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

<b>let </b><font color="Maroon" title="c1">n</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/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">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"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>;<br/>





<b>assume </b><a NAME="E1:81"/><i><font color="Green" title="E47">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="c4">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"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">i</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ;<br/>

<b>thus </b><a NAME="E2:81"/>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <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="c4">C</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:81_1"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:81_1"/><i><font color="Green" title="E49">A3</font></i>: 
( <font color="Maroon" title="c6">x1</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="c2">A</font> &amp; <font color="Maroon" title="c7">x2</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="c3">B</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">x1</font>,<font color="Maroon" title="c7">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x1</font> = <font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c7">x2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="kurato_2.html#E3:81_1"><i><font color="Green" title="E49">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">p</font> = <font color="Maroon" title="c5">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span> <b>by </b><i><a class="txt" href="kurato_2.html#E1:81_1"><i><font color="Green" title="E48">A2</font></i></a></i>;<br/>
<a NAME="E6:81_1"/>
for <font color="Olive" title="b1">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c10">p</font>  ex <font color="Olive" title="b2">k</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>  st <br/>for <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b3">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c10">p</font>;<br/>

<a NAME="E1:81_1_1"/>
<font color="Maroon" title="c11">G</font> is    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>,<span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">x2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:81_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="kurato_2.html#T41" target="_self" title="KURATO_2:th.41">Th41</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">V</font> being    <a href="connsp_2.html#M2" title="CONNSP_2:mode.2">a_neighborhood</a> of <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c8">x1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c13">W</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c9">x2</font><b> such that </b><br/><a NAME="E3:81_1_1"/><i><font color="Green" title="E50">A4</font></i>: 
<span class="p1"><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">V</font>,<font color="Maroon" title="c13">W</font></span><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T67" title="BORSUK_1:th.67">BORSUK_1:67</a></i>;<br/>
<a NAME="E4:81_1_1"/>
<font color="Maroon" title="c12">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c8">x1</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T10" title="CONNSP_2:th.10">CONNSP_2:10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">k1</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> <b> such that </b><br/><a NAME="E6:81_1_1"/><i><font color="Green" title="E51">A5</font></i>: 
for <font color="Olive" title="b1">m</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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">k1</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">V</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:81_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="kurato_2.html#D11" target="_self" title="KURATO_2:def.11">Def11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c15">k2</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> <b> such that </b><br/><a NAME="E8:81_1_1"/><i><font color="Green" title="E52">A6</font></i>: 
for <font color="Olive" title="b1">m</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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">k2</font> holds <br/><font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c13">W</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:81_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="kurato_2.html#D11" target="_self" title="KURATO_2:def.11">Def11</a></i>;<br/>
<b>take </b><font color="Maroon" title="c16">k</font> =  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <font color="Maroon" title="c14">k1</font>,<font color="Maroon" title="c15">k2</font>;<br/>

<a NAME="E9:81_1_1"/><i><font color="Green" title="E53">A7</font></i>: 
( <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c14">k1</font> &amp; <font color="Maroon" title="c16">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c15">k2</font> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<b>let </b><font color="Maroon" title="c17">m</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/>

<b>assume </b><a NAME="E10:81_1_1"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">k</font>
 ;<br/>

<a NAME="E11:81_1_1"/><b>then </b>
<font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c14">k1</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E9:81_1_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:81_1_1"/><b>then </b><i><font color="Green" title="E55">A9</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c17">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c12">V</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E6:81_1_1"><i><font color="Green" title="E51">A5</font></i></a></i>;<br/>
<a NAME="E13:81_1_1"/>
<font color="Maroon" title="c17">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">k2</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E9:81_1_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="kurato_2.html#E10:81_1_1"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:81_1_1"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c17">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c13">W</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E8:81_1_1"><i><font color="Green" title="E52">A6</font></i></a></i>;<br/>
<a NAME="E15:81_1_1"/><b>then </b>
<span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c17">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c17">m</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">V</font>,<font color="Maroon" title="c13">W</font></span><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E12:81_1_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="kurato_2.html#T4" target="_self" title="KURATO_2:th.4">Th4</a></i>;<br/>
<a NAME="E16:81_1_1"/><b>then </b>
<font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c17">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">[:</a><span class="default"><font color="Maroon" title="c12">V</font>,<font color="Maroon" title="c13">W</font></span><a href="borsuk_1.html#K8" title="BORSUK_1:func.8">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:81"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:81_1_1"/>
<font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c17">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">G</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E3:81_1_1"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:81_1"/>
<font color="Maroon" title="c5">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="c4">C</font>
 <b>by </b><i><a class="ref" href="kurato_2.html#D11" target="_self" title="KURATO_2:def.11">Def11</a></i>;<br/>


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

<b>thus </b><a NAME="E3:81"/>
 <a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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="E1:81_2"/><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c5">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="c4">C</font>
 ;<br/>

<a NAME="E2:81_2"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1"><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">[:</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="borsuk_1.html#K4" title="BORSUK_1:func.4">:]</a></span>
 
