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

<b>let </b><font color="Maroon">A</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">B</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">C</font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:91"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font></span>)</span>
 ;<br/>

<a NAME="E2:91"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:91_1"/>
<font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:91"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:91_1"/>
<font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:91"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:91_2"/>
<font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:91"><i><font color="Green">E54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:91_2"/>
<font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:91"/>
<span class="p1">(<span class="default"><a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">B</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:91_3"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">B</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_3_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">A</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">B</font> )
 </span><b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_3_1_1"><b>suppose </b></a><a NAME="E1:91_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">A</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">A1</font> being    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:91_3_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">A1</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/><b>consider </b><font color="Maroon">C1</font> being    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:91_3_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">A1</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 <b>by </b><i>, </i>;<br/><a NAME="E6:91_3_1_1"/>
 <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">C1</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E7:91_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_3_1_2"><b>suppose </b></a><a NAME="E1:91_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">B1</font> being    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E3:91_3_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B1</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/><b>consider </b><font color="Maroon">C1</font> being    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E5:91_3_1_2"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">B1</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:91_3"><i><font color="Green">E56</font></i></a>, </i>;<br/><a NAME="E6:91_3_1_2"/>
 <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">C1</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E7:91_3_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div>
