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

<a NAME="E1:23"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a>   st  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">B</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a> , <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="borsuk_1.html#K20">I[01]</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_1"/><i><font color="Green">E32</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:23_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <a href="borsuk_1.html#K20">I[01]</a>  )
 <b>and </b><br/><a NAME="E3:23_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E4:23_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>assume </b><a NAME="E5:23_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E6:23_1"/><i><font color="Green">E47</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E7:23_1"/><i><font color="Green">E50</font></i>: 
 <a href="borsuk_1.html#K20">I[01]</a>  is  <a href="borsuk_1.html#V1">closed</a>  <a href="pre_topc.html#M1">SubSpace</a> of  <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">A</font>, <font color="Maroon">Q</font> = <font color="Maroon">B</font> as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:23_1"/><i><font color="Green">E51</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <a href="metric_1.html#K8">RealSpace</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<b>reconsider </b><font color="Maroon">A0</font> = <font color="Maroon">P</font>, <font color="Maroon">B0</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<a NAME="E11:23_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">A0</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">B0</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i>, , <a class="ref" href="treal_1.html#T3" target="_self">Th3</a>, <a class="ref" href="tsep_1.html#T12">TSEP_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">P</font>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E1:23_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Q</font>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>thus </b><a NAME="E1:23_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E18:23_1"/>
0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<a NAME="E19:23_1"/><b>then </b><i><font color="Green">E69</font></i>: 
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E20:23_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">P</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">p</font> = 0;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:23_1_3_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:23_1_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E3:23_1_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:23_1_3_1"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>hence </b><a NAME="E5:23_1_3_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1_3"/>
<font color="Maroon">P</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:23_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">Q</font> is <a href="seq_4.html#V2">bounded_below</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">p</font> = 0;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:23_1_4_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:23_1_4_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E3:23_1_4_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:23_1_4_1"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>hence </b><a NAME="E5:23_1_4_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1_4"/>
<font color="Maroon">Q</font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_5_1"/>
( 0 <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or 0 <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 </span><b>by </b><i>, <a class="ref" href="treal_1.html#T2" target="_self">Th2</a>, <a class="txt" href="treal_1.html#E1:23_1"><i><font color="Green">E32</font></i></a>, <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="23_1_5_1_1"><b>suppose </b></a><a NAME="E1:23_1_5_1_1"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><a NAME="E2:23_1_5_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="treal_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>set </b><font color="Maroon">l</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font>;<br/><b>reconsider </b><font color="Maroon">m</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">m</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">B00</font> = <font color="Maroon">B0</font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/><a NAME="E6:23_1_5_1_1"/><i><font color="Green">E76</font></i>: 
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_5_1_1_1"/>
not  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:23_1_5_1_1_1"/><b>then </b>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B00</font> &amp; <font color="Maroon">B00</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:23_1_5_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:23_1_5_1_1_1"/><i><font color="Green">E91</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B0</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:23_1_5_1_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E8:23_1_5_1_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rm</font> = <font color="Maroon">r</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E11:23_1_5_1_1_1"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E2:23_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E12:23_1_5_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E13:23_1_5_1_1_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E14:23_1_5_1_1_1"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E15:23_1_5_1_1_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E16:23_1_5_1_1_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">rm</font>,<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E17:23_1_5_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Maroon">rm</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E18:23_1_5_1_1_1"/><b>then </b>
<font color="Maroon">rm</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> </span>}</span> 
 
;<br/>
<a NAME="E19:23_1_5_1_1_1"/><b>then </b>
<font color="Maroon">rm</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E20:23_1_5_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="treal_1.html#E2:23_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:23_1_5_1_1"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E8:23_1_5_1_1"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E9:23_1_5_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
 ex <font color="Olive">u0</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">u0</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u0</font> &amp; <font color="Olive">u0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_5_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">w0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:23_1_5_1_1_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">w0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w0</font> &amp; <font color="Maroon">w0</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> )
 ;<br/><a NAME="E4:23_1_5_1_1_2"/>
<font color="Maroon">w0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:23_1_5_1_1_2"/><b>then </b>
<font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:23_1_5_1_1_2"/><b>then </b>
