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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>set </b><font color="Maroon" title="c2">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E1:41"/><i><font color="Green" title="E27">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</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 <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 ;<br/><b>let </b><font color="Maroon" title="c3">F</font> be  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c4">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> </span>}</span> ;<br/><a NAME="E2:41_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> </span>}</span> 
 ;<br/>

<a NAME="E2:41_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 
;<br/>
<b>hence </b><a NAME="E3:41_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">G</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">A</font></span>)</span> where <font color="Olive" title="b1">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> </span>}</span>  as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E4:41_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c5">G</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">centered</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">A0</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">F</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">A0</font> = <font color="Maroon" title="c6">A0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="yellow19.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="waybel_7.html#T4" title="WAYBEL_7:th.4">WAYBEL_7:4</a></i>;<br/>
<a NAME="E3:41_1_2"/>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c7">A0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 
;<br/>
<b>hence </b><a NAME="E4:41_1_2"/>
<font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="compts_1.html#D2" title="COMPTS_1:def.2">COMPTS_1:def 2</a><br/>

<b>let </b><font color="Maroon" title="c8">H</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E5:41_1_2"/><i><font color="Green" title="E30">A4</font></i>: 
( <font color="Maroon" title="c8">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">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="c8">H</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c9">H1</font> = <font color="Maroon" title="c8">H</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E7:41_1_2"/>
<font color="Maroon" title="c9">H1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">H1</font>
 ;<br/>

<a NAME="E2:41_1_2_1"/><b>then </b>
<font color="Maroon" title="c10">x</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="yellow19.html#E5:41_1_2"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E4:41_1_2_1"/><i><font color="Green" title="E31">A5</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c11">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 ;<br/>
<a NAME="E5:41_1_2_1"/>
<font color="Maroon" title="c11">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c11">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48" title="PRE_TOPC:th.48">PRE_TOPC:48</a></i>;<br/>
<b>hence </b><a NAME="E6:41_1_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="yellow19.html#E4:41_1_2_1"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="waybel_7.html#T11" title="WAYBEL_7:th.11">WAYBEL_7:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:41_1_2"/><b>then </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">H1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="yellow19.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="waybel_7.html#T15" title="WAYBEL_7:th.15">WAYBEL_7:15</a></i>;<br/>
<a NAME="E9:41_1_2"/><b>then </b>
 <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c9">H1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="yellow19.html#T2" target="_self" title="YELLOW19:th.2">Th2</a></i>;<br/>
<b>hence </b><a NAME="E10:41_1_2"/>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c8">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E5:41_1_2"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="setfam_1.html#D10" title="SETFAM_1:def.10">SETFAM_1:def 10</a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G</font>;<br/><a NAME="E6:41_1"/>
<font color="Maroon" title="c5">G</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a><br/>

<b>assume </b><a NAME="E1:41_1_3"/>
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font>
 ;<br/>

<a NAME="E2:41_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 
;<br/>
<b>hence </b><a NAME="E3:41_1_3"/>
<font color="Maroon" title="c7">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:41_1"/><b>then </b><i><font color="Green" title="E30">A6</font></i>: 
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="yellow19.html#E4:41_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/><a NAME="E8:41_1"/><b>then </b>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c5">G</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>take </b><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font>;<br/><b>thus </b><a NAME="E10:41_1"/>
<font color="Maroon" title="c8">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">T</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5" title="WAYBEL_7:def.5">WAYBEL_7:def 5</a><br/>

<b>assume </b><a NAME="E1:41_1_4"/><i><font color="Green" title="E31">A7</font></i>: 
( <font color="Maroon" title="c9">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> )
 ;<br/>

<a NAME="E2:41_1_4"/><i><font color="Green" title="E32">A8</font></i>: 
<font color="Maroon" title="c3">F</font> is <a href="waybel_7.html#V2" title="WAYBEL_7:attr.2">prime</a>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T26" title="WAYBEL_7:th.26">WAYBEL_7:26</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c9">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_4_1"/>
<font color="Maroon" title="c9">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/><a NAME="E2:41_1_4_1"/><b>then </b>
(  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; <font color="Maroon" title="c9">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41_1_4"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="tops_1.html#T30" title="TOPS_1:th.30">TOPS_1:30</a></i>;<br/><a NAME="E3:41_1_4_1"/><b>then </b>
( <font color="Maroon" title="c9">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">G</font> &amp; not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41_1_4"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E4:41_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow19.html#E7:41_1"><i><font color="Green" title="E30">A6</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:41_1_4"/>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="yellow19.html#E2:41_1_4"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="waybel_7.html#T25" title="WAYBEL_7:th.25">WAYBEL_7:25</a></i>;<br/>


