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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#NM2">Subset</a> of <font color="Maroon">G</font>;<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="E1:13"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> <a href="group_2.html#K1">"</a>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">g</font> <a href="group_1.html#K3">"</a> </span>)</span> where <font color="Olive">g</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> : <font color="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1">GROUP_2:def 1</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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#K10">.:</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</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="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font> <b>by </b><i/>;<br/><a NAME="E5:13_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</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/><b>hence </b><a NAME="E6:13_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K1">"</a> 
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:13"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="group_2.html#K1">"</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E5:13"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="group_1.html#K3">"</a>  &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>
<a NAME="E6:13"/>
<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> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <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/>
<b>hence </b><a NAME="E7:13"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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#K10">.:</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/>


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