<?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">w1</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>, <font color="Maroon">w2</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="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> &amp; not  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> )
 ;<br/>

<b>set </b><font color="Maroon">M</font> =  <a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font>;<br/>
<b>reconsider </b><font color="Maroon">P0</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P</font> as   <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>reconsider </b><font color="Maroon">o</font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</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> ;<br/>
<b>reconsider </b><font color="Maroon">o2</font> =  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:53"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">P0</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i>, <a class="ref" href="sppol_1.html#T28">SPPOL_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Q</font> = <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span> ;<br/>
<a NAME="E8:53"/>
<font color="Maroon">Q</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T41">BORSUK_1:41</a></i>;<br/>
<b>then consider </b><font color="Maroon">x1</font> being   <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>, <font color="Maroon">x2</font> being   <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> such that </b><br/><a NAME="E10:53"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P0</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K9">min_dist_min</a> <font color="Maroon">P0</font>,<font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="ref" href="jordan2c.html#T1" target="_self">Th1</a>, <a class="ref" href="weierstr.html#T36">WEIERSTR:36</a></i>;<br/>
<a NAME="E11:53"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> iff <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:53_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>thus </b><a NAME="E4:53_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:53_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 ;<br/>

<a NAME="E3:53_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:53_1"/>
<font color="Maroon">o2</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="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:53_1"/><b>then </b>
<font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan2c.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:53"/><i><font color="Green">E63</font></i>: 
 <a href="weierstr.html#K3">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E13:53"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font></span>)</span></span><a href="seq_4.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:53"/>
 <a href="seq_4.html#K5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>;<br/>
<a NAME="E15:53"/><b>then </b>
 <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P0</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">o</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T3" target="_self">Th3</a>, <a class="ref" href="jordan2c.html#T4" target="_self">Th4</a>, <a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>;<br/>
<a NAME="E16:53"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="weierstr.html#D9">WEIERSTR:def 9</a></i>;<br/>
<a NAME="E17:53"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">w01</font> = <font color="Maroon">x1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E1:53"><i><font color="Green">E47</font></i></a></i>;<br/>
<i><font color="Green">E67</font></i>: <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w01</font></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"><font color="Maroon">w01</font> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i/>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
<b>by </b><i>, <a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>
;<br/>
<a NAME="E20:53"/><i><font color="Green">E130</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w01</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>
<a NAME="E21:53"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">w01</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="txt" href="jordan2c.html#E1:53"><i><font color="Green">E47</font></i></a>, <a class="ref" href="toprns_1.html#T25">TOPRNS_1:25</a></i>;<br/>
<b>hence </b><a NAME="E22:53"/>
 ex <font color="Olive">w0</font> being  <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> st <br/>( <font color="Olive">w0</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">w0</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">w0</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="txt" href="jordan2c.html#E1:53"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan2c.html#E5"><i><font color="Green">Lemma51</font></i></a>, , <a class="ref" href="jordan2c.html#T7" target="_self">Th7</a></i>;<br/>


</div>
