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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:39"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">n</font>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E5:39"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">H</font> is <a href="group_1.html#V6">finite</a>
 
<b>by </b><i>, <a class="ref" href="group_2.html#T48">GROUP_2:48</a></i>;<br/>
<a NAME="E6:39"/><b>then </b><i><font color="Green">E61</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1"/>
(  <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> 1 or  <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>suppose </b></a><a NAME="E1:39_1_1"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:39_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T82">GROUP_2:82</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1"/>
<font color="Maroon">H</font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_2"><b>suppose </b></a><a NAME="E1:39_1_2"/>
 <a href="group_1.html#K9">ord</a> <font color="Maroon">H</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font><b> such that </b><br/><a NAME="E3:39_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
 <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">a<sub>1</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> );<br/><a NAME="E4:39_1_2"/>
<font color="Maroon">h</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:39_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E7:39_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b><font color="Maroon">h</font> = 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i>, <a class="ref" href="group_1.html#T59">GROUP_1:59</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">H</font>
<b>by </b><i><a class="ref" href="group_2.html#T53">GROUP_2:53</a></i>
;<br/>
<b>hence </b><a NAME="E3:39_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E8:39_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E9:39_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i/>;<br/><a NAME="E10:39_1_2"/>
 ex <font color="Olive">h1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> st <br/>for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>  ex <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">h1</font> <a href="group_1.html#K6">|^</a> <font color="Olive">s</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:39_1_2_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">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="Olive">k</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>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:39_1_2_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> )
 <b>and </b><br/><a NAME="E4:39_1_2_2"/><i><font color="Green">E66</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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; <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> holds <br/><font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font>
 ;<br/>
<b>consider </b><font color="Maroon">h1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font><b> such that </b><br/><a NAME="E6:39_1_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:39"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">h1</font>
;<br/>

<a NAME="E7:39_1_2_2"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>  ex <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="group_1.html#K6">|^</a> <font color="Olive">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/>

<a NAME="E1:39_1_2_2_1"/>
<font color="Maroon">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T51">GROUP_2:51</a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:39_1_2_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">r</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:39_1_2_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E6:39_1_2_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T42">NAT_1:42</a></i>;<br/>
<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font>, <font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:39_1_2_2_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_2_2_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E2:39_1_2_2_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E3:39_1_2_2_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="gr_cy_1.html#T10" target="_self">Th10</a>, <a class="ref" href="group_1.html#T48">GROUP_1:48</a></i>;<br/>
<a NAME="E4:39_1_2_2_1_1"/>
<font color="Maroon">h1</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font>
 
<b>by </b><i>, <a class="ref" href="group_4.html#T3">GROUP_4:3</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K3">"</a>  = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_2.html#T57">GROUP_2:57</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K3">"</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> <b>by </b><i><a class="txt" href="gr_cy_1.html#E2"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p4">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E4:39"><i><font color="Green">E50</font></i></a>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="group_1.html#D5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E9:39_1_2_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="group_2.html#T52">GROUP_2:52</a></i>;<br/>
<b>hence </b><a NAME="E10:39_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D1" target="_self">Def1</a>, <a class="txt" href="gr_cy_1.html#E2:39"><i><font color="Green">E47</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">p</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">n</font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font>
<b>by </b><i><a class="ref" href="gr_cy_1.html#T10" target="_self">Th10</a>, <a class="ref" href="gr_cy_1.html#T13" target="_self">Th13</a>, <a class="ref" href="group_1.html#T50">GROUP_1:50</a></i>
<br/>.= 
<font color="Maroon">h1</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font>
<b>by </b><i>, <a class="ref" href="group_4.html#T3">GROUP_4:3</a></i>
;<br/>
<b>hence </b><a NAME="E10:39_1_2_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E2"><i><font color="Green">Lemma45</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:39_1_2_2"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>  ex <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="group_1.html#K6">|^</a> <font color="Olive">s</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E11:39_1_2"/>
<font color="Maroon">H</font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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