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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">H</font> be    <a href="group_2.html#M1" target="_self">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<a NAME="E1:45"/><i><font color="Green">E35</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">H</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> &amp;  <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; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> )
 
<b>by </b><i><a class="txt" href="group_2.html#E1"><i><font color="Green">Lemma22</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:45"/><b>then </b><i><font color="Green">E36</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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="E1:45_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">H</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">H</font> ;<br/><b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="group_1.html#K4">inverse_op</a> <font color="Maroon">H</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">a</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/>.= 
<font color="Maroon">b</font> <a href="group_1.html#K3">"</a> 
<b>by </b><i/>
<br/>.= 
<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>
<b>by </b><i><a class="ref" href="group_1.html#D7">GROUP_1:def 7</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45"/>
 <a href="group_1.html#K4">inverse_op</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</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="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">H</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>


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