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

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">s</font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:13"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/>

<a NAME="E2:13"/><i><font color="Green">E28</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">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="topmetr.html#T25">TOPMETR:25</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1"/>
( <font color="Maroon">s</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> or  not <font color="Maroon">s</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><b>suppose </b></a><a NAME="E1:13_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">s</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>
 ;<br/><div class="add"><a NAME="E2:13_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>reconsider </b><font color="Maroon">Q</font> = <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></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="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E4:13_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="jordan6.html#T46">JORDAN6:46</a></i>;<br/><b>reconsider </b><font color="Maroon">T</font> = <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></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="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/><a NAME="E6:13_1_1"/><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_2.html#T17">RCOMP_2:17</a></i>;<br/><a NAME="E7:13_1_1"/>
1 <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#T15">RCOMP_1:15</a></i>;<br/><a NAME="E8:13_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</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#T16">RCOMP_1:16</a></i>;<br/><a NAME="E9:13_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</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="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:13_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>then reconsider </b><font color="Maroon">P2</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">T</font></span>)</span> ;<br/><a NAME="E12:13_1_1"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</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> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</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>
 ;<br/>

<a NAME="E2:13_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">x</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i/>;<br/>
<a NAME="E4:13_1_1_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">sx</font> &amp; <font color="Maroon">sx</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 
<b>by </b><i><a class="ref" href="jordan20.html#T1" target="_self">Th1</a>, <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a></i>;<br/>
<a NAME="E5:13_1_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">sx</font> &amp; <font color="Maroon">sx</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="jordan20.html#T1" target="_self">Th1</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E6:13_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:13_1_1"/>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:13_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">sx</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:13_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">sx</font> &amp; <font color="Maroon">sx</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i>, <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E4:13_1_1_2"/><i><font color="Green">E48</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">sx</font>
 
<b>by </b><i>, , <a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<a NAME="E5:13_1_1_2"/>
2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">sx</font>
 
<b>by </b><i><a class="ref" href="jordan20.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:13_1_1_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></span> &amp; <font color="Maroon">x</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="jordan20.html#T1" target="_self">Th1</a>, , <a class="ref" href="rcomp_1.html#T47">RCOMP_1:47</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<b>hence </b><a NAME="E7:13_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:13_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K2">].</a><span class="default"><font color="Maroon">s</font>,2</span><a href="rcomp_1.html#K2">.[</a></span> <a href="subset_1.html#K9">/\</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="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E15:13_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">T</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E16:13_1_1"/>
 <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1 <a href="hidden.html#R1">=</a> <a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="topmetr.html#T26">TOPMETR:26</a></i>;<br/><b>hence </b><a NAME="E17:13_1_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T32">TOPS_2:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2"><b>suppose </b></a><a NAME="E1:13_1_2"/><i><font color="Green">E73</font></i>: 
not <font color="Maroon">s</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>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1_2_1_1"/>
( <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 or <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span><b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_1_1_1"><b>case </b></a><a NAME="E1:13_1_2_1_1_1"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:13_1_2_1_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="rcomp_2.html#T10">RCOMP_2:10</a></i>;<br/><a NAME="E3:13_1_2_1_1_1"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="topmetr.html#K5">I[01]</a> 
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T5">PRE_TOPC:5</a></i>;<br/><b>hence </b><a NAME="E4:13_1_2_1_1_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2_1_1_2"><b>case </b></a><a NAME="E1:13_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">t</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="E3:13_1_2_1_1_2"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T7">XREAL_1:7</a></i>;<br/><a NAME="E4:13_1_2_1_1_2"/><i><font color="Green">E78</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:13_1_2_1_1_2_1"/>
( <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E2:13_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_2.html#K2">].</a><span class="default"><font color="Maroon">s</font>,1</span><a href="rcomp_2.html#K2">.]</a></span>
 
<b>by </b><i><a class="ref" href="rcomp_2.html#T4">RCOMP_2:4</a></i>;<br/>
<b>hence </b><a NAME="E3:13_1_2_1_1_2_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">t</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="E6:13_1_2_1_1_2"/>
<font color="Maroon">t</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:13_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:13_1_2"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
