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

<b>set </b><font color="Maroon">f</font> =  <a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font>;<br/>
<a NAME="E1:18"/><i><font color="Green">E30</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> ;<br/><i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="group_1.html#K3">"</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">g</font> <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#D7">GROUP_1:def 7</a></i>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="group_1.html#T19">GROUP_1:19</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">g</font> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="group_1.html#D7">GROUP_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
<span class="p1">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font></span>)</span> <a href="funct_2.html#K2">"</a>  <a href="funct_2.html#R2">=</a>  <a href="group_1.html#K4">inverse_op</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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