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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="group_1.html#V6">finite</a>
 ;<br/>

<b>thus </b><a NAME="E2:38"/>
( <font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="group_1.html#K7">ord</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">c<sub>1</sub></font> is  <a href="gr_cy_1.html#V1" target="_self">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9" target="_self">Def9</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font>
;<br/>

<b>thus </b><a NAME="E4:38_1"/>
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gr_cy_1.html#E3:38_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" target="_self">Th27</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E3:38"/><i><font color="Green">E35</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E4:38"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>1</sub></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="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:38_2"/><i><font color="Green">E36</font></i>: 
 <a href="group_1.html#K7">ord</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E3:38"><i><font color="Green">E35</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E3:38_2"/>
 <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"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:38"><i><font color="Green">E34</font></i></a>, <a class="txt" href="gr_cy_1.html#E2:38_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="gr_cy_1.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E4:38_2"/>
<font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="gr_cy_1.html#E1:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="group_2.html#T85">GROUP_2:85</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:38"/>
<font color="Maroon">c<sub>1</sub></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><a class="ref" href="gr_cy_1.html#D9" target="_self">Def9</a></i>;<br/>


</div>
