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

<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">Q</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>, <font color="Maroon">R</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a> &amp; <font color="Maroon">Q</font> is <a href="compts_1.html#V6">compact</a> )
 <b>and </b><br/><a NAME="E2:38"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">R</font> is <a href="compts_1.html#V6">compact</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">DPR</font> = <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<i><font color="Green">E36</font></i>:  <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">DPR</font> = 
 <a href="seq_4.html#K4">upper_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">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>
<br/>.= 
 <a href="weierstr.html#K4">upper_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">P</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D4">WEIERSTR:def 4</a></i>
<br/>.= 
 <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">P</font>,<font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="weierstr.html#D10">WEIERSTR:def 10</a></i>
;<br/>
<a NAME="E5:38"/>
for <font color="Olive">w</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DPR</font> holds <br/><font color="Olive">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:38_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">DPR</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="E3:38_2"/>
<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">P</font></span>)</span>
 <b>and </b><br/><a NAME="E4:38_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>and </b><br/><a NAME="E5:38_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">w</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> <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>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="hausdorf.html#T5" target="_self">Th5</a>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E7:38_2"/>
for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Olive">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E1:38_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:38_2_1"/><i><font color="Green">E47</font></i>: 
<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> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">z</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:38_2_1"/>
<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> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="hausdorf.html#T4" target="_self">Th4</a>, <a class="txt" href="hausdorf.html#E4:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="hausdorf.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E4:38_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E5:38_2_1"/><b>then </b>
<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> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E6:38_2_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:38_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<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> <font color="Maroon">y</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:38_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<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> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/>
<a NAME="E10:38_2"/>
<span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">R</font>,<font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="hausdorf.html#T4" target="_self">Th4</a>, , <a class="ref" href="hausdorf.html#T5" target="_self">Th5</a>, <a class="ref" href="hausdorf.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E11:38_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">Q</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E12:38_2"/>
<font color="Maroon">w</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="hausdorf.html#E4:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:38"/>
 <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">P</font>,<font color="Maroon">R</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">Q</font>,<font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/>


</div>
