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

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font>, <font color="Maroon" title="c4">R</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <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> <font color="Maroon" title="c1">M</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c2">P</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> &amp; <font color="Maroon" title="c3">Q</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> )
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c4">R</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">DPR</font> = <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<i><font color="Green" title="E33">A3</font></i>:  <a href="pscomp_1.html#NK2" title="PSCOMP_1:NK.2">sup</a> <font color="Maroon" title="c5">DPR</font> = 
 <a href="seq_4.html#K4" title="SEQ_4:func.4">upper_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="c4">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D3" title="WEIERSTR:def.3">WEIERSTR:def 3</a></i>
<br/>.= 
 <a href="weierstr.html#K4" title="WEIERSTR:func.4">upper_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="c4">R</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">P</font></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D4" title="WEIERSTR:def.4">WEIERSTR:def 4</a></i>
<br/>.= 
 <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c2">P</font>
<b>by </b><i><a class="ref" href="weierstr.html#D10" title="WEIERSTR:def.10">WEIERSTR:def 10</a></i>
;<br/>
<a NAME="E5:39"/>
for <font color="Olive" title="b1">w</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">DPR</font> holds <br/><font color="Olive" title="b1">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">w</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">DPR</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:39_2"/>
<font color="Maroon" title="c7">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="c4">R</font></span>)</span>
 <b>and </b><br/><a NAME="E4:39_2"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 <b>and </b><br/><a NAME="E5:39_2"/><i><font color="Green" title="E35">A5</font></i>: 
<font color="Maroon" title="c6">w</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">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">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>reconsider </b><font color="Maroon" title="c8">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="hausdorf.html#E4:39_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="topmetr.html#T16" title="TOPMETR:th.16">TOPMETR:16</a></i>;<br/>
<a NAME="E7:39_2"/>
for <font color="Olive" title="b1">z</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y</font>,<font color="Olive" title="b1">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>;<br/>

<b>assume </b><a NAME="E1:39_2_1"/><i><font color="Green" title="E36">A6</font></i>: 
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 ;<br/>

<a NAME="E2:39_2_1"/><i><font color="Green" title="E37">A7</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hausdorf.html#T17" target="_self" title="HAUSDORF:th.17">Th17</a></i>;<br/>
<a NAME="E3:39_2_1"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E1:39"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="hausdorf.html#E2:39"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="hausdorf.html#E1:39_2_1"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="hausdorf.html#T34" target="_self" title="HAUSDORF:th.34">Th34</a></i>;<br/>
<a NAME="E4:39_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">z</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:39_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E2:39_2_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:39_2_1"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c9">z</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</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="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:39_2"/><b>then </b><i><font color="Green" title="E36">A8</font></i>: 
<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="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c3">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font>
 
<b>by </b><i><a class="ref" href="hausdorf.html#T16" target="_self" title="HAUSDORF:th.16">Th16</a></i>;<br/>
<a NAME="E9:39_2"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" title="WEIERSTR:func.8">dist_min</a> <font color="Maroon" title="c3">Q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E1:39"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="hausdorf.html#E4:39_2"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="hausdorf.html#T34" target="_self" title="HAUSDORF:th.34">Th34</a></i>;<br/>
<a NAME="E10:39_2"/><b>then </b>
<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="c4">R</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E8:39_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E11:39_2"/>
<font color="Maroon" title="c6">w</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="hausdorf.html#E5:39_2"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="xreal_1.html#T22" title="XREAL_1:th.22">XREAL_1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
 <a href="weierstr.html#K10" title="WEIERSTR:func.10">max_dist_min</a> <font color="Maroon" title="c4">R</font>,<font color="Maroon" title="c2">P</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" title="HAUSDORF:func.1">HausDist</a> <font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c4">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="hausdorf.html#E4:39"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="seq_4.html#T62" title="SEQ_4:th.62">SEQ_4:62</a></i>;<br/>


</div>
