<?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>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</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><b>that </b><br/><a NAME="E1:26"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="pcomps_1.html#V2" title="PCOMPS_1:attr.2">paracompact</a>
 <b>and </b><br/><a NAME="E2:26"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:26"/>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E5:26"/><i><font color="Green" title="E22">A3</font></i>: 
for <font color="Olive" title="b1">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="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> holds <br/> ex <font color="Olive" title="b2">V</font>, <font color="Olive" title="b3">W</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="Olive" title="b2">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">V</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">W</font> &amp; <font color="Olive" title="b2">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b3">W</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">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 <br/>( <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">W</font> ) );<br/>
<b>consider </b><font color="Maroon" title="c4">GX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E7:26"/><i><font color="Green" title="E23">A4</font></i>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">X</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>
<a NAME="E8:26"/><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c4">GX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1"/>
 <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>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">GX</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:26_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 ;<br/><a NAME="E2:26_1_1_1_1"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1_1_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 </span><b>by </b><i><a class="txt" href="pcomps_1.html#E2:26_1_1_1_1"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_1_1_1"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1_1_1_1"/>
 ex <font color="Olive" title="b1">X</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c6">V</font>, <font color="Maroon" title="c7">W</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="E2:26_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E27">A8</font></i>: 
<font color="Maroon" title="c6">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E3:26_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E28">A9</font></i>: 
<font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E4:26_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E29">A10</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font>
 <b>and </b><br/><a NAME="E5:26_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E30">A11</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c6">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c7">W</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E5:26"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="pcomps_1.html#E1:26_1_1_1_1_1_1_1"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c8">X</font> = <font color="Maroon" title="c7">W</font>;<br/>

<b>thus </b><a NAME="E6:26_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E5:26_1_1_1_1_1_1_1_1"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/>

<b>thus </b><a NAME="E7:26_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E7:26"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="pcomps_1.html#E1:26_1_1_1_1_1_1_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="pcomps_1.html#E2:26_1_1_1_1_1_1_1_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="txt" href="pcomps_1.html#E3:26_1_1_1_1_1_1_1_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="txt" href="pcomps_1.html#E4:26_1_1_1_1_1_1_1_1"><i><font color="Green" title="E29">A10</font></i></a>, <a class="txt" href="pcomps_1.html#E5:26_1_1_1_1_1_1_1_1"><i><font color="Green" title="E30">A11</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">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="c4">GX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_1_1_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><div class="add"><a NAME="E2:26_1_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">X</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#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>

<b>thus </b><a NAME="E1:26_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:26_1_1_1_1_1_1_2"><i><font color="Green" title="E26">A12</font></i></a></i>;<br/>

