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

<b>let </b><font color="Maroon">F</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:65_1_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font>
 ;<br/><a NAME="E2:65_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_1"><b>proof </b></a><div class="add">

<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="E1:65_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:65_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">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#T8" target="_self">Th8</a>, <a class="ref" href="kurato_2.html#T7">KURATO_2:7</a></i>;<br/><a NAME="E4:65_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:65_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:65_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:65_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>
<a NAME="E5:65_1_1_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:65_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T8" target="_self">Th8</a>, <a class="ref" href="kurato_2.html#T7">KURATO_2:7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:65_1_1_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:65_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K1">Union</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span>
 ;<br/>

<a NAME="E3:65_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_2"><b>proof </b></a><div class="add">

<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="E1:65_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:65_1_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T8" target="_self">Th8</a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/><a NAME="E4:65_1_1_2_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Olive">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:65_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T7">KURATO_2:7</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:65_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:65_1_1_2"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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">A1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">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#T7">KURATO_2:7</a></i>;<br/>
<a NAME="E5:65_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="setlim_1.html#K1" target="_self">inferior_setsequence</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:65_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="setlim_1.html#T8" target="_self">Th8</a>, <a class="ref" href="prob_1.html#T25">PROB_1:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_1_1"/>
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="kurato_2.html#K4">lim_inf</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
