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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V6">finite</a>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">G</font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 <b>and </b><br/><a NAME="E3:20"/><i><font color="Green">E50</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E4:20"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>(  <a href="group_1.html#K9">ord</a> <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; ( for <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st  <a href="group_1.html#K9">ord</a> <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Olive">G2</font> <a href="group_2.html#R1">=</a> <font color="Olive">G1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:20_1"/>
 ex <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>(  <a href="group_1.html#K9">ord</a> <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st  <a href="group_1.html#K9">ord</a> <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> holds <br/><font color="Olive">G2</font> <a href="group_2.html#R1">=</a> <font color="Olive">G1</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>proof </b></a><div class="add">

<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="E2:20_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="gr_cy_1.html#D9">GR_CY_1:def 9</a></i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E5:20_1_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <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="E6:20_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="group_1.html#K7">ord</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E7:20_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>
<a NAME="E8:20_1_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st  <a href="group_1.html#K9">ord</a> <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> holds <br/><font color="Olive">G2</font> <a href="group_2.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G2</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><a NAME="E1:20_1_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="group_1.html#K9">ord</a> <font color="Maroon">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/>

<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="E3:20_1_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">G2</font> <a href="group_2.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E4:20_1_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , , , , </i>;<br/>
<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:20_1_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K2">*</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E7:20_1_1_1"/>
<font color="Maroon">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:20_1_1_1_1"/>
 ex <font color="Olive">l</font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">l</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as   <a href="int_1.html#NM1">Integer</a> ;<br/>
<a NAME="E2:20_1_1_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , , <a class="ref" href="group_1.html#T90">GROUP_1:90</a></i>;<br/>
<a NAME="E3:20_1_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">p1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xcmplx_1.html#T4">XCMPLX_1:4</a></i>;<br/>
<a NAME="E4:20_1_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default">1 <a href="xcmplx_0.html#K7">/</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/>
<a NAME="E5:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>;<br/>
<b>take </b>
<font color="Maroon">m</font>
;<br/>

<b>thus </b><a NAME="E6:20_1_1_1_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1_1_1_1"/>
<font color="Maroon">s</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:20_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">k</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:20_1_1_1"/>
<font color="Maroon">G2</font> <a href="group_2.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, , , , , </i>;<br/>


</div><b>end;</b></div>
<b>take </b>
 <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
;<br/>

<b>thus </b><a NAME="E9:20_1_1"/>
(  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="group_4.html#K5">gr</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st  <a href="group_1.html#K9">ord</a> <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> holds <br/><font color="Olive">G2</font> <a href="group_2.html#R1">=</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K6">|^</a> <font color="Maroon">s</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> ) )
 <b>by </b><i>, , , <a class="ref" href="gr_cy_1.html#T27">GR_CY_1:27</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20_1"/>
 ex <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>(  <a href="group_1.html#K9">ord</a> <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st  <a href="group_1.html#K9">ord</a> <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> holds <br/><font color="Olive">G2</font> <a href="group_2.html#R1">=</a> <font color="Olive">G1</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:20"/>
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> holds <br/> ex <font color="Olive">G1</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> st <br/>(  <a href="group_1.html#K9">ord</a> <font color="Olive">G1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; ( for <font color="Olive">G2</font> being <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>  st  <a href="group_1.html#K9">ord</a> <font color="Olive">G2</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> holds <br/><font color="Olive">G2</font> <a href="group_2.html#R1">=</a> <font color="Olive">G1</font> ) )
 ;<br/>


</div>