<b>thus </b><a NAME="E2:26_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E7:26"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:26_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">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="c4">GX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:26_1_1_1_1"/>
<font color="Maroon" title="c5">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="c4">GX</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">GX</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1_1"/>
 <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>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c4">GX</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon" title="c4">GX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</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>
<a NAME="E9:26"/>
<font color="Maroon" title="c4">GX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<a NAME="E1:26_2"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font> holds <br/><font color="Olive" title="b1">X</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="26_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">X</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="E1:26_2_1"/><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_2_1_1_1"/>
( <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">W</font> ) )
 </span><b>by </b><i><a class="txt" href="pcomps_1.html#E7:26"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="pcomps_1.html#E1:26_2_1"><i><font color="Green" title="E25">A13</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_2_1_1_1_1"/>
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_2_1_1_1_1"/>
<font color="Maroon" title="c5">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:26"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_2_1_1_1_2"/>
 ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">W</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_2_1_1_1_2"/>
<font color="Maroon" title="c5">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_2_1"/>
<font color="Maroon" title="c5">X</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_2"/>
<font color="Maroon" title="c4">GX</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/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">FX</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E11:26"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c5">FX</font> is <a href="tops_2.html#V1" title="TOPS_2:attr.1">open</a>
 <b>and </b><br/><a NAME="E12:26"/><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c5">FX</font> <a href="pre_topc.html#R1" title="PRE_TOPC:pred.1">is_a_cover_of</a> <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E13:26"/><i><font color="Green" title="E27">A16</font></i>: 
<font color="Maroon" title="c5">FX</font> <a href="setfam_1.html#R1" title="SETFAM_1:pred.1">is_finer_than</a> <font color="Maroon" title="c4">GX</font>
 <b>and </b><br/><a NAME="E14:26"/><i><font color="Green" title="E28">A17</font></i>: 
<font color="Maroon" title="c5">FX</font> is <a href="pcomps_1.html#V1" title="PCOMPS_1:attr.1">locally_finite</a>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E1:26"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="pcomps_1.html#E8:26"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="pcomps_1.html#D4" target="_self" title="PCOMPS_1:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">HX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">W</font> where <font color="Olive" title="b1">W</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">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">FX</font> &amp; <font color="Olive" title="b1">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font> ) </span>}</span> ;<br/>
<a NAME="E15:26"/><i><font color="Green" title="E29">A18</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">W</font> where <font color="Olive" title="b1">W</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">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">FX</font> &amp; <font color="Olive" title="b1">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">FX</font>
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T11" target="_self" title="PCOMPS_1:th.11">Th11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">HX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">W</font> where <font color="Olive" title="b1">W</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">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">FX</font> &amp; <font color="Olive" title="b1">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</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> <b>by </b><i><a class="ref" href="pcomps_1.html#T11" target="_self" title="PCOMPS_1:th.11">Th11</a>, <a class="ref" href="tops_2.html#T3" title="TOPS_2:th.3">TOPS_2:3</a></i>;<br/>
<a NAME="E17:26"/><i><font color="Green" title="E30">A19</font></i>: 
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">HX</font> holds <br/><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">X</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">X</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="E1:26_3"/>
<font color="Maroon" title="c8">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">HX</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">W</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="E3:26_3"/><i><font color="Green" title="E31">A20</font></i>: 
<font color="Maroon" title="c9">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">X</font>
 <b>and </b><br/><a NAME="E4:26_3"/><i><font color="Green" title="E32">A21</font></i>: 
<font color="Maroon" title="c9">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">FX</font>
 <b>and </b><br/><a NAME="E5:26_3"/><i><font color="Green" title="E33">A22</font></i>: 
<font color="Maroon" title="c9">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:26_3"/><i><font color="Green" title="E34">A23</font></i>: 
<font color="Maroon" title="c10">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">GX</font>
 <b>and </b><br/><a NAME="E8:26_3"/><i><font color="Green" title="E35">A24</font></i>: 
<font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Y</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E13:26"><i><font color="Green" title="E27">A16</font></i></a>, <a class="txt" href="pcomps_1.html#E3:26_3"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="pcomps_1.html#E4:26_3"><i><font color="Green" title="E32">A21</font></i></a>, <a class="ref" href="setfam_1.html#D2" title="SETFAM_1:def.2">SETFAM_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Y</font> = <font color="Maroon" title="c10">Y</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="pcomps_1.html#E7:26_3"><i><font color="Green" title="E34">A23</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_1_1"/>
( <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  or  ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">W</font> ) )
 </span><b>by </b><i><a class="txt" href="pcomps_1.html#E7:26"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="pcomps_1.html#E7:26_3"><i><font color="Green" title="E34">A23</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_1_1_1"><b>suppose </b></a><a NAME="E1:26_3_1_1_1"/>
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:26_3_1_1_1"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</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="txt" href="pcomps_1.html#E3:26_3"><i><font color="Green" title="E31">A20</font></i></a>, <a class="txt" href="pcomps_1.html#E5:26_3"><i><font color="Green" title="E33">A22</font></i></a>, <a class="txt" href="pcomps_1.html#E8:26_3"><i><font color="Green" title="E35">A24</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_1_1_2"><b>suppose </b></a><a NAME="E1:26_3_1_1_2"/>
 ex <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">W</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">y</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">W</font> &amp; <font color="Olive" title="b1">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">W</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">V</font>, <font color="Maroon" title="c13">W</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c14">y</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="E3:26_3_1_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E4:26_3_1_1_2"/><i><font color="Green" title="E36">A25</font></i>: 
<font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">W</font>
 <b>and </b><br/><a NAME="E5:26_3_1_1_2"/><i><font color="Green" title="E37">A26</font></i>: 
<font color="Maroon" title="c12">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E6:26_3_1_1_2"/>
<font color="Maroon" title="c13">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E7:26_3_1_1_2"/><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">V</font>
 <b>and </b><br/><a NAME="E8:26_3_1_1_2"/>
<font color="Maroon" title="c14">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">W</font>
 <b>and </b><br/><a NAME="E9:26_3_1_1_2"/><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c12">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c13">W</font>
 ;<br/><a NAME="E10:26_3_1_1_2"/>
<font color="Maroon" title="c12">V</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E8:26_3"><i><font color="Green" title="E35">A24</font></i></a>, <a class="txt" href="pcomps_1.html#E4:26_3_1_1_2"><i><font color="Green" title="E36">A25</font></i></a>, <a class="txt" href="pcomps_1.html#E9:26_3_1_1_2"><i><font color="Green" title="E39">A28</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/><a NAME="E11:26_3_1_1_2"/><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="c12">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E12:26_3_1_1_2"/><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="c12">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c8">X</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="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/><a NAME="E13:26_3_1_1_2"/><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="c12">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</font></span>)</span></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="txt" href="pcomps_1.html#E5:26_3_1_1_2"><i><font color="Green" title="E37">A26</font></i></a>, <a class="ref" href="tops_1.html#T41" title="TOPS_1:th.41">TOPS_1:41</a></i>;<br/><a NAME="E14:26_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">V</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</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="pcomps_1.html#T2" target="_self" title="PCOMPS_1:th.2">Th2</a></i>;<br/><a NAME="E15:26_3_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c12">V</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/><a NAME="E16:26_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E7:26_3_1_1_2"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></i>;<br/><b>hence </b><a NAME="E17:26_3_1_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</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="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:26_3"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">X</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> 
 ;<br/>


