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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:13_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">F</font>
 ;<br/><b>consider </b><font color="Maroon">f</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:13_1"/><i><font color="Green">E49</font></i>: 
(  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K2" target="_self">meet</a> <font color="Maroon">f</font> &amp; ( 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#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Olive">n</font></span>)</span> ) )
 <b>by </b><i/>;<br/><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/><b>set </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">n</font>;<br/><a NAME="E4:13_1"/>
<font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:13_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , </i>;<br/><b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:13_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">n</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E8:13_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:13"/><i><font color="Green">E56</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>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:13"/><i><font color="Green">E57</font></i>: 
(  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K2" target="_self">meet</a> <font color="Maroon">f</font> &amp; ( 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#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Olive">n</font></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E5:13"/>
for <font color="Olive">z</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

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

<b>set </b><font color="Maroon">G</font> = <font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">z</font>;<br/>
<a NAME="E1:13_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1" target="_self">Union</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:13_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i/>;<br/>
<a NAME="E4:13_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="kurato_2.html#K3" target="_self">^\</a> <font color="Maroon">z</font></span>)</span> <a href="prob_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="prob_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">z</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:13_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="prob_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, , <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:13"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K5" target="_self">lim_sup</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


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