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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="group_1.html#V1" title="GROUP_1:attr.1">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>

<b>thus </b><a NAME="E1:89"/>
( <font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> implies  <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:89_1"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">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/><a NAME="E1:89_1_1"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> holds  <font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="group_5.html#E1:89_1"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="group_5.html#E42"><i><font color="Green" title="E23">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:89_1_1"/>
<font color="Maroon" title="c2">a</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="ref" href="group_5.html#T89" target="_self" title="GROUP_5:th.89">Th89</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:89_1"/>
 <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T71" title="GROUP_2:th.71">GROUP_2:71</a></i>;<br/>


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

<b>assume </b><a NAME="E2:89"/><i><font color="Green" title="E45">A2</font></i>: 
 <a href="group_5.html#K10" title="GROUP_5:func.10">center</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E3:89"/>
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">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_1.html#D16" title="GROUP_1:def.16">GROUP_1:def 16</a><br/>

<a NAME="E1:89_2"/>
<font color="Maroon" title="c2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">G</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/>
<b>hence </b><a NAME="E2:89_2"/>
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="group_5.html#E2:89"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="group_5.html#T89" target="_self" title="GROUP_5:th.89">Th89</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:89"/>
<font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/>


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