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<b>let </b><font color="Maroon">G</font> be   <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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>





<b>thus </b><a NAME="E1:88"/>
<span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:88_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:88_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>
<a NAME="E4:88_1"/>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="group_3.html#R2" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a>, <a class="ref" href="group_3.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E6:88_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a></i>;<br/>
<font color="Maroon">h</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K3">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a>, <a class="ref" href="group_3.html#T25" target="_self">Th25</a>, <a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
<br/>.= 
<font color="Maroon">c</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">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">c</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/>.= 
<font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>then </b><i><font color="Green">E40</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E10:88_1"/>
<font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E11:88_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a>, <a class="ref" href="group_3.html#T26" target="_self">Th26</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:88"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K2">*</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:88"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font> )
 ;<br/>
<a NAME="E5:88"/>
<font color="Maroon">c</font>,<font color="Maroon">a</font> <a href="group_3.html#R2" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a>, <a class="ref" href="group_3.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E7:88"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E9:88"/>
<font color="Maroon">h</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a>, <a class="ref" href="group_3.html#T25" target="_self">Th25</a>, </i>;<br/>
<b>then </b><i><font color="Green">E51</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">g</font></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">a</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</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">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">b</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">a</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i/>
;<br/>
<a NAME="E11:88"/>
<font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E12:88"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">a</font></span>)</span> <a href="group_2.html#K2">*</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="group_3.html#T24" target="_self">Th24</a>, <a class="ref" href="group_3.html#T26" target="_self">Th26</a></i>;<br/>


</div>
