<?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="E1:132"/>
( <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  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:132_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>thus </b><a NAME="E2:132_1"/>
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</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="132_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="E1:132_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">H1</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:132_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">H</font>,<font color="Maroon">H1</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E4:132_1_1"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> st <font color="Maroon">H1</font> <a href="group_2.html#R1">=</a> <font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Olive">g</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T59" target="_self">Th59</a>, <a class="txt" href="group_3.html#E38"><i><font color="Green">Lemma75</font></i></a></i>;<br/>
<a NAME="E5:132_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</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#T44" target="_self">Th44</a></i>;<br/>
<b>hence </b><a NAME="E6:132_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</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="E3:132_1"/>
<font color="Maroon">H</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T41" target="_self">Th41</a></i>;<br/>
<b>hence </b><a NAME="E4:132_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</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="E2:132"/><i><font color="Green">E40</font></i>: 
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">H</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:132"/>
<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="132_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="E1:132_2"/>
<font color="Maroon">H</font> <a href="group_3.html#K6" target="_self">|^</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E2:132_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> #)
 <b>by </b><i><a class="ref" href="group_3.html#T56" target="_self">Th56</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:132"/>
<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>
