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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a> <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/>

<b>thus </b><a NAME="E1:80"/>
( <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> implies <font color="Maroon" title="c1">G</font> <a href="group_5.html#K9" title="GROUP_5:func.9">`</a>  <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green" title="E43">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/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_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/><b>assume </b><a NAME="E1:80_1_1"/>
<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> <a href="group_5.html#K9" title="GROUP_5:func.9">`</a> 
 ;<br/><a NAME="E2:80_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="group_5.html#E1:80_1"><i><font color="Green" title="E43">A1</font></i></a>, <a class="ref" href="group_5.html#T56" target="_self" title="GROUP_5:th.56">Th56</a></i>;<br/><a NAME="E3:80_1_1"/><b>then </b>
( <font color="Maroon" title="c2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="ref" href="group_4.html#T44" title="GROUP_4:th.44">GROUP_4:44</a></i>;<br/><a NAME="E4:80_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_4.html#K5" title="GROUP_4:func.5">gr</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E5:80_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_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_4.html#T39" title="GROUP_4:th.39">GROUP_4:39</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:80_1"/><b>then </b>
( <font color="Maroon" title="c1">G</font> <a href="group_5.html#K9" title="GROUP_5:func.9">`</a>  is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font> is    <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <a href="group_5.html#K9" title="GROUP_5:func.9">`</a>  )
 
<b>by </b><i><a class="ref" href="group_2.html#T67" title="GROUP_2:th.67">GROUP_2:67</a>, <a class="ref" href="group_2.html#T77" title="GROUP_2:th.77">GROUP_2:77</a></i>;<br/>
<b>hence </b><a NAME="E4:80_1"/>
<font color="Maroon" title="c1">G</font> <a href="group_5.html#K9" title="GROUP_5:func.9">`</a>  <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="ref" href="group_2.html#T64" title="GROUP_2:th.64">GROUP_2:64</a></i>;<br/>


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

<b>assume </b><a NAME="E2:80"/><i><font color="Green" title="E43">A2</font></i>: 
<font color="Maroon" title="c1">G</font> <a href="group_5.html#K9" title="GROUP_5:func.9">`</a>  <a href="group_2.html#R1" title="GROUP_2:pred.1">=</a>  <a href="group_2.html#K5" title="GROUP_2:func.5">(1).</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<a NAME="E3:80"/>
<font color="Maroon" title="c1">G</font> is <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_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="E1:80_2"/>
<span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c1">G</font> <a href="group_5.html#K9" title="GROUP_5:func.9">`</a> 
 
<b>by </b><i><a class="ref" href="group_5.html#T84" target="_self" title="GROUP_5:th.84">Th84</a></i>;<br/>
<a NAME="E2:80_2"/><b>then </b>
<span class="p1"><a href="group_5.html#K2" title="GROUP_5:func.2">[.</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="group_5.html#K2" title="GROUP_5:func.2">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="txt" href="group_5.html#E2:80"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="group_5.html#T1" target="_self" title="GROUP_5:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E3:80_2"/>
<font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="group_5.html#T39" target="_self" title="GROUP_5:th.39">Th39</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:80"/>
<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/>


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