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<div class="add">

<b>let </b><font color="Maroon">G</font> be  <a href="group_1.html#V1">strict</a> <a href="group_1.html#NM1">Group</a>;<br/>

<b>thus </b><a NAME="E1:81"/>
( <font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a> implies <font color="Maroon">G</font> <a href="group_5.html#K9" target="_self">`</a>  <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:81_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_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>assume </b><a NAME="E1:81_1_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G</font> <a href="group_5.html#K9" target="_self">`</a> 
 ;<br/><a NAME="E2:81_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <span class="p1"><a href="group_5.html#K8" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span></span><a href="group_5.html#K8" target="_self">.]</a></span>
 ;<br/><a NAME="E3:81_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a> <span class="p1"><a href="group_5.html#K7" target="_self">[.</a><span class="default"><span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p3">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p3">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span></span>)</span></span><a href="group_5.html#K7" target="_self">.]</a></span>
 ;<br/><a NAME="E4:81_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><a href="group_5.html#K4" target="_self">commutators</a> <span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p3">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="group_2.html#K7">carr</a> <span class="p3">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="group_2.html#K7">carr</a> <span class="p1">(<span class="default"><a href="group_2.html#K6">(Omega).</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E5:81_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E6:81_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 <b>by </b><i><a class="ref" href="group_4.html#T44">GROUP_4:44</a></i>;<br/><a NAME="E7:81_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_4.html#K5">gr</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E8:81_1_1"/>
<font color="Maroon">a</font> <a href="rlvect_1.html#R1">in</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="group_4.html#T39">GROUP_4:39</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:81_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="group_5.html#K9" target="_self">`</a>  is    <a href="group_2.html#M1">Subgroup</a> of  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font> &amp;  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font> is    <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <a href="group_5.html#K9" target="_self">`</a>  )
 
<b>by </b><i><a class="ref" href="group_2.html#T67">GROUP_2:67</a>, <a class="ref" href="group_2.html#T77">GROUP_2:77</a></i>;<br/>
<b>hence </b><a NAME="E4:81_1"/>
<font color="Maroon">G</font> <a href="group_5.html#K9" target="_self">`</a>  <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T64">GROUP_2:64</a></i>;<br/>


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

<b>assume </b><a NAME="E2:81"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G</font> <a href="group_5.html#K9" target="_self">`</a>  <a href="group_2.html#R1">=</a>  <a href="group_2.html#K5">(1).</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:81"/>
<font color="Maroon">G</font> is <a href="group_1.html#V7">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </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>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="group_1.html#D16">GROUP_1:def 16</a><br/>

<a NAME="E1:81_2"/>
<span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="rlvect_1.html#R1">in</a> <font color="Maroon">G</font> <a href="group_5.html#K9" target="_self">`</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:81_2"/><b>then </b>
<span class="p1"><a href="group_5.html#K2" target="_self">[.</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="group_5.html#K2" target="_self">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:81_2"/>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <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="E4:81"/>
<font color="Maroon">G</font> is  <a href="group_1.html#V7">commutative</a> <a href="group_1.html#NM1">Group</a>
 ;<br/>


</div>
