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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>thus </b><a NAME="E1:133"/>
( <font color="Maroon">H</font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> implies for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Olive">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> holds <br/> <a href="group_3.html#K7" target="_self">con_class</a> <font color="Olive">a</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:133_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">H</font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 ;<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>assume </b><a NAME="E2:133_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 ;<br/>

<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="E3:133_1"/>
<font color="Maroon">x</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/>

<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="E5:133_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="group_3.html#R2" target="_self">are_conjugated</a>  )
 <b>by </b><i><a class="ref" href="group_3.html#T21" target="_self">Th21</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="E7:133_1"/><i><font color="Green">E49</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">c</font>
 <b>by </b><i><a class="ref" href="group_3.html#T61" target="_self">Th61</a>, </i>;<br/>
<a NAME="E8:133_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T59" target="_self">Th59</a>, <a class="ref" href="group_3.html#T61" target="_self">Th61</a>, <a class="ref" href="group_3.html#T63" target="_self">Th63</a>, <a class="ref" href="group_3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E9:133_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T56" target="_self">Th56</a>, <a class="ref" href="group_3.html#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E10:133_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:133"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>  st <font color="Olive">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> holds <br/> <a href="group_3.html#K7" target="_self">con_class</a> <font color="Olive">a</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E3:133"/>
<font color="Maroon">H</font> is <a href="group_3.html#V1" target="_self">normal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_3.html#D13" target="_self">GROUP_3:def 13</a><br/>

<a NAME="E1:133_2"/>
<font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font> <a href="group_2.html#R1">=</a> <font color="Maroon">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2_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>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_2.html#D6">GROUP_2:def 6</a><br/>

<b>thus </b><a NAME="E1:133_2_1"/>
( <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font> implies <font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="133_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:133_2_1_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font>
 ;<br/>

<b>then 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="E3:133_2_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E4:133_2_1_1"/>
( <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">c</font> &amp;  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">c</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="group_3.html#T56" target="_self">Th56</a>, <a class="ref" href="group_3.html#T59" target="_self">Th59</a>, <a class="ref" href="group_3.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E5:133_2_1_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>


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

<b>assume </b><a NAME="E2:133_2_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E3:133_2_1"/><b>then </b>
(  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> &amp; <font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K7" target="_self">con_class</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="group_3.html#T56" target="_self">Th56</a>, <a class="ref" href="group_3.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E4:133_2_1"/><b>then </b>
<font color="Maroon">b</font> <a href="group_3.html#K2" target="_self">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<a NAME="E5:133_2_1"/><b>then </b>
<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">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E6:133_2_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:133_2"/>
<font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#G1">HGrStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>,the <a href="group_1.html#U1">mult</a> of <font color="Maroon">H</font> #)
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:133"/>
<font color="Maroon">H</font> is  <a href="group_3.html#V1" target="_self">normal</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>
 ;<br/>


</div>
