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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:113"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>and </b><br/><a NAME="E2:113"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">p</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E3:113"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:113_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:113_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_1"><b>suppose </b></a><a NAME="E1:113_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">ez</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><a NAME="E3:113_1_1_1"/>
for <font color="Olive">G1</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>  st <font color="Olive">G1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">G1</font> holds <br/><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:113_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">G1</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>thus </b><a NAME="E2:113_1_1_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> implies <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:113_1_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 ;<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="E3:113_1_1_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ez</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T25" target="_self">Th25</a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</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/>
<a NAME="E5:113_1_1_1_1_1"/><i><font color="Green">E63</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="square_1.html#T93">SQUARE_1:93</a>, <a class="ref" href="jordan2c.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E6:113_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<b>set </b><font color="Maroon">p2</font> = <font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">ep2</font> = <font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<i><font color="Green">E65</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T51">EUCLID:51</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p0">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p5">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">p</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2
<b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<a NAME="E9:113_1_1_1_1_1"/>
<font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<a NAME="E10:113_1_1_1_1_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">ez</font>,<font color="Maroon">ep2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E11:113_1_1_1_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">ez</font>,<font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E12:113_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T27" target="_self">Th27</a>, <a class="ref" href="jordan2c.html#T28" target="_self">Th28</a>, <a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/>
<a NAME="E13:113_1_1_1_1_1"/><b>then </b>
not <font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:113_1_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K12">/</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2">(<span class="default"><a href="jordan2c.html#K5" target="_self">1.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E15:113_1_1_1_1_1"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G1</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T26" target="_self">Th26</a>, <a class="ref" href="jordan2c.html#T29" target="_self">Th29</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><b>hence </b><a NAME="E4:113_1_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="pre_topc.html#D13">PRE_TOPC:def 13</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="113_1_1_2"><b>suppose </b></a><a NAME="E1:113_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><a NAME="E2:113_1_1_2"/><b>then </b>
not <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T23" target="_self">Th23</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:113_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:113_1_1_2"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><b>hence </b><a NAME="E5:113_1_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:113"/><b>then </b><i><font color="Green">E130</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
;<br/>
<a NAME="E5:113"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T24" target="_self">Th24</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E6:113"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D3">TOPS_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:113"/>
<font color="Maroon">P</font> is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a></i>;<br/>


</div>