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

<b>assume </b><a NAME="E3:41"/><i><font color="Green" title="E28">A9</font></i>: 
for <font color="Olive" title="b1">F</font> being <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span>  ex <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <font color="Olive" title="b2">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Olive" title="b1">F</font>,<font color="Maroon" title="c1">T</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:41_2"/><i><font color="Green" title="E29">A10</font></i>: 
( <font color="Maroon" title="c3">F</font> is <a href="compts_1.html#V1" title="COMPTS_1:attr.1">centered</a> &amp; <font color="Maroon" title="c3">F</font> is <a href="tops_2.html#V2" title="TOPS_2:attr.2">closed</a> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c4">Y</font> = <font color="Maroon" title="c3">F</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_7.html#T4" title="WAYBEL_7:th.4">WAYBEL_7:4</a></i>;<br/><b>set </b><font color="Maroon" title="c5">L</font> =  <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/><b>set </b><font color="Maroon" title="c6">G</font> =  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">Y</font></span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_2_1"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c7">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="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span><b> such that </b><br/><a NAME="E3:41_2_1"/><i><font color="Green" title="E30">A11</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">Y</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D16" title="WAYBEL_0:def.16">WAYBEL_0:def 16</a></i>;<br/><a NAME="E4:41_2_1"/><i><font color="Green" title="E31">A12</font></i>: 
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_1.html#T18" title="YELLOW_1:th.18">YELLOW_1:18</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">Z</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c4">Y</font><b> such that </b><br/><a NAME="E6:41_2_1"/><i><font color="Green" title="E32">A13</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c8">Z</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> &amp;  <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c8">Z</font>, <a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E3:41_2_1"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">Z</font> = <font color="Maroon" title="c8">Z</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E8:41_2_1"/><i><font color="Green" title="E33">A14</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:41_2_1"><i><font color="Green" title="E30">A11</font></i></a>, <a class="ref" href="yellow_5.html#T22" title="YELLOW_5:th.22">YELLOW_5:22</a></i>;<br/><a NAME="E9:41_2_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="yellow_0.html#K4" title="YELLOW_0:func.4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_7.html#T5" title="WAYBEL_7:th.5">WAYBEL_7:5</a></i>;<br/><a NAME="E10:41_2_1"/><b>then </b><i><font color="Green" title="E34">A15</font></i>: 
<font color="Maroon" title="c9">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E6:41_2_1"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="yellow_0.html#D12" title="YELLOW_0:def.12">YELLOW_0:def 12</a></i>;<br/><a NAME="E11:41_2_1"/><b>then </b>
 <a href="setfam_1.html#K1" title="SETFAM_1:func.1">meet</a> <font color="Maroon" title="c9">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="compts_1.html#D2" title="COMPTS_1:def.2">COMPTS_1:def 2</a></i>;<br/><b>hence </b><a NAME="E12:41_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow19.html#E4:41_2_1"><i><font color="Green" title="E31">A12</font></i></a>, <a class="txt" href="yellow19.html#E6:41_2_1"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="yellow19.html#E8:41_2_1"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="yellow19.html#E10:41_2_1"><i><font color="Green" title="E34">A15</font></i></a>, <a class="ref" href="yellow_1.html#T20" title="YELLOW_1:th.20">YELLOW_1:20</a></i>;<br/></div><b>end;</b></div><a NAME="E4:41_2"/><b>then </b>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">Y</font></span>)</span> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 <b>by </b><i><a class="ref" href="waybel_7.html#T8" title="WAYBEL_7:th.8">WAYBEL_7:8</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">UF</font> being   <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span><b> such that </b><br/><a NAME="E6:41_2"/><i><font color="Green" title="E30">A16</font></i>: 
