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





<b>reconsider </b><font color="Maroon">X</font> = <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</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/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:8_1"/><i><font color="Green">E26</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="hidden.html#R1">=</a> 0
 ;<br/><b>let </b><font color="Maroon">r</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="E2:8_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>assume </b><a NAME="E3:8_1"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E4:8_1"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:8_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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="E3:8_1_1"/><i><font color="Green">E39</font></i>: 
( <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#K6">dist</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</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">z</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="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:8_1_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:8_1_1"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:8_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/> <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> = 
 <a href="seq_4.html#K5">lower_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#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>
<br/>.= 
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>
;<br/><b>hence </b><a NAME="E7:8_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="weierstr.html#D8">WEIERSTR:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:8"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 holds <br/> ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> )
 ;<br/>

<i><font color="Green">E47</font></i>:  <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span> = 
 <a href="seq_4.html#K5">lower_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#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="weierstr.html#D5">WEIERSTR:def 5</a></i>
<br/>.= 
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="weierstr.html#D3">WEIERSTR:def 3</a></i>
;<br/>
<a NAME="E5:8"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:8_3"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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="E3:8_3"/><i><font color="Green">E49</font></i>: 
( <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#K6">dist</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</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">z</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="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E5:8_3"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<b>hence </b><a NAME="E6:8_3"/>
<font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:8"/>
for <font color="Olive">q</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st ( for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q</font> ) holds <br/>0 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</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="E1:8_4"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">z</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">q</font>
 ;<br/>

<b>assume </b><a NAME="E2:8_4"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:8_4"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font> )
 <b>by </b><i/>;<br/>
<a NAME="E5:8_4"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="weierstr.html#D6">WEIERSTR:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>;<br/>
<a NAME="E6:8_4"/>
<font color="Maroon">p</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">M</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:8_4"/><b>then </b>
<font color="Maroon">p</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#K6">dist</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:8_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E9:8_4"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:8"/><b>then </b>
 <a href="weierstr.html#K5">lower_bound</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="weierstr.html#K6">dist</a> <font color="Maroon">x</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="seq_4.html#T61">SEQ_4:61</a></i>;<br/>
<b>hence </b><a NAME="E8:8"/>
<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="weierstr.html#D8">WEIERSTR:def 8</a></i>;<br/>


</div>
