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

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ;<br/>
<a NAME="E1:62_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:62_1_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="newton.html#T65">NEWTON:65</a></i>;<br/>
<b>hence </b><a NAME="E3:62_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:62_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:62_1_1_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><a href="euler_1.html#K1">Euler</a> <font color="Maroon">p</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
 
<b>by </b><i>, <a class="txt" href="pepin.html#E2:62_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="euler_2.html#T33">EULER_2:33</a></i>;<br/>
<a NAME="E2:62_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:62_1_1_2"/><b>then </b>
 <a href="euler_1.html#K1">Euler</a> <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#E2"><i><font color="Green">Lemma46</font></i></a></i>;<br/>
<a NAME="E4:62_1_1_2"/><b>then </b>
 <a href="euler_1.html#K1">Euler</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E5:62_1_1_2"/><b>then </b>
 <a href="euler_1.html#K1">Euler</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:62_1_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  is  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:62_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:62_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:62_1_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 )
 ;<br/>
<b>thus </b><a NAME="E4:62_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="pepin.html#E3:62_1_1_3"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <font color="Olive">k</font> where <font color="Olive">k</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">A</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>;<br/>
<a NAME="E6:62_1_1"/>
<font color="Maroon">a</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<a NAME="E7:62_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
 ex <font color="Olive">kk</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">kk</font>]
 
;<br/>
<b>consider </b><font color="Maroon">kk</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:62_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">kk</font>]
 <b>and </b><br/><a NAME="E10:62_1_1"/><i><font color="Green">E98</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:62_1_1"/><i><font color="Green">E100</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="pepin.html#E7:62_1_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">kk</font>
;<br/>

<a NAME="E13:62_1_1"/>
for <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="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 holds <br/><font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:62_1_1_4"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></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="E2:62_1_1_4"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
;<br/>
<b>hence </b><a NAME="E3:62_1_1_4"/>
<font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="pepin.html#E9:62_1_1"><i><font color="Green">E53</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:62_1_1"/>
( <font color="Maroon">kk</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">kk</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 &amp; ( for <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="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span> <a href="nat_1.html#K6">mod</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1 holds <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">kk</font> &amp; <font color="Maroon">kk</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> ) ) )
 <b>by </b><i><a class="txt" href="pepin.html#E4"><i><font color="Green">Lemma54</font></i></a></i>;<br/>


</div>
