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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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> <font color="Maroon">n</font></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> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><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"><a href="borsuk_1.html#K5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:82"/><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"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">i</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:82"/>
<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">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="tarski.html#R1">c=</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">C</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><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="E1:82_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</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">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:82_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font>, <font color="Maroon">x2</font> = <font color="Maroon">x2</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</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> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span> <b>by </b><i/>;<br/>
<a NAME="E6:82_1"/>
for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p</font>;<br/>

<a NAME="E1:82_1_1"/>
<font color="Maroon">G</font> is    <a href="connsp_2.html#M2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x2</font></span><a href="domain_1.html#K6">}</a></span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being    <a href="connsp_2.html#M2">a_neighborhood</a> of <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x1</font></span><a href="domain_1.html#K6">}</a></span>, <font color="Maroon">W</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x2</font><b> such that </b><br/><a NAME="E3:82_1_1"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="borsuk_1.html#K9">[:</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">W</font></span><a href="borsuk_1.html#K9">:]</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T67">BORSUK_1:67</a></i>;<br/>
<a NAME="E4:82_1_1"/>
<font color="Maroon">V</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x1</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T10">CONNSP_2:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">k1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:82_1_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font> holds <br/><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">k2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:82_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font> holds <br/><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">k</font> =  <a href="limfunc1.html#K1">max</a> <font color="Maroon">k1</font>,<font color="Maroon">k2</font>;<br/>

<a NAME="E9:82_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k1</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/>
<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E10:82_1_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E11:82_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k1</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:82_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:82_1_1"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:82_1"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E13:82_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k2</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:82_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="kurato_2.html#E6:82_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:82_1_1"/><b>then </b>
<font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">W</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:82_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K6">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K9">[:</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">W</font></span><a href="borsuk_1.html#K9">:]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E16:82_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K9">[:</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">W</font></span><a href="borsuk_1.html#K9">:]</a></span>
 
<b>by </b><i><a class="ref" href="kurato_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E17:82_1_1"/>
<font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:82_1"/>
<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">C</font>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E3:82"/>
 <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</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">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><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="E1:82_2"/><i><font color="Green">E84</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">C</font>
 ;<br/>

<a NAME="E2:82_2"/><b>then </b>
<font color="Maroon">x</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> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="borsuk_1.html#K5">:]</a></span>
 
;<br/>
<a NAME="E3:82_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">x</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> <font color="Maroon">n</font></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> <font color="Maroon">n</font></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 reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">p2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E5:82_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="borsuk_1.html#K7">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="borsuk_1.html#K7">]</a></span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p2</font>;<br/>
<b>consider </b><font color="Maroon">F</font> being    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p1</font>;<br/>
<a NAME="E8:82_2"/>
for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p1</font>  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p1</font>;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:82_2_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K8">[:</a><span class="default"><font color="Maroon">G</font>,<font color="Maroon">H</font></span><a href="borsuk_1.html#K8">:]</a></span>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

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

<b>assume </b><a NAME="E3:82_2_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E4:82_2_1"/><b>then </b>
<font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K8">[:</a><span class="default"><font color="Maroon">G</font>,<font color="Maroon">H</font></span><a href="borsuk_1.html#K8">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:82_2_1"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K8">[:</a><span class="default"><font color="Maroon">G</font>,<font color="Maroon">H</font></span><a href="borsuk_1.html#K8">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:82_2_1"/>
<font color="Maroon">y</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"><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="kurato_2.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E8:82_2_1"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E9:82_2_1"/>
<font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:82_2_1"/>
<font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:82_2"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:82_2"/>
for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p2</font>  ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">k</font> holds <br/><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">p2</font>;<br/>

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:82_2_2"/><i><font color="Green">E127</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> holds <br/><font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Olive">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K8">[:</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">G</font></span><a href="borsuk_1.html#K8">:]</a></span>
 <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

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

<b>assume </b><a NAME="E3:82_2_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E4:82_2_2"/><b>then </b>
<font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <span class="p1"><a href="borsuk_1.html#K8">[:</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">G</font></span><a href="borsuk_1.html#K8">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:82_2_2"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="borsuk_1.html#K8">[:</a><span class="default"><font color="Maroon">F</font>,<font color="Maroon">G</font></span><a href="borsuk_1.html#K8">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:82_2_2"/>
<font color="Maroon">y</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"><font color="Maroon">A</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 
<b>by </b><i><a class="ref" href="kurato_2.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E8:82_2_2"/><b>then </b><i><font color="Green">E184</font></i>: 
<font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E9:82_2_2"/>
<font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:82_2_2"/>
<font color="Maroon">B</font> <a href="kurato_2.html#K7" target="_self">.</a> <font color="Maroon">m</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:82_2"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:82_2"/>
<font color="Maroon">x</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">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" target="_self">Lim_inf</a> <font color="Maroon">B</font></span>)</span></span><a href="borsuk_1.html#K6">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>


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


</div>
