<?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>thus </b><a NAME="E1:30"/>
( <font color="Maroon">M</font> is <a href="topmetr.html#V2" target="_self">compact</a> implies for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>  st <font color="Olive">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  &amp; <font color="Olive">F</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> &amp; <font color="Olive">G</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V1">finite</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">M</font> is <a href="topmetr.html#V2" target="_self">compact</a>
 ;<br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a> 
 <b>and </b><br/><a NAME="E4:30_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">F</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font>
 ;<br/>

<b>set </b><font color="Maroon">TM</font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font>;<br/>
<b>reconsider </b><font color="Maroon">TF</font> = <font color="Maroon">F</font> 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">M</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:30_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E7:30_1"/><b>then </b>
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> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">TF</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:30_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">TF</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:30_1"/>
<font color="Maroon">TF</font> is <a href="tops_2.html#V1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</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">M</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a><br/>

<b>assume </b><a NAME="E1:30_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">TF</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<a NAME="E3:30_1_1"/>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">P'</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:30_1_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">TG</font> being   <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">M</font></span>)</span><b> such that </b><br/><a NAME="E11:30_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">TG</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">TF</font> &amp; <font color="Maroon">TG</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font> &amp; <font color="Maroon">TG</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, , <a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">TG</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<a NAME="E13:30_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">M</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">TG</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:30_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:30_1"/>
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">G</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i>, </i>;<br/>


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

<b>thus </b><a NAME="E2:30"/>
( ( for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>  st <font color="Olive">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  &amp; <font color="Olive">F</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> &amp; <font color="Olive">G</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V1">finite</a> ) ) implies <font color="Maroon">M</font> is <a href="topmetr.html#V2" target="_self">compact</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">F</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font>  st <font color="Olive">F</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a>  &amp; <font color="Olive">F</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font> holds <br/> ex <font color="Olive">G</font> being  <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font> st <br/>( <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Olive">F</font> &amp; <font color="Olive">G</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font> &amp; <font color="Olive">G</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<b>set </b><font color="Maroon">TM</font> =  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F</font> be   <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">M</font></span>)</span>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:30_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">F</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:30_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V1">open</a>
 ;<br/><b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> :  ex <font color="Olive">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">M</font></span>)</span> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> ) </span>}</span> ;<br/><a NAME="E3:30_2_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> :  ex <font color="Olive">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">M</font></span>)</span> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> ) </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">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:30_2_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> :  ex <font color="Olive">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">M</font></span>)</span> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:30_2_1_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> &amp;  ex <font color="Olive">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">M</font></span>)</span> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:30_2_1_1"/>
<font color="Maroon">a</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">M</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font></span>)</span> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>, <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> :  ex <font color="Olive">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">M</font></span>)</span> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font> ;<br/><a NAME="E5:30_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">Z</font> <a href="topmetr.html#NR2" target="_self">is_ball-family</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topmetr.html#D4" target="_self">TOPMETR:def 4</a><br/>

<b>assume </b><a NAME="E1:30_2_1_2"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:30_2_1_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</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">p</font>,<font color="Olive">r</font> &amp;  ex <font color="Olive">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">M</font></span>)</span> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:30_2_1_2"/>
 ex <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">p</font>,<font color="Olive">r</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:30_2_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:30_2_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font> ;<br/>
<a NAME="E3:30_2_1_3"/>
( 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> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> &amp; 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> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:30_2_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:30_2_1_3"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E7:30_2_1_3"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</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">M</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E9:30_2_1_3"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T1" target="_self">Th1</a>, <a class="ref" href="topmetr.html#T2" target="_self">Th2</a>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E11:30_2_1_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E13:30_2_1_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:30_2_1_3"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#T2" target="_self">Th2</a>, </i>;<br/>
<b>hence </b><a NAME="E15:30_2_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">Z</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="E7:30_2_1"/><b>then </b>
<font color="Maroon">Z</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E9:30_2_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E10:30_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">G</font> <a href="topmetr.html#R1" target="_self">is_a_cover_of</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E11:30_2_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as  non <a href="xboole_0.html#V1">empty</a> <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">M</font></span>)</span> <b>by </b><i>, <a class="ref" href="tops_2.html#T5">TOPS_2:5</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/><a NAME="E13:30_2_1"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">u</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:30_2_1_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:30_2_1_4"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 
<b>by </b><i/>;<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>, <font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:30_2_1_4"/><i><font color="Green">E107</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:30_2_1_4"/><i><font color="Green">E108</font></i>: 
 ex <font color="Olive">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">M</font></span>)</span> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> )
 ;<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">M</font></span>)</span><b> such that </b><br/><a NAME="E7:30_2_1_4"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>thus </b><a NAME="E8:30_2_1_4"/>
( <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F'</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">G</font>,<font color="Maroon">F'</font><b> such that </b><br/><a NAME="E15:30_2_1"/><i><font color="Green">E110</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i><b>reconsider </b><font color="Maroon">GG</font> =  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> 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">M</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T3">TOPS_2:3</a></i>;<br/><b>take </b><font color="Maroon">GG</font> = <font color="Maroon">GG</font>;<br/><b>thus </b><a NAME="E17:30_2_1"/>
<font color="Maroon">GG</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E18:30_2_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">M</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GG</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:30_2_1_5"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">a</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>
 ;<br/>

<a NAME="E2:30_2_1_5"/>
( 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> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="topmetr.html#T2" target="_self">Th2</a>, , </i>;<br/>
<b>then consider </b><font color="Maroon">P</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:30_2_1_5"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:30_2_1_5"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:30_2_1_5"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">GG</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">P</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E7:30_2_1_5"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">GG</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E19:30_2_1"/>
<font color="Maroon">GG</font> <a href="pre_topc.html#R1">is_a_cover_of</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/><a NAME="E20:30_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>hence </b><a NAME="E21:30_2_1"/>
<font color="Maroon">GG</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T26">FINSET_1:26</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:30_2"/><b>then </b>
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <font color="Maroon">M</font> is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#D3">COMPTS_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:30_2"/>
<font color="Maroon">M</font> is <a href="topmetr.html#V2" target="_self">compact</a>
 <b>by </b><i/>;<br/>


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


</div>
