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<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="E1:29"/><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/>

<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="E2:29"/><i><font color="Green">E32</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E3:29"/><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  <span class="p1">(<span class="default"><a href="dynkin.html#K6" target="_self">disjointify</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="29_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="E1:29_1"/><i><font color="Green">E41</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E2:29_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:29_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T87">RELAT_1:87</a></i>;<br/>
<a NAME="E4:29_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E5:29_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:29_1"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font></span>)</span></span>)</span> is  <a href="finset_1.html#V1">finite</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">Omega</font>
 
<b>by </b><i>, , <a class="ref" href="finset_1.html#T26">FINSET_1:26</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:29_1"/><b>then </b><i><font color="Green">E46</font></i>: 
 <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"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:29_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:29_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">urf</font> =  <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"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K2">PSeg</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">Omega</font> <b>by </b><i/>;<br/>
<a NAME="E11:29_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#K7">\</a> <font color="Maroon">urf</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/>
<b>hence </b><a NAME="E12:29_1"/>
<span class="p1">(<span class="default"><a href="dynkin.html#K6" target="_self">disjointify</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/>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="dynkin.html#K6" target="_self">disjointify</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="dynkin.html#K6" target="_self">disjointify</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K6" target="_self">disjointify</a> <font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><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">x</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="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:29_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="dynkin.html#K6" target="_self">disjointify</a> <font color="Maroon">f</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:29_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:29"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="dynkin.html#K6" target="_self">disjointify</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/>


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