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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>;<br/>





<b>thus </b><a NAME="E1:88"/>
<span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:88_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:88_1"/><i><font color="Green" title="E60">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> )
 ;<br/>
<a NAME="E4:88_1"/>
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="ref" href="group_3.html#T96" target="_self" title="GROUP_3:th.96">Th96</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E6:88_1"/><i><font color="Green" title="E61">A2</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="ref" href="group_3.html#D7" target="_self" title="GROUP_3:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green" title="E60">A1</font></i></a></i>;<br/>
<font color="Maroon" title="c8">h</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="group_3.html#E6:88_1"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="group_1.html#T19" title="GROUP_1:th.19">GROUP_1:19</a></i>
<br/>.= 
<font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T1" target="_self" title="GROUP_3:th.1">Th1</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E62">A3</font></i>: <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">c</font>
<b>by </b><i><a class="ref" href="group_3.html#T30" target="_self" title="GROUP_3:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T28" target="_self" title="GROUP_3:th.28">Th28</a></i>
<br/>.= 
<font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T25" target="_self" title="GROUP_3:th.25">Th25</a></i>
<br/>.= 
<font color="Maroon" title="c6">c</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T29" target="_self" title="GROUP_3:th.29">Th29</a></i>
;<br/>
<a NAME="E10:88_1"/>
<font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">c</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T97" target="_self" title="GROUP_3:th.97">Th97</a></i>;<br/>
<b>hence </b><a NAME="E11:88_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="group_3.html#E3:88_1"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="group_3.html#E9:88_1"><i><font color="Green" title="E62">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:88"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</font>, <font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:88"/><i><font color="Green" title="E60">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">c</font> &amp; <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font> )
 ;<br/>
<a NAME="E5:88"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a> 
 
<b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="group_3.html#T96" target="_self" title="GROUP_3:th.96">Th96</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:88"/><i><font color="Green" title="E61">A5</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font>
 <b>by </b><i><a class="ref" href="group_3.html#D7" target="_self" title="GROUP_3:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">h</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green" title="E60">A4</font></i></a></i>;<br/>
<a NAME="E9:88"/>
<font color="Maroon" title="c8">h</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="group_3.html#E7:88"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="group_3.html#T1" target="_self" title="GROUP_3:th.1">Th1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E62">A6</font></i>: <font color="Maroon" title="c4">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T30" target="_self" title="GROUP_3:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c7">g</font></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T28" target="_self" title="GROUP_3:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T29" target="_self" title="GROUP_3:th.29">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font>
<b>by </b><i><a class="ref" href="group_3.html#T26" target="_self" title="GROUP_3:th.26">Th26</a></i>
;<br/>
<a NAME="E11:88"/>
<font color="Maroon" title="c2">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">b</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T97" target="_self" title="GROUP_3:th.97">Th97</a></i>;<br/>
<b>hence </b><a NAME="E12:88"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_2.html#K2" title="GROUP_2:func.2">*</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="group_3.html#E4:88"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="group_3.html#E10:88"><i><font color="Green" title="E62">A6</font></i></a></i>;<br/>


</div>
