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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">w1</font>, <font color="Maroon" title="c3">w2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:52"/><i><font color="Green" title="E44">A1</font></i>: 
( <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">w1</font>,<font color="Maroon" title="c3">w2</font> &amp; not  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">w1</font>,<font color="Maroon" title="c3">w2</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">M</font> =  <a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">P0</font> = <font color="Maroon" title="c4">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">P1</font> = <font color="Maroon" title="c4">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">o</font> =  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">o2</font> =  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:52"/><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c6">P0</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:52"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="sppol_1.html#T28" title="SPPOL_1:th.28">SPPOL_1:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">Q</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> ;<br/>
<a NAME="E8:52"/>
<font color="Maroon" title="c10">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T41" title="BORSUK_1:th.41">BORSUK_1:41</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">x1</font>, <font color="Maroon" title="c12">x2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E10:52"/><i><font color="Green" title="E46">A3</font></i>: 
( <font color="Maroon" title="c11">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P0</font> &amp; <font color="Maroon" title="c12">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q</font> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K9" title="WEIERSTR:func.9">min_dist_min</a> <font color="Maroon" title="c6">P0</font>,<font color="Maroon" title="c10">Q</font> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:52"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="jordan2c.html#E6:52"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="weierstr.html#T36" title="WEIERSTR:th.36">WEIERSTR:36</a></i>;<br/>
<a NAME="E11:52"/><i><font color="Green" title="E47">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:52_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c14">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:52_1_1"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> &amp; <font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q</font> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>thus </b><a NAME="E4:52_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E3:52_1_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:52_1"/><i><font color="Green" title="E48">A6</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font>
 ;<br/>

<a NAME="E3:52_1"/><i><font color="Green" title="E49">A7</font></i>: 
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Q</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:52_1"/>
<font color="Maroon" title="c9">o2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E5:52_1"/><b>then </b>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:52_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font>
 <b>by </b><i><a class="txt" href="jordan2c.html#E2:52_1"><i><font color="Green" title="E48">A6</font></i></a>, <a class="txt" href="jordan2c.html#E3:52_1"><i><font color="Green" title="E49">A7</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:52"/><i><font color="Green" title="E48">A8</font></i>: 
 <a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font>
 
<b>by </b><i><a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>;<br/>
<a NAME="E13:52"/><i><font color="Green" title="E49">A9</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font></span>)</span></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E11:52"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:52"/>
 <a href="seq_4.html#K5" title="SEQ_4:func.5">lower_bound</a> <span class="p1">(<span class="default"><a href="weierstr.html#K3" title="WEIERSTR:func.3">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="weierstr.html#K5" title="WEIERSTR:func.5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="weierstr.html#D5" title="WEIERSTR:def.5">WEIERSTR:def 5</a></i>;<br/>
<a NAME="E15:52"/><b>then </b>
 <a href="weierstr.html#K5" title="WEIERSTR:func.5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c10">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c6">P0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">o</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E12:52"><i><font color="Green" title="E48">A8</font></i></a>, <a class="txt" href="jordan2c.html#E13:52"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="seq_4.html#T22" title="SEQ_4:th.22">SEQ_4:22</a></i>;<br/>
<a NAME="E16:52"/><b>then </b><i><font color="Green" title="E50">A10</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E10:52"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="weierstr.html#D9" title="WEIERSTR:def.9">WEIERSTR:def 9</a></i>;<br/>
<a NAME="E17:52"/><i><font color="Green" title="E51">A11</font></i>: 
<font color="Maroon" title="c12">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="jordan2c.html#E10:52"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">w01</font> = <font color="Maroon" title="c11">x1</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="txt" href="jordan2c.html#E10:52"><i><font color="Green" title="E46">A3</font></i></a></i>;<br/>
<i><font color="Green" title="E52">A12</font></i>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c13">w01</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">w01</font> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="jordan2c.html#T13" target="_self" title="JORDAN2C:th.13">Th13</a></i>
<br/>.= 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font>
<b>by </b><i><a class="txt" href="jordan2c.html#E17:52"><i><font color="Green" title="E51">A11</font></i></a>, <a class="ref" href="jgraph_1.html#T45" title="JGRAPH_1:th.45">JGRAPH_1:45</a></i>
;<br/>
<a NAME="E20:52"/><i><font color="Green" title="E53">A13</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c13">w01</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E21:52"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Maroon" title="c13">w01</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="jordan2c.html#E1:52"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="jordan2c.html#E10:52"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="toprns_1.html#T25" title="TOPRNS_1:th.25">TOPRNS_1:25</a></i>;<br/>
<b>hence </b><a NAME="E22:52"/>
 ex <font color="Olive" title="b1">w0</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> st <br/>( <font color="Olive" title="b1">w0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c2">w1</font>,<font color="Maroon" title="c3">w2</font> &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">w0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><font color="Olive" title="b1">w0</font></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan2c.html#E1:52"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="jordan2c.html#E10:52"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="jordan2c.html#E16:52"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="jordan2c.html#E19:52"><i><font color="Green" title="E52">A12</font></i></a>, <a class="txt" href="jordan2c.html#E20:52"><i><font color="Green" title="E53">A13</font></i></a></i>;<br/>


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