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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><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="E1:146"/>
( <font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">H</font> implies  ex <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Olive">h</font> <a href="group_2.html#R1">=</a> <font color="Maroon">H</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="146_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:146_1"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">H</font>
 ;<br/>

<a NAME="E2:146_1"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_3.html#K10" target="_self">Normalizator</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>
 
;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E4:146_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="group_3.html#K4" target="_self">|^</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</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#T63" target="_self">Th63</a></i>;<br/>
<a NAME="E5:146_1"/>
(  <a href="group_2.html#K7">carr</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="group_3.html#K4" target="_self">|^</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E6:146_1"/><b>then </b>
<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>
 
<b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a>, <a class="ref" href="group_2.html#T68">GROUP_2:68</a></i>;<br/>
<b>hence </b><a NAME="E7:146_1"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> &amp; <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Olive">h</font> <a href="group_2.html#R1">=</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a></i>;<br/>


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

<b>given </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><a NAME="E2:146"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> &amp; <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">h</font> <a href="group_2.html#R1">=</a> <font color="Maroon">H</font> )
 ;<br/>

<a NAME="E3:146"/>
<span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span> <a href="group_3.html#K4" target="_self">|^</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</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#T71" target="_self">Th71</a>, <a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a></i>;<br/>
<a NAME="E4:146"/><b>then </b>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_3.html#K10" target="_self">Normalizator</a> <span class="p1">(<span class="default"><a href="group_2.html#K7">carr</a> <font color="Maroon">H</font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_3.html#T71" target="_self">Th71</a>, <a class="ref" href="group_3.html#T63" target="_self">Th63</a></i>;<br/>
<b>hence </b><a NAME="E5:146"/>
<font color="Maroon">x</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_3.html#K11" target="_self">Normalizator</a> <font color="Maroon">H</font>
 ;<br/>


</div>
