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

<b>let </b><font color="Maroon" title="c1">N</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/>

<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon" title="c1">N</font> is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>
 ;<br/>

<b>assume </b><a NAME="E2:16"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c1">N</font> is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">TM</font> =  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font>;<br/>
<a NAME="E3:16"/><i><font color="Green" title="E13">A3</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font>,<span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">Family_open_set</a> <font color="Maroon" title="c1">N</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="pcomps_1.html#D6" title="PCOMPS_1:def.6">PCOMPS_1:def 6</a></i>;<br/>
<b>assume </b><a NAME="E4:16"/><i><font color="Green" title="E14">A4</font></i>: 
 <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="tbsp_1.html#D1" target="_self" title="TBSP_1:def.1">TBSP_1:def 1</a><br/>

<b>assume </b><a NAME="E5:16"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>] <b>means </b> ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">r</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">G</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">N</font><b> such that </b><br/><a NAME="E7:16"/><i><font color="Green" title="E16">A6</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> holds <br/> ( <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">G</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">G</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <b>by </b><i><a class="txt" href="tbsp_1.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<a NAME="E9:16"/>
for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> holds <br/><font color="Olive" title="b1">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E17">A7</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">C</font> = <font color="Maroon" title="c6">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="tbsp_1.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<a NAME="E3:16_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <font color="Maroon" title="c7">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E7:16"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="tbsp_1.html#E1:16_1"><i><font color="Green" title="E17">A7</font></i></a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b>
<font color="Maroon" title="c7">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E2:16"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="tbsp_1.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="pcomps_1.html#T33" title="PCOMPS_1:th.33">PCOMPS_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1"/>
<font color="Maroon" title="c6">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:16"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
<font color="Maroon" title="c5">G</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D1" title="TOPS_2:def.1">TOPS_2:def 1</a></i>;<br/>
<a NAME="E11:16"/>
<font color="Maroon" title="c5">G</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2"/><i><font color="Green" title="E18">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> holds  <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="tbsp_1.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<a NAME="E2:16_2_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E1:16"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/>
<a NAME="E3:16_2_1"/><b>then </b><i><font color="Green" title="E19">A10</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E5:16"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="metric_1.html#T12" title="METRIC_1:th.12">METRIC_1:12</a></i>;<br/>
<a NAME="E4:16_2_1"/>
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E7:16"><i><font color="Green" title="E16">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:16_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">G</font>
 <b>by </b><i><a class="txt" href="tbsp_1.html#E3:16_2_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">G</font>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E1:16_2"><i><font color="Green" title="E18">A9</font></i></a>, <a class="ref" href="subset_1.html#T49" title="SUBSET_1:th.49">SUBSET_1:49</a></i>;<br/>
<b>hence </b><a NAME="E3:16_2"/>
<font color="Maroon" title="c5">G</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c6">H</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><b> such that </b><br/><a NAME="E13:16"/><i><font color="Green" title="E18">A11</font></i>: 
( <font color="Maroon" title="c6">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c6">H</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a>  <a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> &amp; <font color="Maroon" title="c6">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="tbsp_1.html#E4:16"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="tbsp_1.html#E10:16"><i><font color="Green" title="E17">A8</font></i></a>, <a class="ref" href="compts_1.html#D3" title="COMPTS_1:def.3">COMPTS_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">H</font> = <font color="Maroon" title="c6">H</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">N</font> <b>by </b><i><a class="txt" href="tbsp_1.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">H</font>
;<br/>

 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">H</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>
<b>by </b><i><a class="txt" href="tbsp_1.html#E13:16"><i><font color="Green" title="E18">A11</font></i></a>, <a class="ref" href="pre_topc.html#D8" title="PRE_TOPC:def.8">PRE_TOPC:def 8</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4" title="PCOMPS_1:func.4">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>

;<br/>
<b>hence </b><a NAME="E16:16"/>
( <font color="Maroon" title="c7">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">N</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">H</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">H</font> holds <br/> ex <font color="Olive" title="b2">w</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">N</font> st <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> <font color="Olive" title="b2">w</font>,<font color="Maroon" title="c3">r</font> ) )
 <b>by </b><i><a class="txt" href="tbsp_1.html#E3:16"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="tbsp_1.html#E7:16"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="tbsp_1.html#E13:16"><i><font color="Green" title="E18">A11</font></i></a></i>;<br/>


</div>
