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<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>assume </b><a NAME="E1:86"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">H</font> is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_5.html#K10" target="_self">center</a> <font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><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><a NAME="E1:86_1"/>
<font color="Maroon">H</font> <a href="group_2.html#K13">*</a> <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> <a href="group_2.html#K12">*</a> <font color="Maroon">H</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_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="E1:86_1_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>
 ;<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="E3:86_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:86_1_1"/><i><font color="Green">E63</font></i>: 
<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="group_2.html#T126">GROUP_2:126</a></i>;<br/>
<a NAME="E5:86_1_1"/>
<font color="Maroon">b</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_5.html#K10" target="_self">center</a> <font color="Maroon">G</font>
 
<b>by </b><i>, , <a class="ref" href="group_2.html#T49">GROUP_2:49</a></i>;<br/>
<a NAME="E6:86_1_1"/><b>then </b>
<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">b</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:86_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_2.html#T125">GROUP_2:125</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:86"/>
<font color="Maroon">H</font> is  <a href="group_3.html#V1">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#T142">GROUP_3:142</a></i>;<br/>


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