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<b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="group_1.html#L1">HGrStr</a> ;<br/><b>let </b><font color="Maroon">H</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:83"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">G</font> is <a href="group_1.html#V7">commutative</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_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">H</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font>;<br/><a NAME="E1:83_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H</font>) <a href="tarski.html#R1">c=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">G</font>)
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">b'</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>thus </b><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> = 
<font color="Maroon">a'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">b'</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a'</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:83"/>
<font color="Maroon">H</font> is <a href="group_1.html#V7">commutative</a>
 <b>by </b><i/>;<br/>


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