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

<b>let </b><font color="Maroon">Gc</font> be  <a href="gr_cy_1.html#V1">cyclic</a> <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Gc</font>;<br/>



<b>assume </b><a NAME="E1:21"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Gc</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">g</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E2:21"/>
for <font color="Olive">G</font> being  <a href="group_1.html#NM1">Group</a><br/> for <font color="Olive">f</font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Olive">G</font>,<font color="Maroon">Gc</font>  st <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <font color="Olive">f</font> holds <br/><font color="Olive">f</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><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">f</font> be   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">Gc</font>;<br/>



<b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:21_1"/><b>then </b>
 <a href="group_6.html#K12">Image</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Gc</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">f</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 <b>by </b><i><a class="ref" href="group_6.html#T67">GROUP_6:67</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
for <font color="Olive">G</font> being  <a href="group_1.html#NM1">Group</a><br/> for <font color="Olive">f</font> being  <a href="group_6.html#NM2">Homomorphism</a> of <font color="Olive">G</font>,<font color="Maroon">Gc</font>  st <font color="Maroon">g</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_6.html#K12">Image</a> <font color="Olive">f</font> holds <br/><font color="Olive">f</font> <a href="group_6.html#NR2" target="_self">is_epimorphism</a> 
 ;<br/>


</div>
