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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="prob_1.html#NM2">SetSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>



<b>assume </b><a NAME="E1:94"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></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="E2:94"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E4:94"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="kurato_2.html#D12" target="_self">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> being   <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><b> such that </b><br/><a NAME="E6:94"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> is    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is    <a href="kurato_2.html#M1" target="_self">subsequence</a> of <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>5</sub></font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> ) )
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:94"><i><font color="Green">E55</font></i></a>, <a class="ref" href="kurato_2.html#T74" target="_self">Th74</a></i>;<br/>
<a NAME="E7:94"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E4:94"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E8:94"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5">BORSUK_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:94"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T102">ZFMISC_1:102</a></i>;<br/>
<a NAME="E11:94"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E4:94"><i><font color="Green">E56</font></i></a>, <a class="txt" href="kurato_2.html#E6:94"><i><font color="Green">E57</font></i></a>, <a class="ref" href="kurato_2.html#T64" target="_self">Th64</a></i>;<br/>
<a NAME="E12:94"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="kurato_2.html#E10:94"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E13:94"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="kurato_2.html#E6:94"><i><font color="Green">E57</font></i></a>, <a class="ref" href="kurato_2.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E14:94"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K9" target="_self">Lim_sup</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E10:94"><i><font color="Green">E58</font></i></a>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


</div>
