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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/>

<b>thus </b><a NAME="E1:53"/>
( <font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a> implies for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E2:53"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font><br/> for <font color="Olive">g</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> holds  <font color="Olive">A</font> <a href="group_3.html#K4" target="_self">|^</a> <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_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>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> = 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">a</font></span><a href="domain_1.html#K6">}</a></span> <a href="group_3.html#K4" target="_self">|^</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="group_3.html#E2"><i><font color="Green">Lemma41</font></i></a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">b</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E2:53_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_3.html#K2" target="_self">|^</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T6">ZFMISC_1:6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:53"/>
<font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 <b>by </b><i/>;<br/>


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