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

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 &amp; <font color="Maroon" title="c1">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="c2">n</font> implies  ex <font color="Olive" title="b1">k</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>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>



<b>assume </b><a NAME="E1:73"/><i><font color="Green" title="E57">A1</font></i>: 
( <font color="Maroon" title="c1">p</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp; <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 2 &amp; <font color="Maroon" title="c1">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="c2">n</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">k</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>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/>

<b>set </b><font color="Maroon" title="c3">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="c2">n</font></span>)</span>;<br/>
<a NAME="E2:73"/><i><font color="Green" title="E58">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">t</font> = <span class="p1">(<span class="default"><a href="pepin.html#K5" title="PEPIN:func.5">Fermat</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font>;<br/>
<a NAME="E3:73"/><i><font color="Green" title="E59">A3</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="nat_d.html#T3" title="NAT_D:th.3">NAT_D:3</a></i>;<br/>
<a NAME="E4:73"/><i><font color="Green" title="E60">A4</font></i>: 
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="c2">n</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="pepin.html#T26" target="_self" title="PEPIN:th.26">Th26</a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<a NAME="E5:73"/><b>then </b>
<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="c2">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c2">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/>
<a NAME="E6:73"/><b>then </b><i><font color="Green" title="E61">A5</font></i>: 
<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="c2">n</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="#E4:73"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:73"/><i><font color="Green" title="E62">A6</font></i>: 
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 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="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T32" title="INT_1:th.32">INT_1:32</a></i>;<br/>
<a NAME="E8:73"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
;<br/>
<a NAME="E9:73"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font></span>)</span>, <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#D3" title="INT_1:def.3">INT_1:def 3</a></i>;<br/>
<a NAME="E10:73"/><b>then </b><i><font color="Green" title="E63">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</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="c2">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.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="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.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="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="#E7:73"><i><font color="Green" title="E62">A6</font></i></a>, <a class="ref" href="int_1.html#T37" title="INT_1:th.37">INT_1:37</a></i>;<br/>
<a NAME="E11:73"/><b>then </b>
<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="c2">n</font></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> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="#E3:73"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="int_1.html#T39" title="INT_1:th.39">INT_1:39</a></i>;<br/>
<a NAME="E12:73"/><b>then </b>
<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="c2">n</font></span>)</span></span>)</span> <a href="pepin.html#K2" title="PEPIN:func.2">^2</a> </span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="#E2:73"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E6:73"><i><font color="Green" title="E61">A5</font></i></a>, <a class="txt" href="pepin.html#E53"><i><font color="Green" title="E36">Lm15</font></i></a></i>;<br/>
<a NAME="E13:73"/><b>then </b><i><font color="Green" title="E64">A8</font></i>: 
<span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="pepin.html#E36"><i><font color="Green" title="E21">Lm12</font></i></a></i>;<br/>
<a NAME="E14:73"/><i><font color="Green" title="E65">A9</font></i>: 
<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="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/>
<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="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:73_1"/><b>then </b>
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="c2">n</font></span>)</span>,1 <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="#E2:73"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="pepin.html#T40" target="_self" title="PEPIN:th.40">Th40</a></i>;<br/>
<a NAME="E3:73_1"/><b>then </b>
1,2 <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="c2">n</font></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">are_congruent_mod</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="int_1.html#T35" title="INT_1:th.35">INT_1:35</a></i>;<br/>
<a NAME="E4:73_1"/><b>then </b>
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="c1">p</font>
 
<b>by </b><i><a class="txt" href="#E3:73"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="#E10:73"><i><font color="Green" title="E63">A7</font></i></a>, <a class="ref" href="int_1.html#T36" title="INT_1:th.36">INT_1:36</a></i>;<br/>
<a NAME="E5:73_1"/><b>then </b>
<font color="Maroon" title="c1">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>
 
<b>by </b><i><a class="ref" href="int_2.html#T19" title="INT_2:th.19">INT_2:19</a></i>;<br/>
<b>hence </b><a NAME="E6:73_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E15:73"/><i><font color="Green" title="E66">A10</font></i>: 
2,<font color="Maroon" title="c1">p</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></i>;<br/>
<a NAME="E16:73"/><b>then </b><i><font color="Green" title="E67">A11</font></i>: 
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 2,<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:73"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E13:73"><i><font color="Green" title="E64">A8</font></i></a>, <a class="ref" href="pepin.html#T52" target="_self" title="PEPIN:th.52">Th52</a>, <a class="ref" href="prepower.html#T13" title="PREPOWER:th.13">PREPOWER:13</a></i>;<br/>
<a NAME="E17:73"/>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 2,<font color="Maroon" title="c1">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> 
 
<b>by </b><i><a class="txt" href="#E2:73"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E15:73"><i><font color="Green" title="E66">A10</font></i></a>, <a class="ref" href="pepin.html#D2" target="_self" title="PEPIN:def.2">Def2</a></i>;<br/>
<a NAME="E18:73"/><b>then </b>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 2,<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E19:73"/><b>then </b><i><font color="Green" title="E68">A12</font></i>: 
(  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 2,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 2,<font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>
<span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2 = 
<span class="p1">(<span class="default">2 <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> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
<b>by </b><i><a class="ref" href="newton.html#T11" title="NEWTON:th.11">NEWTON:11</a></i>
<br/>.= 
2 <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c2">n</font>
<b>by </b><i><a class="ref" href="nat_d.html#T18" title="NAT_D:th.18">NAT_D:18</a></i>
;<br/>
<a NAME="E21:73"/><b>then </b>
not  <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 2,<font color="Maroon" title="c1">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#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> 2
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="#E2:73"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="#E14:73"><i><font color="Green" title="E65">A9</font></i></a>, <a class="ref" href="pepin.html#T53" target="_self" title="PEPIN:th.53">Th53</a>, <a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></i>;<br/>
<a NAME="E22:73"/><b>then </b>
 <a href="pepin.html#K4" title="PEPIN:func.4">order</a> 2,<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="#E16:73"><i><font color="Green" title="E67">A11</font></i></a>, <a class="txt" href="#E19:73"><i><font color="Green" title="E68">A12</font></i></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>, <a class="ref" href="nat_d.html#T6" title="NAT_D:th.6">NAT_D:6</a></i>;<br/>
<a NAME="E23:73"/><b>then </b>
2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<b>by </b><i><a class="txt" href="#E1:73"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="pepin.html#T54" target="_self" title="PEPIN:th.54">Th54</a>, <a class="ref" href="int_2.html#T44" title="INT_2:th.44">INT_2:44</a>, <a class="ref" href="int_2.html#T47" title="INT_2:th.47">INT_2:47</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">t2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E25:73"/><i><font color="Green" title="E69">A13</font></i>: 
<font color="Maroon" title="c1">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">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">t2</font>
 <b>by </b><i><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">t2</font> = <font color="Maroon" title="c5">t2</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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E27:73"/>
<font color="Maroon" title="c1">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
 
<b>by </b><i><a class="txt" href="#E2:73"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E28:73"/><b>then </b>
<font color="Maroon" title="c1">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> <span class="p1">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">t2</font>
 
<b>by </b><i><a class="txt" href="#E25:73"><i><font color="Green" title="E69">A13</font></i></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></i>;<br/>
<a NAME="E29:73"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t2</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<b>hence </b><a NAME="E30:73"/>
 ex <font color="Olive" title="b1">k</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>  st <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="newton.html#K16" title="NEWTON:func.16">|^</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