(  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">Y</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">UF</font> &amp; <font color="Maroon" title="c7">UF</font> is <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T30" title="WAYBEL_7:th.30">WAYBEL_7:30</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E8:41_2"/><i><font color="Green" title="E31">A17</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c7">UF</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="yellow19.html#E3:41"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="yellow19.html#E6:41_2"><i><font color="Green" title="E30">A16</font></i></a></i>;<br/><a NAME="E9:41_2"/><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="ref" href="compts_1.html#D2" title="COMPTS_1:def.2">COMPTS_1:def 2</a></i>;<br/><a NAME="E10:41_2"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K12" title="WAYBEL_0:func.12">fininfs</a> <font color="Maroon" title="c4">Y</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_0.html#T62" title="WAYBEL_0:th.62">WAYBEL_0:62</a></i>;<br/><a NAME="E11:41_2"/><b>then </b><i><font color="Green" title="E33">A19</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">UF</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E6:41_2"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:41_2_2"/><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c10">B</font> = <font color="Maroon" title="c9">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><div><i><font color="Green" title="E35">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:41_2_2_1"/>
( <font color="Maroon" title="c11">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">C</font> )
 ;<br/><a NAME="E2:41_2_2_1"/><b>then </b><i><font color="Green" title="E36">A22</font></i>: 
( <font color="Maroon" title="c11">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">UF</font> &amp; <font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">UF</font> )
 <b>by </b><i><a class="txt" href="yellow19.html#E8:41_2"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="yellow19.html#E11:41_2"><i><font color="Green" title="E33">A19</font></i></a>, <a class="txt" href="yellow19.html#E1:41_2_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="waybel_7.html#D5" title="WAYBEL_7:def.5">WAYBEL_7:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">c</font> = <font color="Maroon" title="c11">C</font>, <font color="Maroon" title="c13">a</font> = <font color="Maroon" title="c9">A</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> ;<br/><a NAME="E4:41_2_2_1"/>
( <font color="Maroon" title="c13">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">UF</font> &amp; <font color="Maroon" title="c7">UF</font> is  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font></span>)</span> )
 <b>by </b><i><a class="txt" href="yellow19.html#E6:41_2"><i><font color="Green" title="E30">A16</font></i></a>, <a class="txt" href="yellow19.html#E2:41_2_2_1"><i><font color="Green" title="E36">A22</font></i></a>, <a class="ref" href="waybel_0.html#T41" title="WAYBEL_0:th.41">WAYBEL_0:41</a></i>;<br/><a NAME="E5:41_2_2_1"/><b>then </b>
<font color="Maroon" title="c13">a</font> <a href="lattice3.html#K12" title="LATTICE3:func.12">"/\"</a> <font color="Maroon" title="c12">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow19.html#T2" target="_self" title="YELLOW19:th.2">Th2</a></i>;<br/><a NAME="E6:41_2_2_1"/><b>then </b>
<font color="Maroon" title="c9">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="yellow_1.html#T17" title="YELLOW_1:th.17">YELLOW_1:17</a></i>;<br/><b>hence </b><a NAME="E7:41_2_2_1"/>
<font color="Maroon" title="c9">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c11">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div><a NAME="E4:41_2_2"/>
<font color="Maroon" title="c10">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:41_2"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="yellow19.html#E1:41_2_2"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="tops_2.html#D2" title="TOPS_2:def.2">TOPS_2:def 2</a></i>;<br/><a NAME="E5:41_2_2"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">B</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/><b>hence </b><a NAME="E6:41_2_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:41_2_2"><i><font color="Green" title="E35">A21</font></i></a>, <a class="ref" href="pre_topc.html#T54" title="PRE_TOPC:th.54">PRE_TOPC:54</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E13:41_2"/>
 <a href="setfam_1.html#K6" title="SETFAM_1:func.6">meet</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="yellow19.html#E9:41_2"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="setfam_1.html#D1" title="SETFAM_1:def.1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:41"/>
<font color="Maroon" title="c1">T</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 <b>by </b><i><a class="ref" href="compts_1.html#T13" title="COMPTS_1:th.13">COMPTS_1:13</a></i>;<br/>


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