<?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="E1:38"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>3</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>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/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:38"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> iff ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E5:38"/><i><font color="Green">E33</font></i>: 
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>6</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is    <a href="dynkin.html#M1" target="_self">Dynkin_System</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:38_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E4:38"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1"/>
<font color="Maroon">c<sub>7</sub></font> is    <a href="dynkin.html#M1" target="_self">Dynkin_System</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:38"/>
not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">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">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E2:38_2"/>
 <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E4:38"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_2"/>
not <font color="Maroon">c<sub>6</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E8:38"/><i><font color="Green">E34</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</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="txt" href="dynkin.html#E5:38"><i><font color="Green">E33</font></i></a>, <a class="ref" href="dynkin.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E9:38"/>
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="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:38_3"/>
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>7</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></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">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_3_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:38_3_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="dynkin.html#K9" target="_self">DynSys</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dynkin.html#E4:38"><i><font color="Green">E32</font></i></a>, <a class="txt" href="dynkin.html#E1:38_3_1"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E4:38_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E1:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="dynkin.html#E1:38_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="dynkin.html#T24" target="_self">Th24</a></i>;<br/>
<b>hence </b><a NAME="E5:38_3_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="dynkin.html#E1:38_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="dynkin.html#E3:38_3_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="dynkin.html#D9" target="_self">Def9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:38_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:38"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E11:38"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E8:38"><i><font color="Green">E34</font></i></a>, <a class="ref" href="dynkin.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E12:38"/>
 <a href="dynkin.html#K8" target="_self">DynSys</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E1:38"><i><font color="Green">E31</font></i></a>, <a class="txt" href="dynkin.html#E4:38"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E13:38"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="dynkin.html#K8" target="_self">DynSys</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/>
<a NAME="E14:38"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="dynkin.html#K8" target="_self">DynSys</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E11:38"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E15:38"/>
<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#E1:38"><i><font color="Green">E31</font></i></a>, <a class="ref" href="dynkin.html#D9" target="_self">Def9</a></i>;<br/>


</div>
