def2</a></span>;<br/><a NAME="E26:80_1_2"/><span class="lab"><font color="Green" title="E75">A22</font></span>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:80_1_2_4"/><span class="lab"><font color="Green" title="E76">A23</font></span>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:80_1_2_4_1_1"/>
(  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 or  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:80_1_2_4"><span class="lab"><font color="Green" title="E76">A23</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:80_1_2_4_1_1_1"/>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:80_1_2_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:80_1_2"><span class="lab"><font color="Green" title="E74">A21</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="80_1_2_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:80_1_2_4_1_1_2"/>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:80_1_2_4_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:80_1_2"><span class="lab"><font color="Green" title="E71">A18</font></span></a>, <a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:80_1_2_4"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E27:80_1_2"/>
3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="pepin.html#E77"><span class="lab"><font color="Green" title="E51">Lm14</font></span></a>, <a class="ref" href="pepin.html#T26" target="_self" title="PEPIN:th.26">Th26</a></span>;<br/><a NAME="E28:80_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">3 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_2"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E16:80_1_2"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="txt" href="#E13:80_1_2"><span class="lab"><font color="Green" title="E65">A11</font></span></a>, <a class="txt" href="pepin.html#E51"><span class="lab"><font color="Green" title="E34">Lm11</font></span></a>, <a class="ref" href="int_1.html#T41" title="INT_1:th.41">INT_1:41</a></span>;<br/><a NAME="E29:80_1_2"/><span class="kw">then </span>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <span class="p1">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:80_1_2"><span class="lab"><font color="Green" title="E73">A20</font></span></a>, <a class="txt" href="#E16:80_1_2"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="pepin.html#T52" target="_self" title="PEPIN:th.52">Th52</a></span>;<br/><a NAME="E30:80_1_2"/><span class="kw">then </span>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_1_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><span class="kw">then </span> <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> = 
2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:80_1_2"><span class="lab"><font color="Green" title="E72">A19</font></span></a>, <a class="txt" href="#E26:80_1_2"><span class="lab"><font color="Green" title="E75">A22</font></span></a>, <a class="ref" href="pepin.html#T39" target="_self" title="PEPIN:th.39">Th39</a>, <a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_1_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
;<br/><a NAME="E32:80_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A24</font></span>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:80_1_2"><span class="lab"><font color="Green" title="E62">A8</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E33:80_1_2"/><span class="lab"><font color="Green" title="E77">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_2"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="pepin.html#E78"><span class="lab"><font color="Green" title="E52">Lm15</font></span></a></span>;<br/><a NAME="E34:80_1_2"/>
<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:80_1_2"><span class="lab"><font color="Green" title="E67">A13</font></span></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></span>;<br/><a NAME="E35:80_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A26</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <span class="p1">(<span class="default"><a href="pepin.html#K4" title="PEPIN:func.4">order</a> 3,<font color="Maroon" title="c2">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:80_1_2"><span class="lab"><font color="Green" title="E69">A16</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/><a NAME="E36:80_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A27</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:80_1_2"><span class="lab"><font color="Green" title="E76">A24</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c2">p</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> = 
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" title="NAT_D:th.21">NAT_D:21</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="pepin.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a>, <a class="ref" href="nat_d.html#T24" title="NAT_D:th.24">NAT_D:24</a></span>
;<br/><a NAME="E38:80_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:80_1_2"><span class="lab"><font color="Green" title="E77">A25</font></span></a>, <a class="ref" href="euler_2.html#T10" title="EULER_2:th.10">EULER_2:10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">y</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E40:80_1_2"/><span class="lab"><font color="Green" title="E80">A28</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">and </span><br/><a NAME="E41:80_1_2"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D2" title="NAT_D:def.2">NAT_D:def 2</a></span>;<br/><a NAME="E42:80_1_2"/><span class="lab"><font color="Green" title="E81">A29</font></span>: 
2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="pepin.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><a NAME="E43:80_1_2"/><span class="lab"><font color="Green" title="E82">A30</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:80_1_2_7"/>
<font color="Maroon" title="c7">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:80_1_2_7"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:80_1_2"><span class="lab"><font color="Green" title="E79">A27</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:80_1_2_7"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c8">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/> <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">y</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:80_1_2"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E32:80_1_2"><span class="lab"><font color="Green" title="E76">A24</font></span></a>, <a class="txt" href="#E35:80_1_2"><span class="lab"><font color="Green" title="E78">A26</font></span></a>, <a class="txt" href="#E40:80_1_2"><span class="lab"><font color="Green" title="E80">A28</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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" title="c7">x</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A31</font></span>: 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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" title="c7">x</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p0">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">y</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:80_1_2"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="ref" href="pepin.html#T48" target="_self" title="PEPIN:th.48">Th48</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p4">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c9">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:80_1_2"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></span>
;<br/><a NAME="E47:80_1_2"/>
2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p1">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="pepin.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c2">p</font> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <span class="p3">(<span class="default">2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:80_1_2"><span class="lab"><font color="Green" title="E79">A27</font></span></a>, <a class="txt" href="#E43:80_1_2"><span class="lab"><font color="Green" title="E82">A30</font></span></a>, <a class="txt" href="#E46:80_1_2"><span class="lab"><font color="Green" title="E83">A31</font></span></a>, <a class="txt" href="pepin.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>
<br/>.= 
 <a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c1">n</font>

;<br/><span class="kw">hence </span><a NAME="E49:80_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:80_1_2"><span class="lab"><font color="Green" title="E66">A12</font></span></a>, <a class="txt" href="#E4:80_1_2"><span class="lab"><font color="Green" title="E58">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
