<?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>;<br/>



<b>assume </b><a NAME="E1:41"/><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> )
 ;<br/>

<a NAME="E2:41"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="compts_1.html#T16">COMPTS_1:16</a></i>;<br/>
<a NAME="E3:41"/><i><font color="Green">E36</font></i>: 
(  <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="txt" href="hausdorf.html#E1:41"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:41"/>
 <a href="hausdorf.html#K1" target="_self">HausDist</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E5:41"/><b>then </b>
 <a href="square_1.html#K2">max</a> <span class="p1">(<span class="default"><a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font></span>)</span>,<span class="p1">(<span class="default"><a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E6:41"/><b>then </b>
(  <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="weierstr.html#K10">max_dist_min</a> <font color="Maroon">Q</font>,<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:41"/><b>then </b><i><font color="Green">E39</font></i>: 
(  <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">Q</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <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">Q</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="weierstr.html#D10">WEIERSTR:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon">X</font> being  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> such that </b><br/><a NAME="E9:41"/><i><font color="Green">E45</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="relset_1.html#K10">.:</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Y</font> being  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> such that </b><br/><a NAME="E11:41"/><i><font color="Green">E46</font></i>: 
( <span class="p1">(<span class="default"><a href="weierstr.html#K8">dist_min</a> <font color="Maroon">Q</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#NK2" target="_self">sup</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E12:41"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:41"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i>, <a class="txt" href="hausdorf.html#E7:41"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E14:41"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:41_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E3:41_1"/>
 <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">Q</font></span>)</span> <a href="hidden.html#R1">=</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> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:41_1"/><b>then </b>
<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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="hausdorf.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E5:41_1"/><b>then </b><i><font color="Green">E50</font></i>: 
<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">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="hausdorf.html#T7" target="_self">Th7</a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E6:41_1"/>
<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">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="hausdorf.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan1k.html#T9">JORDAN1K:9</a></i>;<br/>
<a NAME="E7:41_1"/><b>then </b>
<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">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="hausdorf.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E8:41_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E3:41"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E9:41_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</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="E15:41"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E17:41"/>
 <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> <a href="hidden.html#R1">=</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> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:41"/><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">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E7:41"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hausdorf.html#T8" target="_self">Th8</a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E19:41"/><b>then </b><i><font color="Green">E52</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">x</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="hausdorf.html#E7:41"><i><font color="Green">E39</font></i></a>, , <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>
<a NAME="E20:41"/>
<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">x</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="hausdorf.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan1k.html#T9">JORDAN1K:9</a></i>;<br/>
<a NAME="E21:41"/><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">x</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="hausdorf.html#T9" target="_self">Th9</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<a NAME="E22:41"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="hausdorf.html#E3:41"><i><font color="Green">E36</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E23:41"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>


</div>
