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

<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">f1</font> = <span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="partfun1.html#K1">*</a> <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>;<br/>
<b>set </b><font color="Maroon">f2</font> =  <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>;<br/>
<a NAME="E1:31"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

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

<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#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span> = 
<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#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2">|^</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="group_3.html#K2">|^</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_3.html#T30">GROUP_3:30</a></i>
;<br/>
<b>hence </b><a NAME="E2:31_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

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

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> = 
<font color="Maroon">b</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="autgroup.html#E2"><i><font color="Green">Lemma27</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
<span class="p1">(<span class="default"><a href="autgroup.html#K6" target="_self">Conjugate</a> <font color="Maroon">a</font></span>)</span> <a href="partfun1.html#K1">*</a> <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="funct_2.html#R2">=</a>  <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><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
