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

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V6">finite</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>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>;<br/>





<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E2:15"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E3:15"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="group_3.html#R2">are_conjugated</a> 
 <b>by </b><i><a class="ref" href="group_3.html#T95">GROUP_3:95</a></i>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:15"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">d</font>
 <b>by </b><i>, , <a class="ref" href="group_3.html#T88">GROUP_3:88</a></i>;<br/>
<a NAME="E6:15"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Cad</font> = <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K13">*</a> <font color="Maroon">d</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> ;<br/>
<b>consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">C</font> being  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:15"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="group_2.html#K12">*</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span>
 <b>and </b><br/><a NAME="E10:15"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Cad</font>
 <b>and </b><br/><a NAME="E11:15"/>
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E12:15"/><i><font color="Green">E50</font></i>: 
 <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="group_2.html#T156">GROUP_2:156</a></i>;<br/>
<a NAME="E13:15"/>
for <font color="Olive">g</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> iff <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cad</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><i><font color="Green">E51</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:15_1_1"/><b>then </b>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/><a NAME="E3:15_1_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E5:15_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E6:15_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font>
 <b>by </b><i>, </i>;<br/><i><font color="Green">E56</font></i>: <font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
;<br/><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="group_3.html#T1">GROUP_3:1</a></i>
;<br/><a NAME="E9:15_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E10:15_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#T1">GROUP_3:1</a></i>;<br/><a NAME="E11:15_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E13:15_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span>
 <b>and </b><br/><a NAME="E14:15_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/><a NAME="E15:15_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E16:15_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/><a NAME="E18:15_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font>
 <b>by </b><i>, <a class="ref" href="group_3.html#T1">GROUP_3:1</a></i>;<br/><b>hence </b><a NAME="E19:15_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cad</font>
 <b>by </b><i>, <a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cad</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:15_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font>
 <b>and </b><br/><a NAME="E4:15_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">z</font> <a href="rlvect_1.html#R1">in</a>  <a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">g</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E6:15_1_2"/>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="group_3.html#T1">GROUP_3:1</a></i>;<br/><a NAME="E7:15_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/><a NAME="E8:15_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span>
 ;<br/><a NAME="E9:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E11:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_3.html#T1">GROUP_3:1</a></i>;<br/><a NAME="E12:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E13:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E14:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E15:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_3.html#T1">GROUP_3:1</a></i>;<br/><a NAME="E16:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">d</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><a NAME="E17:15_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/><a NAME="E18:15_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E19:15_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E20:15_1_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E21:15_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> iff <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Cad</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:15"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="weddwitt.html#K2" target="_self">-con_map</a> </span>)</span> <a href="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K9">ord</a> <span class="p1">(<span class="default"><a href="weddwitt.html#K1" target="_self">Centralizer</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
