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

<b>let </b><font color="Maroon">Omega</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">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">D</font> is    <a href="dynkin.html#M1" target="_self">Dynkin_System</a> of <font color="Maroon">Omega</font> &amp; <font color="Maroon">D</font> is <a href="finsub_1.html#V2">intersection_stable</a> )
 ;<br/>

<a NAME="E2:33"/><b>then </b><i><font color="Green">E32</font></i>: 
for <font color="Olive">X</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font>  st <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Olive">X</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:33"/>
for <font color="Olive">f</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>  st ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Olive">f</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> ) holds <br/> <a href="prob_1.html#K4">Intersection</a> <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">Omega</font>;<br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:33_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:33_1_1"/>
<font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:33_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:33_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:33_1_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="prob_1.html#D4">PROB_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E2:33_1_2"/>
<span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="dynkin.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:33_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_3"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="txt" href="dynkin.html#E2:33"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="dynkin.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:33_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="dynkin.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E6:33_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="txt" href="dynkin.html#E1:33"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:33_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:33_1"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:33_1"/><b>then </b>
 <a href="prob_1.html#K2">Union</a> <span class="p1">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="card_3.html#D4">CARD_3:def 4</a></i>;<br/>
<a NAME="E8:33_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="prob_1.html#K2">Union</a> <span class="p2">(<span class="default"><a href="prob_1.html#K3">Complement</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:33_1"/>
 <a href="prob_1.html#K4">Intersection</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="prob_1.html#D5">PROB_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33"/>
<font color="Maroon">D</font> is   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">Omega</font>
 <b>by </b><i>, <a class="ref" href="prob_1.html#T32">PROB_1:32</a></i>;<br/>


</div>
