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

<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">H</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>thus </b><a NAME="E1:132"/>
( <font color="Maroon" title="c2">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> implies for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> holds <br/> <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> )
 <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" title="E85">A1</font></i>: 
<font color="Maroon" title="c2">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/>

<b>assume </b><a NAME="E2:132_1"/><i><font color="Green" title="E86">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:132_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c3">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E5:132_1"/><i><font color="Green" title="E87">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font> <a href="group_3.html#R2" title="GROUP_3:pred.2">are_conjugated</a>  )
 <b>by </b><i><a class="ref" href="group_3.html#T95" target="_self" title="GROUP_3:th.95">Th95</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:132_1"/><i><font color="Green" title="E88">A4</font></i>: 
<font color="Maroon" title="c5">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="group_3.html#E5:132_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="ref" href="group_3.html#T88" target="_self" title="GROUP_3:th.88">Th88</a></i>;<br/>
<a NAME="E8:132_1"/>
<font color="Maroon" title="c4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E2:132_1"><i><font color="Green" title="E86">A2</font></i></a>, <a class="txt" href="group_3.html#E5:132_1"><i><font color="Green" title="E87">A3</font></i></a>, <a class="txt" href="group_3.html#E7:132_1"><i><font color="Green" title="E88">A4</font></i></a>, <a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></i>;<br/>
<a NAME="E9:132_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="txt" href="group_3.html#E1:132_1"><i><font color="Green" title="E85">A1</font></i></a>, <a class="ref" href="group_3.html#D13" target="_self" title="GROUP_3:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E10:132_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:132"/><i><font color="Green" title="E85">A5</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b1">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> holds <br/> <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Olive" title="b1">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font>
 ;<br/>

<a NAME="E3:132"/>
<font color="Maroon" title="c2">H</font> is <a href="group_3.html#V1" title="GROUP_3:attr.1">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" title="c3">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_3.html#D13" target="_self" title="GROUP_3:def.13">GROUP_3:def 13</a><br/>

<a NAME="E1:132_2"/>
<font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a> <font color="Maroon" title="c2">H</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_2.html#D6" title="GROUP_2:def.6">GROUP_2:def 6</a><br/>

<b>thus </b><a NAME="E1:132_2_1"/>
( <font color="Maroon" title="c4">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font> implies <font color="Maroon" title="c4">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="132_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:132_2_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:132_2_1_1"/><i><font color="Green" title="E86">A6</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">c</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> )
 <b>by </b><i><a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></i>;<br/>
<a NAME="E4:132_2_1_1"/>
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c5">c</font> &amp;  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c5">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> )
 
<b>by </b><i><a class="txt" href="group_3.html#E2:132"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="group_3.html#E3:132_2_1_1"><i><font color="Green" title="E86">A6</font></i></a>, <a class="ref" href="group_3.html#T97" target="_self" title="GROUP_3:th.97">Th97</a></i>;<br/>
<b>hence </b><a NAME="E5:132_2_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>


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

<b>assume </b><a NAME="E2:132_2_1"/>
<font color="Maroon" title="c4">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 ;<br/>

<a NAME="E3:132_2_1"/><b>then </b>
(  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_2.html#K7" title="GROUP_2:func.7">carr</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c4">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K7" title="GROUP_3:func.7">con_class</a> <font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="group_3.html#E2:132"><i><font color="Green" title="E85">A5</font></i></a>, <a class="ref" href="group_3.html#T97" target="_self" title="GROUP_3:th.97">Th97</a></i>;<br/>
<a NAME="E4:132_2_1"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font>
 
<b>by </b><i><a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/>
<a NAME="E5:132_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">b</font> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a> </span>)</span></span>)</span> <a href="group_3.html#K2" title="GROUP_3:func.2">|^</a> <font color="Maroon" title="c3">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T70" target="_self" title="GROUP_3:th.70">Th70</a></i>;<br/>
<b>hence </b><a NAME="E6:132_2_1"/>
<font color="Maroon" title="c4">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="ref" href="group_3.html#T30" target="_self" title="GROUP_3:th.30">Th30</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:132_2"/>
<font color="Maroon" title="c2">H</font> <a href="group_3.html#K6" title="GROUP_3:func.6">|^</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#G3" title="ALGSTR_0:aggr.3">multMagma</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">H</font>,the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c2">H</font> #)
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:132"/>
<font color="Maroon" title="c2">H</font> is  <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>
 ;<br/>


</div>
