<?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_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>thus </b><a NAME="E1:129"/>
( <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#NM2">Subset</a> of <font color="Maroon">G</font> holds  <font color="Olive">A</font> <a href="group_2.html#K10">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K11">*</a> <font color="Olive">A</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:129_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#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>

<b>thus </b><a NAME="E2:129_1"/>
<font color="Maroon">A</font> <a href="group_2.html#K10">*</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> <a href="group_2.html#K11">*</a> <font color="Maroon">A</font>
  <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="129_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="E1:129_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K10">*</a> <font color="Maroon">H</font>
 ;<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>, <font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:129_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">h</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T114">GROUP_2:114</a></i>;<br/>
<a NAME="E4:129_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T59" target="_self">Th59</a>, <a class="ref" href="group_2.html#T125">GROUP_2:125</a></i>;<br/>
<a NAME="E5:129_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T56" target="_self">Th56</a>, <a class="ref" href="group_3.html#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E6:129_1_1"/><b>then </b>
 ex <font color="Olive">g</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">g</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> &amp; <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/>
<b>hence </b><a NAME="E7:129_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K11">*</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="group_3.html#T59" target="_self">Th59</a>, <a class="ref" href="group_2.html#T115">GROUP_2:115</a></i>;<br/>


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

<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="E3:129_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K11">*</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>, <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="E5:129_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font> &amp; <font color="Maroon">h</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="group_2.html#T115">GROUP_2:115</a></i>;<br/>
<a NAME="E6:129_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="group_3.html#T59" target="_self">Th59</a>, <a class="ref" href="group_2.html#T126">GROUP_2:126</a></i>;<br/>
<a NAME="E7:129_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</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#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E8:129_1"/><b>then </b>
 ex <font color="Olive">g</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="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">g</font> &amp; <font color="Olive">g</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="group_2.html#T125">GROUP_2:125</a></i>;<br/>
<b>hence </b><a NAME="E9:129_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K10">*</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="group_3.html#T59" target="_self">Th59</a>, <a class="ref" href="group_2.html#T114">GROUP_2:114</a></i>;<br/>


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

<b>assume </b><a NAME="E2:129"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font> holds  <font color="Olive">A</font> <a href="group_2.html#K10">*</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="group_2.html#K11">*</a> <font color="Olive">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="129_2"><b>now </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>;<br/><b>thus </b><font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font> = 
<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="group_2.html#K10">*</a> <font color="Maroon">H</font>

<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K11">*</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="group_3.html#T56" target="_self">Th56</a></i>
<br/>.= 
<font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font>

;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:129"/>
<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>
 <b>by </b><i><a class="ref" href="group_3.html#T51" target="_self">Th51</a></i>;<br/>


</div>
