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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:46"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="compts_1.html#V6">compact</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6">compact</a> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:46"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E3:46"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:46"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E5:46"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E6:46"/><b>then </b>
<span class="p1">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:46"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:46"/><i><font color="Green">E34</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:46"><i><font color="Green">E30</font></i></a>, <a class="txt" href="weierstr.html#E2:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="weierstr.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E10:46"/><i><font color="Green">E35</font></i>: 
 <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E8:46"><i><font color="Green">E33</font></i></a>, <a class="ref" href="weierstr.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E11:46"/>
 <a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:46"><i><font color="Green">E30</font></i></a>, <a class="txt" href="weierstr.html#E2:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="weierstr.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E12:46"/><b>then </b>
<span class="p1">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:46"><i><font color="Green">E30</font></i></a>, <a class="ref" href="weierstr.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E13:46"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="weierstr.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E14:46"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E2:46"><i><font color="Green">E31</font></i></a>, <a class="txt" href="weierstr.html#E3:46"><i><font color="Green">E32</font></i></a>, <a class="txt" href="weierstr.html#E8:46"><i><font color="Green">E33</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E15:46"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<a NAME="E1:46_1"/><i><font color="Green">E38</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="weierstr.html#E13:46"><i><font color="Green">E36</font></i></a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:46_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="weierstr.html#E14:46"><i><font color="Green">E37</font></i></a>, <a class="txt" href="weierstr.html#E1:46_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:46"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K7" target="_self">dist_max</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E9:46"><i><font color="Green">E34</font></i></a>, <a class="txt" href="weierstr.html#E10:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E17:46"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:46"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E19:46"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E20:46"/><b>then </b>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E22:46"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E23:46"/><i><font color="Green">E41</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="weierstr.html#K4" target="_self">upper_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:46"><i><font color="Green">E30</font></i></a>, <a class="txt" href="weierstr.html#E2:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="weierstr.html#T25" target="_self">Th25</a></i>;<br/>
<a NAME="E24:46"/><i><font color="Green">E42</font></i>: 
 <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6" target="_self">dist</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E22:46"><i><font color="Green">E40</font></i></a>, <a class="ref" href="weierstr.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E25:46"/>
 <a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:46"><i><font color="Green">E30</font></i></a>, <a class="txt" href="weierstr.html#E2:46"><i><font color="Green">E31</font></i></a>, <a class="ref" href="weierstr.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E26:46"/><b>then </b>
<span class="p1">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="txt" href="weierstr.html#E1:46"><i><font color="Green">E30</font></i></a>, <a class="ref" href="weierstr.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E27:46"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_4.html#V3">bounded</a>
 
<b>by </b><i><a class="ref" href="weierstr.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E28:46"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="weierstr.html#K3" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="weierstr.html#E2:46"><i><font color="Green">E31</font></i></a>, <a class="txt" href="weierstr.html#E17:46"><i><font color="Green">E39</font></i></a>, <a class="txt" href="weierstr.html#E22:46"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E29:46"/>
 <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<a NAME="E1:46_2"/><i><font color="Green">E45</font></i>: 
 <a href="weierstr.html#K3" target="_self">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="txt" href="weierstr.html#E27:46"><i><font color="Green">E43</font></i></a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/>
<b>thus </b><a NAME="E2:46_2"/>
 <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="weierstr.html#E28:46"><i><font color="Green">E44</font></i></a>, <a class="txt" href="weierstr.html#E1:46_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E30:46"/><b>then </b>
 <a href="weierstr.html#K5" target="_self">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K8" target="_self">dist_min</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="weierstr.html#E23:46"><i><font color="Green">E41</font></i></a>, <a class="txt" href="weierstr.html#E24:46"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E31:46"/>
(  <a href="weierstr.html#K9" target="_self">min_dist_min</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="weierstr.html#K12" target="_self">max_dist_max</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="weierstr.html#E16:46"><i><font color="Green">E38</font></i></a></i>;<br/>


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