<?xml version="1.0"?>
<div class="add">

<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/>





<b>thus </b><a NAME="E1:102"/>
(  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font> implies  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:102_1"/>
 <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<a NAME="E2:102_1"/><b>then </b>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> )
 
;<br/>
<b>hence </b><a NAME="E3:102_1"/>
 <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:102"/>
 <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:102"/><i><font color="Green" title="E73">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_3.html#E4:102"><i><font color="Green" title="E73">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:102"/>
 <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</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="E1:102_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:102_2"/><i><font color="Green" title="E74">A2</font></i>: 
( <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )
 ;<br/>
<a NAME="E4:102_2"/>
( <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">A</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  &amp; <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">B</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E4:102"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="group_3.html#T113" target="_self" title="GROUP_3:th.113">Th113</a></i>;<br/>
<a NAME="E5:102_2"/><b>then </b>
( <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">C</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  &amp; <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E3:102_2"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th107</a></i>;<br/>
<a NAME="E6:102_2"/><b>then </b>
<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">C</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th107</a></i>;<br/>
<b>hence </b><a NAME="E7:102_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="group_3.html#E3:102_2"><i><font color="Green" title="E74">A2</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c6">y</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="E7:102"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:102"/><i><font color="Green" title="E74">A3</font></i>: 
( <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c7">C</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )
 ;<br/>
<a NAME="E10:102"/>
( <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">B</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  &amp; <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">A</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E4:102"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="group_3.html#T113" target="_self" title="GROUP_3:th.113">Th113</a></i>;<br/>
<a NAME="E11:102"/><b>then </b>
( <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  &amp; <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">C</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E9:102"><i><font color="Green" title="E74">A3</font></i></a>, <a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th107</a></i>;<br/>
<a NAME="E12:102"/><b>then </b>
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c7">C</font> <a href="group_3.html#R4" title="GROUP_3:pred.4">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T107" target="_self" title="GROUP_3:th.107">Th107</a></i>;<br/>
<b>hence </b><a NAME="E13:102"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K8" title="GROUP_3:func.8">con_class</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="group_3.html#E9:102"><i><font color="Green" title="E74">A3</font></i></a></i>;<br/>


</div>
