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<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="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>;<br/>



<b>assume </b><a NAME="E1:85"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c2">H</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>now </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>;<br/><b>thus </b><a NAME="E1:85_1"/>
<font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</a> <font color="Maroon" title="c3">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1_1"><b>proof </b></a><div class="add">

<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="E1:85_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">H</font> <a href="group_2.html#K13" title="GROUP_2:func.13">*</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="E3:85_1_1"/><i><font color="Green" title="E46">A2</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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">a</font>
 <b>and </b><br/><a NAME="E4:85_1_1"/><i><font color="Green" title="E47">A3</font></i>: 
<font color="Maroon" title="c5">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="group_2.html#T126" title="GROUP_2:th.126">GROUP_2:126</a></i>;<br/>
<a NAME="E5:85_1_1"/>
<font color="Maroon" title="c5">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_5.html#E1:85"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="group_5.html#E4:85_1_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="group_2.html#T49" title="GROUP_2:th.49">GROUP_2:49</a></i>;<br/>
<a NAME="E6:85_1_1"/><b>then </b>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">b</font>
 
<b>by </b><i><a class="txt" href="group_5.html#E3:85_1_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="group_5.html#T89" target="_self" title="GROUP_5:th.89">Th89</a></i>;<br/>
<b>hence </b><a NAME="E7:85_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">a</font> <a href="group_2.html#K12" title="GROUP_2:func.12">*</a> <font color="Maroon" title="c2">H</font>
 <b>by </b><i><a class="txt" href="group_5.html#E4:85_1_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="group_2.html#T125" title="GROUP_2:th.125">GROUP_2:125</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:85"/>
<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>
 <b>by </b><i><a class="ref" href="group_3.html#T142" title="GROUP_3:th.142">GROUP_3:142</a></i>;<br/>


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