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

<b>let </b><font color="Maroon">N</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>, <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">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">N</font>,<font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span>,<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="E1:8"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">f</font> &amp;  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font> is <a href="compts_1.html#V2">compact</a> &amp; <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 ;<br/>

<a NAME="E2:8"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">u1</font>, <font color="Olive">u2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st  <a href="metric_1.html#K4">dist</a> <font color="Olive">u1</font>,<font color="Olive">u2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">r</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:8_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:8_1"/><b>then </b><i><font color="Green">E36</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>set </b><font color="Maroon">G</font> = <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <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">N</font></span>)</span> :  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) ) </span>}</span> ;<br/>
<a NAME="E3:8_1"/><i><font color="Green">E37</font></i>: 
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">N</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E4:8_1"/>
<span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <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">N</font></span>)</span> :  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</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">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="E1:8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <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">N</font></span>)</span> :  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <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">N</font></span>)</span><b> such that </b><br/><a NAME="E3:8_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp;  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) ) )
 ;<br/>
<b>thus </b><a NAME="E4:8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">G1</font> = <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <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">N</font></span>)</span> :  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:8_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><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="E1:8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">w0</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<a NAME="E3:8_1_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w0</font>
 
<b>by </b><i>, <a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">w4</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w0</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> ;<br/>
<b>consider </b><font color="Maroon">s4</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:8_1_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">s4</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:8_1_2"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">u5</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w5</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w5</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u5</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w0</font>,<font color="Olive">u5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s4</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Maroon">w4</font>,<font color="Olive">w5</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, , </i>;<br/>
<a NAME="E8:8_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">w0</font>,<font color="Maroon">s4</font>
 
<b>by </b><i>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P2</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">w0</font>,<font color="Maroon">s4</font> as   <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">N</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E10:8_1_2"/>
for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w0</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w0</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s4</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:8_1_2"/><b>then </b>
 ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) )
 
;<br/>
<a NAME="E12:8_1_2"/><b>then </b>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">w0</font>,<font color="Maroon">s4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 
;<br/>
<b>hence </b><a NAME="E13:8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G1</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:8_1"/><b>then </b><i><font color="Green">E60</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E8:8_1"/>
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">N</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/>
<a NAME="E9:8_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G1</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E10:8_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">G1</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D8">PRE_TOPC:def 8</a></i>;<br/>
<a NAME="E11:8_1"/>
for <font color="Olive">P3</font> being  <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">N</font></span>)</span>  st <font color="Olive">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> holds <br/><font color="Olive">P3</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P3</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">N</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:8_1_3"/>
<font color="Maroon">P3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <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">N</font></span>)</span><b> such that </b><br/><a NAME="E3:8_1_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">P3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:8_1_3"/><i><font color="Green">E82</font></i>: 
 ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) )
 ;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:8_1_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">w</font>,<font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E7:8_1_3"/>
for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E8:8_1_3"/>
<font color="Maroon">P3</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="topmetr.html#T21">TOPMETR:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:8_1"/><b>then </b>
<font color="Maroon">G1</font> is <a href="tops_2.html#V1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">s1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E14:8_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">s1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E15:8_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">w1</font>, <font color="Olive">w2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st  <a href="metric_1.html#K4">dist</a> <font color="Olive">w1</font>,<font color="Olive">w2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> holds <br/> ex <font color="Olive">Ga</font> being  <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">N</font></span>)</span> st <br/>( <font color="Olive">w1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Ga</font> &amp; <font color="Olive">w2</font> <a href="hidden.html#R2">in</a> <font color="Olive">Ga</font> &amp; <font color="Olive">Ga</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E16:8_1"/>
for <font color="Olive">u1</font>, <font color="Olive">u2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st  <a href="metric_1.html#K4">dist</a> <font color="Olive">u1</font>,<font color="Olive">u2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">u2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>

<b>assume </b><a NAME="E1:8_1_4"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">u1</font>,<font color="Maroon">u2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Ga</font> being   <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">N</font></span>)</span><b> such that </b><br/><a NAME="E3:8_1_4"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ga</font> &amp; <font color="Maroon">u2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Ga</font> &amp; <font color="Maroon">Ga</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G1</font> )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <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">N</font></span>)</span><b> such that </b><br/><a NAME="E5:8_1_4"/><i><font color="Green">E90</font></i>: 
( <font color="Maroon">Ga</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> &amp;  ex <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ex <font color="Olive">s2</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">w</font>,<font color="Olive">s2</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s2</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>, <font color="Maroon">s2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:8_1_4"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">w</font>,<font color="Maroon">s2</font> &amp; ( for <font color="Olive">w1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><br/> for <font color="Olive">w2</font>, <font color="Olive">w3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">w2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> &amp; <font color="Olive">w3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w1</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">w</font>,<font color="Olive">w1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">w2</font>,<font color="Olive">w3</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 ) )
 <b>by </b><i/>;<br/>
<a NAME="E8:8_1_4"/><i><font color="Green">E94</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">w</font>,<font color="Maroon">u1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, , , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E9:8_1_4"/><i><font color="Green">E95</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">w</font>,<font color="Maroon">u2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s2</font>
 
<b>by </b><i>, , , <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E10:8_1_4"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/>
<a NAME="E11:8_1_4"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u1</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/>
<a NAME="E12:8_1_4"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">u2</font>
 
<b>by </b><i><a class="ref" href="cat_3.html#D1">CAT_3:def 1</a></i>;<br/>
<a NAME="E13:8_1_4"/><i><font color="Green">E100</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E14:8_1_4"/><i><font color="Green">E101</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E15:8_1_4"/><i><font color="Green">E102</font></i>: 
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u2</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> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E16:8_1_4"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">w</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<b>hence </b><a NAME="E17:8_1_4"/>
 <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:8_1"/>
 ex <font color="Olive">s</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> &amp; ( for <font color="Olive">u1</font>, <font color="Olive">u2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st  <a href="metric_1.html#K4">dist</a> <font color="Olive">u1</font>,<font color="Olive">u2</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">s</font> holds <br/> <a href="metric_1.html#K4">dist</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">u2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8"/>
<font color="Maroon">f</font> is <a href="uniform1.html#V1" target="_self">uniformly_continuous</a>
 <b>by </b><i/>;<br/>


</div>
