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

<b>let </b><font color="Maroon" title="c1">Omega</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">E</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">Omega</font>;<br/><b>let </b><font color="Maroon" title="c3">X</font>, <font color="Maroon" title="c4">Y</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font>;<br/>





<b>assume </b><a NAME="E1:38"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K7" title="DYNKIN:func.7">generated_Dynkin_System</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K7" title="DYNKIN:func.7">generated_Dynkin_System</a> <font color="Maroon" title="c2">E</font> &amp; <font color="Maroon" title="c2">E</font> is <a href="finsub_1.html#V2" title="FINSUB_1:attr.2">intersection_stable</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">G</font> =  <a href="dynkin.html#K7" title="DYNKIN:func.7">generated_Dynkin_System</a> <font color="Maroon" title="c2">E</font> as    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">X</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">G</font> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c5">G</font>;<br/>
<b>consider </b><font color="Maroon" title="c6">h</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:38"/><i><font color="Green" title="E32">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">h</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">Omega</font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:38"/><i><font color="Green" title="E33">A3</font></i>: 
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">h</font> holds <br/><font color="Olive" title="b1">Y</font> is    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon" title="c7">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">h</font>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
 ex <font color="Olive" title="b1">X</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">G</font> st <font color="Maroon" title="c7">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E4:38"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon" title="c7">Y</font> is    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:38"/>
not <font color="Maroon" title="c6">h</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c7">X</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">G</font>;<br/>
<a NAME="E2:38_2"/>
 <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Maroon" title="c7">X</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">h</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E4:38"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_2"/>
not <font color="Maroon" title="c6">h</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c6">h</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E8:38"/><i><font color="Green" title="E34">A4</font></i>: 
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">h</font> is    <a href="dynkin.html#M1" title="DYNKIN:mode.1">Dynkin_System</a> of <font color="Maroon" title="c1">Omega</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E5:38"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="dynkin.html#T15" target="_self" title="DYNKIN:th.15">Th15</a></i>;<br/>
<a NAME="E9:38"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_3"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 ;<br/>

<a NAME="E2:38_3"/>
for <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font> holds <br/><font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_3_1"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c10">X</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">G</font><b> such that </b><br/><a NAME="E3:38_3_1"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c9">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Maroon" title="c10">X</font>,<font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="dynkin.html#E4:38"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="dynkin.html#E1:38_3_1"><i><font color="Green" title="E36">A6</font></i></a></i>;<br/>
<a NAME="E4:38_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c10">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="dynkin.html#E1:38_3"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="dynkin.html#T24" target="_self" title="DYNKIN:th.24">Th24</a></i>;<br/>
<b>hence </b><a NAME="E5:38_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font>
 <b>by </b><i><a class="txt" href="dynkin.html#E1:38_3"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="dynkin.html#E3:38_3_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="dynkin.html#D9" target="_self" title="DYNKIN:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_3"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">h</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:38"/><b>then </b>
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">h</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:38"/><b>then </b><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">h</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E8:38"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="dynkin.html#D8" target="_self" title="DYNKIN:def.8">Def8</a></i>;<br/>
<a NAME="E12:38"/>
 <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c5">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">h</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="dynkin.html#E4:38"><i><font color="Green" title="E32">A2</font></i></a></i>;<br/>
<a NAME="E13:38"/><b>then </b>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/>
<a NAME="E14:38"/><b>then </b>
<font color="Maroon" title="c5">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E11:38"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E15:38"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K7" title="DYNKIN:func.7">generated_Dynkin_System</a> <font color="Maroon" title="c2">E</font>
 <b>by </b><i><a class="txt" href="dynkin.html#E1:38"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="dynkin.html#D9" target="_self" title="DYNKIN:def.9">Def9</a></i>;<br/>


</div>
