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

<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">H1</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>, <font color="Maroon">H2</font> be  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font>;<br/>



<b>thus </b><a NAME="E1:118"/>
(  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font> implies  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:118_1"/>
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> <a href="hidden.html#R1">=</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font>
 ;<br/>

<a NAME="E2:118_1"/><b>then </b>
( <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font> &amp; <font color="Maroon">H1</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> )
 
<b>by </b><i><a class="ref" href="group_3.html#T41" target="_self">Th41</a></i>;<br/>
<b>hence </b><a NAME="E3:118_1"/>
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:118"/>
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</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="E4:118"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="group_3.html#T42" target="_self">Th42</a>, <a class="ref" href="group_3.html#T38" target="_self">Th38</a></i>;<br/>
<b>thus </b><a NAME="E6:118"/>
 <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font> <a href="tarski.html#R1">c=</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="118_2"><b>proof </b></a><div class="add">

<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="E1:118_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">H3</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E3:118_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">H3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">H1</font>,<font color="Maroon">H3</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E4:118_2"/>
( <font color="Maroon">x</font>,<font color="Maroon">H1</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">H2</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  )
 
<b>by </b><i><a class="ref" href="group_3.html#T42" target="_self">Th42</a>, <a class="ref" href="group_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E5:118_2"/><b>then </b>
( <font color="Maroon">x</font>,<font color="Maroon">H3</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  &amp; <font color="Maroon">H2</font>,<font color="Maroon">x</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  )
 
<b>by </b><i><a class="ref" href="group_3.html#T44" target="_self">Th44</a>, <a class="ref" href="group_3.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E6:118_2"/><b>then </b>
<font color="Maroon">H2</font>,<font color="Maroon">H3</font> <a href="group_3.html#R6" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T36" target="_self">Th36</a></i>;<br/>
<b>hence </b><a NAME="E7:118_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font>
 <b>by </b><i><a class="ref" href="group_3.html#T44" target="_self">Th44</a>, </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="E7:118"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">H3</font> being  <a href="group_1.html#V1">strict</a>  <a href="group_2.html#M1">Subgroup</a> of <font color="Maroon">G</font><b> such that </b><br/><a NAME="E9:118"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">H3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">H2</font>,<font color="Maroon">H3</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E10:118"/>
( <font color="Maroon">x</font>,<font color="Maroon">H2</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">H1</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  )
 
<b>by </b><i><a class="ref" href="group_3.html#T42" target="_self">Th42</a>, <a class="ref" href="group_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E11:118"/><b>then </b>
( <font color="Maroon">H1</font>,<font color="Maroon">x</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  &amp; <font color="Maroon">x</font>,<font color="Maroon">H3</font> <a href="group_3.html#R6" target="_self">are_conjugated</a>  )
 
<b>by </b><i><a class="ref" href="group_3.html#T44" target="_self">Th44</a>, <a class="ref" href="group_3.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E12:118"/><b>then </b>
<font color="Maroon">H1</font>,<font color="Maroon">H3</font> <a href="group_3.html#R6" target="_self">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T36" target="_self">Th36</a></i>;<br/>
<b>hence </b><a NAME="E13:118"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="group_3.html#K9" target="_self">con_class</a> <font color="Maroon">H1</font>
 <b>by </b><i><a class="ref" href="group_3.html#T44" target="_self">Th44</a>, </i>;<br/>


</div>
