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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">Si</font> be   <a href="prob_1.html#NM3">SigmaField</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">S</font> be    <a href="prob_1.html#M1">SetSequence</a> of <font color="Maroon">Si</font>;<br/>







<a NAME="E1:85"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">A1</font> being  <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font> holds <br/> ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Olive">A1</font></span>)</span> iff for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">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="Olive">A1</font> <a href="funct_2.html#K3">.</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="prob_1.html#NM2">SetSequence</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:85_1"/>
 <a href="kurato_2.html#K5">lim_sup</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E2:85_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="prob_1.html#K4">Intersection</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K2" target="_self">superior_setsequence</a> <font color="Maroon">A1</font></span>)</span> iff for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">A1</font> <a href="funct_2.html#K3">.</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> )
 <b>by </b><i><a class="ref" href="kurato_2.html#T8">KURATO_2:8</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:85"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setlim_1.html#K7" target="_self">lim_sup</a> <font color="Maroon">S</font> iff for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">S</font> <a href="funct_2.html#K3">.</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> )
 <b>by </b><i><a class="ref" href="setlim_1.html#T16" target="_self">Th16</a></i>;<br/>


</div>
