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

<span class="kw">let </span><font color="Maroon" title="c1">N</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp; <font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  &amp;  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  implies <font color="Maroon" title="c1">N</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:16"/><span class="lab"><font color="Green" title="E7">A1</font></span>: 
<font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  or  not  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  or <font color="Maroon" title="c1">N</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">TM</font> =  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font>;<br/>
<span class="kw">assume </span><a NAME="E2:16"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
<font color="Maroon" title="c1">N</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a>  or <font color="Maroon" title="c1">N</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:16"/><span class="lab"><font color="Green" title="E9">A3</font></span>: 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font> is  <a href="compts_1.html#V1" title="COMPTS_1:attr.1">compact</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">N</font> is  <a href="tbsp_1.html#V1" title="TBSP_1:attr.1">totally_bounded</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tbsp_1.html#D1" target="_self" onmouseover="rs('tbsp_1/D1')" onmouseout="rh()">TBSP_1:def 1</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">G</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="Olive" title="b1">G</font> &amp; (  for <font color="Olive" title="b2">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="b2">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/> ex <font color="Olive" title="b3">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="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c3">r</font>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:16"/><span class="lab"><font color="Green" title="E10">A4</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">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> st <br/>( <font color="Olive" title="b1">G</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="Olive" title="b1">G</font> &amp; (  for <font color="Olive" title="b2">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="b2">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/> ex <font color="Olive" title="b3">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="b2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c3">r</font>) ) )</span><br/>

<span class="kw">defpred </span><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>] <span class="kw">means </span> 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#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">r</font>);<br/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:16"/><span class="lab"><font color="Green" title="E11">A5</font></span>: 
 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="tarski.html#R2" title="TARSKI: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>] )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="subset_1.html#S3" onmouseover="rs('subset_1/S3')" onmouseout="rh()">SUBSET_1:sch 3</a>();<br/></span>
<a NAME="E7:16"/><span class="lab"><font color="Green" title="E12">A6</font></span>: 
 <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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#K2" title="PCOMPS_1:func.2">Family_open_set</a> <font color="Maroon" title="c1">N</font></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pcomps_1.html#D5" onmouseover="rs('pcomps_1/D5')" onmouseout="rh()">PCOMPS_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> ;<br/>
<a NAME="E9:16"/>
 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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span> holds  <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">G</font></span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<a NAME="E2:16_1"/>
 <a href="metric_1.html#K1" title="METRIC_1:func.1">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>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:16"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="ref" href="metric_1.html#T1" onmouseover="rs('metric_1/T1')" onmouseout="rh()">METRIC_1:1</a></span>;<br/>
<a NAME="E3:16_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A7</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:16"><span class="lab"><font color="Green" title="E10">A4</font></span></a>, <a class="ref" href="metric_1.html#T11" onmouseover="rs('metric_1/T11')" onmouseout="rh()">METRIC_1:11</a></span>;<br/>
<a NAME="E4:16_1"/>
 <a href="metric_1.html#K8" title="METRIC_1:func.8">Ball</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c3">r</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E11">A5</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_1"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c5">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16_1"><span class="lab"><font color="Green" title="E13">A7</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:16"/><span class="kw">then </span>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T28" onmouseover="rs('subset_1/T28')" onmouseout="rh()">SUBSET_1:28</a></span>;<br/>
<a NAME="E11:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A8</font></span>: 
<font color="Maroon" title="c5">G</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T45" onmouseover="rs('setfam_1/T45')" onmouseout="rh()">SETFAM_1:45</a></span>;<br/>
<a NAME="E12:16"/>
 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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>  st <font color="Olive" title="b1">C</font> <a href="tarski.html#R2" title="TARSKI: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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">G</font> implies <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:16_2"/><span class="lab"><font color="Green" title="E14">A9</font></span>: 
<font color="Maroon" title="c6">C</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<a NAME="E3:16_2"/>
 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#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E1:16_2"><span class="lab"><font color="Green" title="E14">A9</font></span></a></span>;<br/>
<a NAME="E4:16_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">C</font> <a href="tarski.html#R2" title="TARSKI: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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="ref" href="pcomps_1.html#T29" onmouseover="rs('pcomps_1/T29')" onmouseout="rh()">PCOMPS_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:16_2"/>
<font color="Maroon" title="c6">C</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" onmouseover="rs('pre_topc/D2')" onmouseout="rh()">PRE_TOPC:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:16"/><span class="kw">then </span>
<font color="Maroon" title="c5">G</font> is  <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#D1" onmouseover="rs('tops_2/D1')" onmouseout="rh()">TOPS_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:16"/><span class="lab"><font color="Green" title="E14">A10</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E16:16"/><span class="lab"><font color="Green" title="E15">A11</font></span>: 
<font color="Maroon" title="c6">H</font> is   <a href="struct_0.html#NM13" title="STRUCT_0:NM.13">Cover</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:16"/><span class="lab"><font color="Green" title="E16">A12</font></span>: 
<font color="Maroon" title="c6">H</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:16"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E13">A8</font></span></a>, <a class="ref" href="compts_1.html#D1" onmouseover="rs('compts_1/D1')" onmouseout="rh()">COMPTS_1:def 1</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E12">A6</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">H</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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="tarski.html#R2" title="TARSKI: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#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c3">r</font>) ) )</span><br/>

<a NAME="E19:16"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <font color="Maroon" title="c1">N</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E15">A11</font></span></a>, <a class="ref" href="setfam_1.html#T45" onmouseover="rs('setfam_1/T45')" onmouseout="rh()">SETFAM_1:45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:16"/>
( <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="tarski.html#R2" title="TARSKI: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#K8" title="METRIC_1:func.8">Ball</a> (<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c3">r</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E12">A6</font></span></a>, <a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E11">A5</font></span></a>, <a class="txt" href="#E15:16"><span class="lab"><font color="Green" title="E14">A10</font></span></a>, <a class="txt" href="#E17:16"><span class="lab"><font color="Green" title="E16">A12</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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