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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:25"/>
<font color="Maroon">c<sub>2</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:25"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</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>4</sub></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/>
<a NAME="E4:25"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> st  <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</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="25_1"><b>proof </b></a><div class="add">

<a NAME="E1:25_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E3:25_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>ex <font color="Olive">b<sub>2</sub></font> being  <a href="int_1.html#NM1">Integer</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K6">|^</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/>

<a NAME="E1:25_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="group_6.html#T54">GROUP_6:54</a></i>;<br/>
<a NAME="E4:25_1_1"/>
<font color="Maroon">c<sub>7</sub></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"><font color="Maroon">c<sub>4</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:25"><i><font color="Green">E19</font></i></a>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="int_1.html#NM1">Integer</a><b> such that </b><br/><a NAME="E6:25_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#T25">GR_CY_1:25</a></i>;<br/>
<i><font color="Green">E23</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="gr_cy_2.html#E3:25_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="gr_cy_2.html#E6:25_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="group_6.html#T46">GROUP_6:46</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="group_4.html#T4">GROUP_4:4</a></i>
;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>thus </b><a NAME="E8:25_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K6">|^</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E7:25_1_1"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E4:25_1"/>
 <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</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>5</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="gr_cy_2.html#E3:25_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="gr_cy_2.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:25"/>
 <a href="group_6.html#K12">Image</a> <font color="Maroon">c<sub>3</sub></font> is  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i><a class="ref" href="gr_cy_1.html#D9">GR_CY_1:def 9</a></i>;<br/>


</div>
