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

<a NAME="E1:36_1_1"/>
( <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="ref" href="dynkin.html#D7" target="_self" title="DYNKIN:def.7">Def7</a></i>;<br/>
<a NAME="E2:36_1_1"/><b>then </b><i><font color="Green" title="E29">A1</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="dynkin.html#D9" target="_self" title="DYNKIN:def.9">Def9</a></i>;<br/>
<a NAME="E3:36_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
for <font color="Olive" title="b1">f</font> being  <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>  st  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f</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="c2">G</font> &amp; <font color="Olive" title="b1">f</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> holds <br/> <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">f</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of <font color="Maroon" title="c1">Omega</font>;<br/>

<b>assume </b><a NAME="E1:36_1_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</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="c2">G</font> &amp; <font color="Maroon" title="c4">f</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">X1</font> = <font color="Maroon" title="c3">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font> ;<br/>
<a NAME="E3:36_1_1_1"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="dynkin.html#K4" title="DYNKIN:func.4">seqIntersection</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c4">f</font> is <a href="prob_2.html#V1" title="PROB_2:attr.1">disjoint_valued</a>
 
<b>by </b><i><a class="txt" href="dynkin.html#E1:36_1_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="dynkin.html#T12" target="_self" title="DYNKIN:th.12">Th12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:36_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="dynkin.html#K4" title="DYNKIN:func.4">seqIntersection</a> <font color="Maroon" title="c5">X1</font>,<font color="Maroon" title="c4">f</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:36_1_1_1_1"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K4" title="DYNKIN:func.4">seqIntersection</a> <font color="Maroon" title="c5">X1</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font>
 <b>by </b><i><a class="ref" href="dynkin.html#T1" target="_self" title="DYNKIN:th.1">Th1</a></i>;<br/><a NAME="E4:36_1_1_1_1"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="dynkin.html#E3:36_1_1_1_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="dynkin.html#D2" target="_self" title="DYNKIN:def.2">Def2</a></i>;<br/><a NAME="E5:36_1_1_1_1"/>
<font color="Maroon" title="c4">f</font> <a href="prob_1.html#K1" title="PROB_1:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="dynkin.html#T1" target="_self" title="DYNKIN:th.1">Th1</a></i>;<br/><b>hence </b><a NAME="E6:36_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="dynkin.html#E1:36_1_1_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="dynkin.html#E4:36_1_1_1_1"><i><font color="Green" title="E34">A6</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>
<a NAME="E5:36_1_1_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="dynkin.html#K4" title="DYNKIN:func.4">seqIntersection</a> <font color="Maroon" title="c5">X1</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:36_1_1_1"/><b>then </b>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <span class="p1">(<span class="default"><a href="dynkin.html#K4" title="DYNKIN:func.4">seqIntersection</a> <font color="Maroon" title="c5">X1</font>,<font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E3:36_1_1_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="dynkin.html#D7" target="_self" title="DYNKIN:def.7">Def7</a></i>;<br/>
<a NAME="E7:36_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="dynkin.html#T13" target="_self" title="DYNKIN:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E8:36_1_1_1"/>
 <a href="prob_1.html#K2" title="PROB_1:func.2">Union</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="dynkin.html#D9" target="_self" title="DYNKIN:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:36_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">Omega</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</font> holds <br/><font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">A</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="E1:36_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</font>
 ;<br/>

<a NAME="E2:36_1_1_2"/><b>then </b><i><font color="Green" title="E31">A7</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="dynkin.html#D9" target="_self" title="DYNKIN:def.9">Def9</a></i>;<br/>
<a NAME="E3:36_1_1_2"/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E4:36_1_1_2"/><b>then </b><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E2:36_1_1_2"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="dynkin.html#T22" target="_self" title="DYNKIN:th.22">Th22</a></i>;<br/>
<a NAME="E5:36_1_1_2"/>
<font color="Maroon" title="c3">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E6:36_1_1_2"/><b>then </b><i><font color="Green" title="E33">A9</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">A</font></span>)</span> = 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">A</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="ref" href="subset_1.html#T32" title="SUBSET_1:th.32">SUBSET_1:32</a></i>
<br/>.= 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23" title="XBOOLE_1:th.23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">X</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
<b>by </b><i><a class="txt" href="dynkin.html#E6:36_1_1_2"><i><font color="Green" title="E33">A9</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:36_1_1_2"/>
<font color="Maroon" title="c4">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="dynkin.html#E4:36_1_1_2"><i><font color="Green" title="E32">A8</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="E5:36_1_1"/>
 <a href="dynkin.html#K8" title="DYNKIN:func.8">DynSys</a> <font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">G</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#E2:36_1_1"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="dynkin.html#E3:36_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="dynkin.html#D7" target="_self" title="DYNKIN:def.7">Def7</a></i>;<br/>


</div>
