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





<b>thus </b><a NAME="E1:38"/>
( <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> implies <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1"/>
<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>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:38_1"/><b>then </b>
<font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K3">"</a> 
 
<b>by </b><i><a class="ref" href="group_1.html#T20">GROUP_1:20</a></i>;<br/>
<b>hence </b><a NAME="E4:38_1"/>
<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><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>;<br/>


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

<b>assume </b><a NAME="E2:38"/>
<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>
 ;<br/>

<a NAME="E3:38"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span> &amp; <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="group_1.html#T26">GROUP_1:26</a></i>;<br/>
<b>hence </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> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K3">"</a> </span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_1.html#K3">"</a> </span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font>
<b>by </b><i><a class="ref" href="group_3.html#T1">GROUP_3:1</a></i>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="group_1.html#D6">GROUP_1:def 6</a></i>
;<br/><br/>


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