<font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="treal_1.html#T2" target="_self">Th2</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E7:23_1_5_1_1_2"/><b>then </b>
( <font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E8:23_1_5_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E11:23_1_5_1_1"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="treal_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:23_1_5_1_1"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_1_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>assume </b><a NAME="E1:23_1_5_1_1_3_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>assume </b><a NAME="E2:23_1_5_1_1_3_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">e</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:23_1_5_1_1_3_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:23_1_5_1_1_3_1"/><i><font color="Green">E120</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">e</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:23_1_5_1_1_3_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>set </b><font color="Maroon">e0</font> =  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>;<br/><a NAME="E8:23_1_5_1_1_3_1"/><i><font color="Green">E123</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E9:23_1_5_1_1_3_1"/><i><font color="Green">E124</font></i>: 
(  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_5_1_1_3_1_1"/><i><font color="Green">E125</font></i>: 
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:23_1_5_1_1_3_1_1"/><b>then </b>
(  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="treal_1.html#E6"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="treal_1.html#T4" target="_self">Th4</a>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>hence </b><a NAME="E3:23_1_5_1_1_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="treal_1.html#E4:23_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="treal_1.html#E5:23_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/></div><b>end;</b></div><a NAME="E11:23_1_5_1_1_3_1"/><b>then </b><i><font color="Green">E126</font></i>: 
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">e1</font> =  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_1_3_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1_1_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_5_1_1_3_1_2_1"/>
(  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1_1_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_5_1_1_3_1_2_1_1"/><i><font color="Green">E128</font></i>: 
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_5_1_1_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E3:23_1_5_1_1_3_1_2_1_1"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E4:23_1_5_1_1_3_1_2_1_1"/><b>then </b>
(  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="txt" href="treal_1.html#E3:23_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:23_1_5_1_1_3_1_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E6:23_1"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1_1_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_5_1_1_3_1_2_1_2"/>
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><div class="add"><a NAME="E2:23_1_5_1_1_3_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i>, <a class="txt" href="treal_1.html#E3:23_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>hence </b><a NAME="E3:23_1_5_1_1_3_1_2_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E14:23_1_5_1_1_3_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E15:23_1_5_1_1_3_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E16:23_1_5_1_1_3_1"/><b>then </b>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> </span>}</span> 
 ;<br/><a NAME="E17:23_1_5_1_1_3_1"/><b>then </b>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><b>hence </b><a NAME="E18:23_1_5_1_1_3_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="treal_1.html#E5:23_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1_5_1_1_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:23_1_5_1_1"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A0</font> )
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><b>hence </b><a NAME="E15:23_1_5_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="treal_1.html#T1" target="_self">Th1</a>, <a class="ref" href="treal_1.html#T4" target="_self">Th4</a>, <a class="txt" href="treal_1.html#E2:23_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1_2"><b>suppose </b></a><a NAME="E1:23_1_5_1_2"/>
0 <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div class="add"><a NAME="E2:23_1_5_1_2"/><b>then </b><i><font color="Green">E130</font></i>: 
not 0 <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="treal_1.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>set </b><font color="Maroon">l</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font>;<br/><b>reconsider </b><font color="Maroon">m</font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">m</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">A00</font> = <font color="Maroon">A0</font> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/><a NAME="E6:23_1_5_1_2"/><i><font color="Green">E132</font></i>: 
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_1_5_1_2_1"/>
not  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:23_1_5_1_2_1"/><b>then </b>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A00</font> &amp; <font color="Maroon">A00</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<b>then consider </b><font color="Maroon">s</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:23_1_5_1_2_1"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:23_1_5_1_2_1"/><i><font color="Green">E134</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A0</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:23_1_5_1_2_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E8:23_1_5_1_2_1"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E5"><i><font color="Green">Lemma30</font></i></a>, , , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">rm</font> = <font color="Maroon">r</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E11:23_1_5_1_2_1"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="txt" href="treal_1.html#E2:23_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<a NAME="E12:23_1_5_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="real_2.html#T106">REAL_2:106</a></i>;<br/>
<a NAME="E13:23_1_5_1_2_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E14:23_1_5_1_2_1"/><b>then </b>
(  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> &amp; <font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/>
<a NAME="E15:23_1_5_1_2_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="seq_2.html#T9">SEQ_2:9</a></i>;<br/>
<a NAME="E16:23_1_5_1_2_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">r</font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/>
<a NAME="E17:23_1_5_1_2_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">rm</font>,<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/>
<a NAME="E18:23_1_5_1_2_1"/><b>then </b>
<font color="Maroon">rm</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">q</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> </span>}</span> 
 
;<br/>
<a NAME="E19:23_1_5_1_2_1"/><b>then </b>
<font color="Maroon">rm</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/>
<b>hence </b><a NAME="E20:23_1_5_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="treal_1.html#E2:23_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E7:23_1_5_1_2"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/><a NAME="E8:23_1_5_1_2"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E9:23_1_5_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