;<br/>
<a NAME="E3:81_2"/><b>then </b><i><font color="Green" title="E49">A11</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D5" title="BORSUK_1:def.5">BORSUK_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c7">p2</font> = <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E5:81_2"/><i><font color="Green" title="E50">A12</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">[</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font></span><a href="borsuk_1.html#K6" title="BORSUK_1:func.6">]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E3:81_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">H</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">p2</font>;<br/>
<b>consider </b><font color="Maroon" title="c9">F</font> being    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">p1</font>;<br/>
<a NAME="E8:81_2"/>
for <font color="Olive" title="b1">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">p1</font>  ex <font color="Olive" title="b2">k</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>  st <br/>for <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b3">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c6">p1</font>;<br/>

<b>consider </b><font color="Maroon" title="c11">k</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> <b> such that </b><br/><a NAME="E2:81_2_1"/><i><font color="Green" title="E51">A13</font></i>: 
for <font color="Olive" title="b1">m</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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> holds <br/><font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c10">G</font>,<font color="Maroon" title="c8">H</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:81_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="kurato_2.html#E5:81_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="kurato_2.html#D11" target="_self" title="KURATO_2:def.11">Def11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">k</font>
;<br/>

<b>let </b><font color="Maroon" title="c12">m</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/>

<b>assume </b><a NAME="E3:81_2_1"/>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font>
 ;<br/>

<a NAME="E4:81_2_1"/><b>then </b>
<font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c10">G</font>,<font color="Maroon" title="c8">H</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E2:81_2_1"><i><font color="Green" title="E51">A13</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:81_2_1"/><i><font color="Green" title="E52">A14</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c10">G</font>,<font color="Maroon" title="c8">H</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:81_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:81"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="kurato_2.html#E6:81_2_1"><i><font color="Green" title="E52">A14</font></i></a></i>;<br/>
<a NAME="E8:81_2_1"/><b>then </b><i><font color="Green" title="E53">A15</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E9:81_2_1"/>
<font color="Maroon" title="c13">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E6:81_2_1"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:81_2_1"/>
<font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E8:81_2_1"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:81_2"/><b>then </b><i><font color="Green" title="E51">A16</font></i>: 
<font color="Maroon" title="c6">p1</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="c2">A</font>
 
<b>by </b><i><a class="ref" href="kurato_2.html#D11" target="_self" title="KURATO_2:def.11">Def11</a></i>;<br/>
<a NAME="E10:81_2"/>
for <font color="Olive" title="b1">G</font> being   <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">p2</font>  ex <font color="Olive" title="b2">k</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>  st <br/>for <font color="Olive" title="b3">m</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>   st <font color="Olive" title="b3">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">k</font> holds <br/><font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b3">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">G</font> be    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c7">p2</font>;<br/>

<b>consider </b><font color="Maroon" title="c11">k</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> <b> such that </b><br/><a NAME="E2:81_2_2"/><i><font color="Green" title="E52">A17</font></i>: 
for <font color="Olive" title="b1">m</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>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font> holds <br/><font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Olive" title="b1">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">G</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E1:81_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="kurato_2.html#E5:81_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="kurato_2.html#D11" target="_self" title="KURATO_2:def.11">Def11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">k</font>
;<br/>

<b>let </b><font color="Maroon" title="c12">m</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/>

<b>assume </b><a NAME="E3:81_2_2"/>
<font color="Maroon" title="c12">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">k</font>
 ;<br/>

<a NAME="E4:81_2_2"/><b>then </b>
<font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">G</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E2:81_2_2"><i><font color="Green" title="E52">A17</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c13">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:81_2_2"/><i><font color="Green" title="E53">A18</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font> &amp; <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">[:</a><span class="default"><font color="Maroon" title="c9">F</font>,<font color="Maroon" title="c10">G</font></span><a href="borsuk_1.html#K7" title="BORSUK_1:func.7">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E7:81_2_2"/>
<font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E1:81"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="kurato_2.html#E6:81_2_2"><i><font color="Green" title="E53">A18</font></i></a></i>;<br/>
<a NAME="E8:81_2_2"/><b>then </b><i><font color="Green" title="E54">A19</font></i>: 
<font color="Maroon" title="c13">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E9:81_2_2"/>
<font color="Maroon" title="c13">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 
<b>by </b><i><a class="txt" href="kurato_2.html#E6:81_2_2"><i><font color="Green" title="E53">A18</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>hence </b><a NAME="E10:81_2_2"/>
<font color="Maroon" title="c3">B</font> <a href="kurato_2.html#K7" title="KURATO_2:func.7">.</a> <font color="Maroon" title="c12">m</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="kurato_2.html#E8:81_2_2"><i><font color="Green" title="E54">A19</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:81_2"/><b>then </b>
<font color="Maroon" title="c7">p2</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="c3">B</font>
 
<b>by </b><i><a class="ref" href="kurato_2.html#D11" target="_self" title="KURATO_2:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E12:81_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">[:</a><span class="default"><span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c2">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="kurato_2.html#K8" title="KURATO_2:func.8">Lim_inf</a> <font color="Maroon" title="c3">B</font></span>)</span></span><a href="borsuk_1.html#K5" title="BORSUK_1:func.5">:]</a></span>
 <b>by </b><i><a class="txt" href="kurato_2.html#E5:81_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="txt" href="kurato_2.html#E9:81_2"><i><font color="Green" title="E51">A16</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


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


</div>
