<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">H1</font>, <font color="Olive" title="b2">H2</font> being <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> holds <br/> (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b1">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b2">H2</font> iff  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b1">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Olive" title="b2">H2</font> )</span><br/><b>let </b><font color="Maroon" title="c2">H1</font>, <font color="Maroon" title="c3">H2</font> be  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> iff  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )</span><br/>



<b>thus </b><a NAME="E1:118"/>
(  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> implies  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> implies  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )</span><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" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font></span><br/>

<a NAME="E2:118_1"/><b>then </b>
( <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> &amp; <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> )
 
<b>by </b><i><a class="ref" href="group_3.html#T130" target="_self" title="GROUP_3:th.130">Th130</a></i>;<br/>
<b>hence </b><a NAME="E3:118_1"/>
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:118"/>
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font></span><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="E4:118"/><i><font color="Green" title="E79">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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</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#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</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="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="group_3.html#E4:118"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="group_3.html#T127" target="_self" title="GROUP_3:th.127">Th127</a></i>;<br/>
<b>thus </b><a NAME="E6:118"/>
 <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font></span><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" 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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> )</span><br/>

<b>assume </b><a NAME="E1:118_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">H3</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:118_2"/><i><font color="Green" title="E80">A2</font></i>: 
( <font color="Maroon" title="c7">H3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  )
 <b>by </b><i><a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></i>;<br/>
<a NAME="E4:118_2"/>
( <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">H1</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  &amp; <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H2</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E4:118"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="group_3.html#T128" target="_self" title="GROUP_3:th.128">Th128</a></i>;<br/>
<a NAME="E5:118_2"/><b>then </b>
( <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  &amp; <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E3:118_2"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="group_3.html#T124" target="_self" title="GROUP_3:th.124">Th124</a></i>;<br/>
<a NAME="E6:118_2"/><b>then </b>
<font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T124" target="_self" title="GROUP_3:th.124">Th124</a></i>;<br/>
<b>hence </b><a NAME="E7:118_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 <b>by </b><i><a class="txt" href="group_3.html#E3:118_2"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font> or <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font> )</span><br/>

<b>assume </b><a NAME="E7:118"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c3">H2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font></span><br/>

<b>then consider </b><font color="Maroon" title="c7">H3</font> being  <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>  <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E9:118"/><i><font color="Green" title="E80">A3</font></i>: 
( <font color="Maroon" title="c7">H3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c3">H2</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  )
 <b>by </b><i><a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></i>;<br/>
<a NAME="E10:118"/>
( <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">H2</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  &amp; <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">H1</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E4:118"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="group_3.html#T128" target="_self" title="GROUP_3:th.128">Th128</a></i>;<br/>
<a NAME="E11:118"/><b>then </b>
( <font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c5">x</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  &amp; <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a>  )
 
<b>by </b><i><a class="txt" href="group_3.html#E9:118"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="group_3.html#T124" target="_self" title="GROUP_3:th.124">Th124</a></i>;<br/>
<a NAME="E12:118"/><b>then </b>
<font color="Maroon" title="c2">H1</font>,<font color="Maroon" title="c7">H3</font> <a href="group_3.html#R6" title="GROUP_3:pred.6">are_conjugated</a> 
 
<b>by </b><i><a class="ref" href="group_3.html#T124" target="_self" title="GROUP_3:th.124">Th124</a></i>;<br/>
<b>hence </b><a NAME="E13:118"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="group_3.html#K9" title="GROUP_3:func.9">con_class</a> <font color="Maroon" title="c2">H1</font>
 <b>by </b><i><a class="txt" href="group_3.html#E9:118"><i><font color="Green" title="E80">A3</font></i></a>, <a class="ref" href="group_3.html#D12" target="_self" title="GROUP_3:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
