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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="dynkin.html#K7" target="_self">generated_Dynkin_System</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="finsub_1.html#V2">intersection_stable</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="dynkin.html#K7" target="_self">generated_Dynkin_System</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="dynkin.html#M1" target="_self">Dynkin_System</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:37"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="dynkin.html#K7" target="_self">generated_Dynkin_System</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="dynkin.html#D8" target="_self">Def8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="dynkin.html#E1:37"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E5:37"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:37_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E1:37"><i><font color="Green">E30</font></i></a>, <a class="txt" href="dynkin.html#E1:37_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>
<a NAME="E4:37_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="dynkin.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dynkin.html#E3:37_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="dynkin.html#D9" target="_self">Def9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> is    <a href="dynkin.html#M1" target="_self">Dynkin_System</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:37"/><b>then </b>
 <a href="dynkin.html#K7" target="_self">generated_Dynkin_System</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="dynkin.html#D8" target="_self">Def8</a></i>;<br/>
<b>hence </b><a NAME="E8:37"/>
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="dynkin.html#K7" target="_self">generated_Dynkin_System</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dynkin.html#E1:37"><i><font color="Green">E30</font></i></a>, <a class="ref" href="dynkin.html#D9" target="_self">Def9</a></i>;<br/>


</div>
