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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>thus </b><a NAME="E1:88"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_5.html#K10" target="_self">center</a> <font color="Maroon">G</font> implies  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:88_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_5.html#K10" target="_self">center</a> <font color="Maroon">G</font>
 ;<br/>

<b>thus </b><a NAME="E2:88_1"/>
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></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_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_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</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><b> such that </b><br/><a NAME="E3:88_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:88_1_1"/><i><font color="Green">E63</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">c</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_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">c</font>
 <b>by </b><i>, <a class="ref" href="group_3.html#D7">GROUP_3:def 7</a></i>;<br/>
<a NAME="E7:88_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c</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">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/>
<a NAME="E8:88_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T21">GROUP_1:21</a></i>;<br/>
<a NAME="E9:88_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:88_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="group_1.html#T14">GROUP_1:14</a></i>;<br/>
<b>hence </b><a NAME="E11:88_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<a NAME="E3:88_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T98">GROUP_3:98</a></i>;<br/>
<b>hence </b><a NAME="E4:88_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>


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

<b>assume </b><a NAME="E2:88"/><i><font color="Green">E65</font></i>: 
 <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </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/><a NAME="E1:88_2"/>
<font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7">con_class</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="group_3.html#T97">GROUP_3:97</a></i>;<br/><a NAME="E2:88_2"/><b>then </b>
<font color="Maroon">a</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:88_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">b</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">b</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>;<br/><b>hence </b><a NAME="E4:88_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="group_1.html#T21">GROUP_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:88"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_5.html#K10" target="_self">center</a> <font color="Maroon">G</font>
 <b>by </b><i/>;<br/>


</div>
