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



<b>set </b><font color="Maroon">f</font> =  <a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span>;<br/>
<a NAME="E1:33"/><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:33"/><i><font color="Green">E24</font></i>: 
 <a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="autgroup.html#K1" target="_self">Aut</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E25</font></i>:  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self">Def1</a>, </i>
;<br/>
<a NAME="E4:33"/>
 <a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font> is   <a href="group_6.html#NM2">Homomorphism</a> of <font color="Maroon">G</font>,<font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E5:33"/><b>then </b>
 <a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="autgroup.html#D1" target="_self">Def1</a>, </i>;<br/>
<b>hence </b><a NAME="E6:33"/>
 <a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i>, , <a class="ref" href="funct_2.html#T36">FUNCT_2:36</a></i>;<br/>


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