 ex <font color="Olive">u0</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">u0</font> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u0</font> &amp; <font color="Olive">u0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> ) </span>}</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:23_1_5_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">w0</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:23_1_5_1_2_2"/><i><font color="Green">E138</font></i>: 
( <font color="Maroon">w0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">w0</font> &amp; <font color="Maroon">w0</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E4:23_1_5_1_2_2"/>
<font color="Maroon">w0</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:23_1_5_1_2_2"/><b>then </b>
<font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E6:23_1_5_1_2_2"/><b>then </b>
<font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="treal_1.html#T2" target="_self">Th2</a>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E7:23_1_5_1_2_2"/><b>then </b>
( <font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or <font color="Maroon">w0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E8:23_1_5_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/></div><b>end;</b></div><a NAME="E11:23_1_5_1_2"/><b>then </b><i><font color="Green">E139</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="txt" href="treal_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E13:23_1_5_1_2"/><i><font color="Green">E140</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_2_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/><b>assume </b><a NAME="E1:23_1_5_1_2_3_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><b>assume </b><a NAME="E2:23_1_5_1_2_3_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><b>then consider </b><font color="Maroon">e</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:23_1_5_1_2_3_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">e</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:23_1_5_1_2_3_1"/><i><font color="Green">E143</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">e</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a>, <a class="ref" href="topmetr.html#D7">TOPMETR:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:23_1_5_1_2_3_1"/><i><font color="Green">E144</font></i>: 
<font color="Maroon">e</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> <a href="real_1.html#K4">*</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><b>set </b><font color="Maroon">e0</font> =  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>;<br/><a NAME="E8:23_1_5_1_2_3_1"/><i><font color="Green">E145</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E9:23_1_5_1_2_3_1"/><i><font color="Green">E146</font></i>: 
(  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_5_1_2_3_1_1"/><i><font color="Green">E147</font></i>: 
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:23_1_5_1_2_3_1_1"/><b>then </b>
(  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="treal_1.html#E6"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="treal_1.html#T4" target="_self">Th4</a>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>hence </b><a NAME="E3:23_1_5_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="treal_1.html#E4:23_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="treal_1.html#E5:23_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/></div><b>end;</b></div><a NAME="E11:23_1_5_1_2_3_1"/><b>then </b><i><font color="Green">E148</font></i>: 
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">w</font> where <font color="Olive">w</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">w</font> &amp; <font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">e1</font> =  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5_1_2_3_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1_2_3_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_5_1_2_3_1_2_1"/>
(  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 or  <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1_2_3_1_2_1_1"><b>suppose </b></a><a NAME="E1:23_1_5_1_2_3_1_2_1_1"/><i><font color="Green">E149</font></i>: 
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:23_1_5_1_2_3_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><a NAME="E3:23_1_5_1_2_3_1_2_1_1"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T52">XREAL_1:52</a></i>;<br/><a NAME="E4:23_1_5_1_2_3_1_2_1_1"/><b>then </b>
(  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font> )
 <b>by </b><i>, <a class="txt" href="treal_1.html#E3:23_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:23_1_5_1_2_3_1_2_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E6:23_1"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_5_1_2_3_1_2_1_2"><b>suppose </b></a><a NAME="E1:23_1_5_1_2_3_1_2_1_2"/>
 <a href="square_1.html#K2">max</a> 0,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 ;<br/><div class="add"><a NAME="E2:23_1_5_1_2_3_1_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> &amp; <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i>, <a class="txt" href="treal_1.html#E3:23_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>hence </b><a NAME="E3:23_1_5_1_2_3_1_2_1_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E14:23_1_5_1_2_3_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="square_1.html#K2">max</a> 0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">P</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">e</font> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D13">METRIC_1:def 13</a></i>;<br/><a NAME="E15:23_1_5_1_2_3_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Maroon">e1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a>, <a class="ref" href="metric_1.html#D14">METRIC_1:def 14</a></i>;<br/><a NAME="E16:23_1_5_1_2_3_1"/><b>then </b>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">z</font> where <font color="Olive">z</font> is   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#K8">RealSpace</a>  :  <a href="metric_1.html#K4">dist</a> <font color="Maroon">m</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">e</font> </span>}</span> 
 ;<br/><a NAME="E17:23_1_5_1_2_3_1"/><b>then </b>
<font color="Maroon">e1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">m</font>,<font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T18">METRIC_1:18</a></i>;<br/><b>hence </b><a NAME="E18:23_1_5_1_2_3_1"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="txt" href="treal_1.html#E5:23_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:23_1_5_1_2_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:23_1_5_1_2"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B0</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B0</font> )
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><b>hence </b><a NAME="E15:23_1_5_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="treal_1.html#T1" target="_self">Th1</a>, <a class="ref" href="treal_1.html#T4" target="_self">Th4</a>, <a class="txt" href="treal_1.html#E2:23_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E23:23_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23"/>
 <a href="borsuk_1.html#K20">I[01]</a>  is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T11">CONNSP_1:11</a></i>;<br/>


</div>
