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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="E1:50"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c1">G</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>
 ;<br/>

<b>thus </b><a NAME="E2:50"/>
for <font color="Olive" title="b1">h1</font>, <font color="Olive" title="b2">h2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> holds  <font color="Olive" title="b1">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b1">h1</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">h1</font>, <font color="Maroon" title="c4">h2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font>;<br/>



<b>reconsider </b><font color="Maroon" title="c5">g1</font> = <font color="Maroon" title="c3">h1</font>, <font color="Maroon" title="c6">g2</font> = <font color="Maroon" title="c4">h2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="group_2.html#T51" target="_self" title="GROUP_2:th.51">Th51</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">h1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c4">h2</font> = 
<font color="Maroon" title="c5">g1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">g2</font>
<b>by </b><i><a class="ref" href="group_2.html#T52" target="_self" title="GROUP_2:th.52">Th52</a></i>
<br/>.= 
<font color="Maroon" title="c6">g2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">g1</font>
<b>by </b><i><a class="txt" href="group_2.html#E1:50"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="group_2.html#E32"><i><font color="Green" title="E11">Lm2</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">h2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">h1</font>
<b>by </b><i><a class="ref" href="group_2.html#T52" target="_self" title="GROUP_2:th.52">Th52</a></i>
;<br/>


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


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