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<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="E1:37"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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/>
<a NAME="E3:37"/>
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="dynkin.html#D8" target="_self" title="DYNKIN:def.8">Def8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c3">X</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">G</font> <b>by </b><i><a class="txt" href="dynkin.html#E1:37"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E5:37"/>
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="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c5">G</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" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:37_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">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="E3:37_1"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">E</font>
 
<b>by </b><i><a class="txt" href="dynkin.html#E1:37"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="dynkin.html#E1:37_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="finsub_1.html#D2" title="FINSUB_1:def.2">FINSUB_1:def 2</a></i>;<br/>
<a NAME="E4:37_1"/>
<font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="dynkin.html#D8" target="_self" title="DYNKIN:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="dynkin.html#E3:37_1"><i><font color="Green" title="E32">A3</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="E6:37"/><b>then </b>
( <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c5">G</font> &amp;  <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c5">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="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E7:37"/><b>then </b>
 <a href="dynkin.html#K7" title="DYNKIN:func.7">generated_Dynkin_System</a> <font color="Maroon" title="c2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dynkin.html#K9" title="DYNKIN:func.9">DynSys</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="ref" href="dynkin.html#D8" target="_self" title="DYNKIN:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E8:37"/>
<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#E1:37"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="dynkin.html#D9" target="_self" title="DYNKIN:def.9">Def9</a></i>;<br/>


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