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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font>;<br/>





<a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1" title="GROUP_2:def.1">GROUP_2:def 1</a></i>;<br/>
<a NAME="E2:12"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1" title="GROUP_2:def.1">GROUP_2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:12"/>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:12_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E4:12_1"/><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:12_1"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 <b>by </b><i><a class="txt" href="topgrp_1.html#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="topgrp_1.html#E1:12_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="topgrp_1.html#E4:12_1"><i><font color="Green" title="E11">A4</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E4:12"/><i><font color="Green" title="E10">A5</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> 
 ;<br/>

<a NAME="E5:12"/><i><font color="Green" title="E11">A6</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  )
 
<b>by </b><i><a class="ref" href="topgrp_1.html#T8" target="_self" title="TOPGRP_1:th.8">Th8</a></i>;<br/>
<a NAME="E6:12"/><i><font color="Green" title="E12">A7</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1" title="GROUP_2:def.1">GROUP_2:def 1</a></i>;<br/>
<a NAME="E7:12"/><i><font color="Green" title="E13">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a> </span>)</span> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> </span>)</span> where <font color="Olive" title="b1">g</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> : <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  </span>}</span> 
 
<b>by </b><i><a class="ref" href="group_2.html#D1" title="GROUP_2:def.1">GROUP_2:def 1</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E8:12"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">g</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E10:12"/><i><font color="Green" title="E14">A9</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="group_2.html#K1" title="GROUP_2:func.1">"</a>  )
 <b>by </b><i><a class="txt" href="topgrp_1.html#E5:12"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="topgrp_1.html#E6:12"><i><font color="Green" title="E12">A7</font></i></a></i>;<br/>
<b>thus </b><a NAME="E11:12"/>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="topgrp_1.html#E4:12"><i><font color="Green" title="E10">A5</font></i></a>, <a class="txt" href="topgrp_1.html#E5:12"><i><font color="Green" title="E11">A6</font></i></a>, <a class="txt" href="topgrp_1.html#E7:12"><i><font color="Green" title="E13">A8</font></i></a>, <a class="txt" href="topgrp_1.html#E10:12"><i><font color="Green" title="E14">A9</font></i></a></i>;<br/>


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