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<div class="add">

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font>, <font color="Maroon" title="c3">C</font> be   <a href="prob_1.html#NM2" title="PROB_1:NM.2">SetSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E54">A1</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>
 ;<br/>

<a NAME="E2:91"/><i><font color="Green" title="E55">A2</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">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" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:91_1"/>
<font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:91"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:91_1"/>
<font color="Maroon" title="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:91"/><i><font color="Green" title="E56">A3</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">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" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:91_2"/>
<font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:91"><i><font color="Green" title="E54">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:91_2"/>
<font color="Maroon" title="c2">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">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" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c3">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" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:91_3"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c2">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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c2">B</font> )
 </span><b>by </b><i><a class="txt" href="kurato_2.html#E1:91_3"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c1">A</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">A1</font> being    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:91_3_1_1"/><i><font color="Green" title="E58">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c5">A1</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#D12" target="_self" title="KURATO_2:def.12">Def12</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">C1</font> being    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c3">C</font><b> such that </b><br/><a NAME="E5:91_3_1_1"/><i><font color="Green" title="E59">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">A1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E2:91"><i><font color="Green" title="E55">A2</font></i></a>, <a class="ref" href="kurato_2.html#T69" target="_self" title="KURATO_2:th.69">Th69</a></i>;<br/><a NAME="E6:91_3_1_1"/>
 <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c5">A1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c6">C1</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E5:91_3_1_1"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="kurato_2.html#T54" target="_self" title="KURATO_2:th.54">Th54</a></i>;<br/><b>hence </b><a NAME="E7:91_3_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:91_3_1_1"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="kurato_2.html#D12" target="_self" title="KURATO_2:def.12">Def12</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" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c2">B</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">B1</font> being    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c2">B</font><b> such that </b><br/><a NAME="E3:91_3_1_2"/><i><font color="Green" title="E58">A7</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c5">B1</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#D12" target="_self" title="KURATO_2:def.12">Def12</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">C1</font> being    <a href="kurato_2.html#M1" title="KURATO_2:mode.1">subsequence</a> of <font color="Maroon" title="c3">C</font><b> such that </b><br/><a NAME="E5:91_3_1_2"/><i><font color="Green" title="E59">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c5">B1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C1</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:91"><i><font color="Green" title="E56">A3</font></i></a>, <a class="ref" href="kurato_2.html#T69" target="_self" title="KURATO_2:th.69">Th69</a></i>;<br/><a NAME="E6:91_3_1_2"/>
 <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c5">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c6">C1</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E5:91_3_1_2"><i><font color="Green" title="E59">A8</font></i></a>, <a class="ref" href="kurato_2.html#T54" target="_self" title="KURATO_2:th.54">Th54</a></i>;<br/><b>hence </b><a NAME="E7:91_3_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="kurato_2.html#K9" title="KURATO_2:func.9">Lim_sup</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:91_3_1_2"><i><font color="Green" title="E58">A7</font></i></a>, <a class="ref" href="kurato_2.html#D12" target="_self" title="KURATO_2:def.12">Def12</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