</div><b>end;</b></div>
<b>take </b><font color="Maroon" title="c8">Y</font> = <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p2">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">HX</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> ;<br/>

<b>take </b><font color="Maroon" title="c9">Z</font> =  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">HX</font>;<br/>

<a NAME="E18:26"/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">HX</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c7">HX</font></span>)</span>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E14:26"><i><font color="Green" title="E28">A17</font></i></a>, <a class="txt" href="pcomps_1.html#E15:26"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="pcomps_1.html#T12" target="_self" title="PCOMPS_1:th.12">Th12</a>, <a class="ref" href="pcomps_1.html#T23" target="_self" title="PCOMPS_1:th.23">Th23</a></i>;<br/>
<b>hence </b><a NAME="E19:26"/>
<font color="Maroon" title="c8">Y</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/>

<b>thus </b><a NAME="E20:26"/>
<font color="Maroon" title="c9">Z</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E11:26"><i><font color="Green" title="E25">A14</font></i></a>, <a class="txt" href="pcomps_1.html#E15:26"><i><font color="Green" title="E29">A18</font></i></a>, <a class="ref" href="tops_2.html#T18" title="TOPS_2:th.18">TOPS_2:18</a>, <a class="ref" href="tops_2.html#T26" title="TOPS_2:th.26">TOPS_2:26</a></i>;<br/>

<a NAME="E21:26"/>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">HX</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_4"/>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">HX</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_4"/><i><font color="Green" title="E31">A29</font></i>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>and </b><br/><a NAME="E4:26_4"/><i><font color="Green" title="E32">A30</font></i>: 
<font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">HX</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="pcomps_1.html#E3:26_4"><i><font color="Green" title="E31">A29</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_4_1"/>
<font color="Maroon" title="c11">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> <span class="p1">(<span class="default"><a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">HX</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_4_1"/><i><font color="Green" title="E33">A31</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">X</font>
 <b>and </b><br/><a NAME="E4:26_4_1"/><i><font color="Green" title="E34">A32</font></i>: 
<font color="Maroon" title="c12">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pcomps_1.html#K2" title="PCOMPS_1:func.2">clf</a> <font color="Maroon" title="c7">HX</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">X</font> = <font color="Maroon" title="c12">X</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="pcomps_1.html#E4:26_4_1"><i><font color="Green" title="E34">A32</font></i></a></i>;<br/><a NAME="E6:26_4_1"/>
 ex <font color="Olive" title="b1">W</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="c13">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">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">HX</font> )
 <b>by </b><i><a class="txt" href="pcomps_1.html#E4:26_4_1"><i><font color="Green" title="E34">A32</font></i></a>, <a class="ref" href="pcomps_1.html#D3" target="_self" title="PCOMPS_1:def.3">Def3</a></i>;<br/><a NAME="E7:26_4_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c13">X</font> <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="txt" href="pcomps_1.html#E17:26"><i><font color="Green" title="E30">A19</font></i></a></i>;<br/><a NAME="E8:26_4_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c13">X</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/><b>hence </b><a NAME="E9:26_4_1"/>
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:26_4_1"><i><font color="Green" title="E33">A31</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:26_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:26_4"><i><font color="Green" title="E31">A29</font></i></a>, <a class="txt" href="pcomps_1.html#E4:26_4"><i><font color="Green" title="E32">A30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E22:26"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>

<b>thus </b><a NAME="E23:26"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:26_5_1"/><i><font color="Green" title="E31">A33</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/><a NAME="E2:26_5_1"/>
 ex <font color="Olive" title="b1">W</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="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">W</font> &amp; <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">HX</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_5_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c11">W</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="E2:26_5_1_1"/><i><font color="Green" title="E32">A34</font></i>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W</font>
 <b>and </b><br/><a NAME="E3:26_5_1_1"/><i><font color="Green" title="E33">A35</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">FX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E12:26"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="pcomps_1.html#T5" target="_self" title="PCOMPS_1:th.5">Th5</a></i>;<br/>
<a NAME="E4:26_5_1_1"/><i><font color="Green" title="E34">A36</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="pcomps_1.html#E1:26_5_1"><i><font color="Green" title="E31">A33</font></i></a>, <a class="txt" href="pcomps_1.html#E2:26_5_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c11">W</font>
;<br/>

<b>thus </b><a NAME="E5:26_5_1_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">W</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E2:26_5_1_1"><i><font color="Green" title="E32">A34</font></i></a></i>;<br/>

<b>thus </b><a NAME="E6:26_5_1_1"/>
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">HX</font>
 <b>by </b><i><a class="txt" href="pcomps_1.html#E3:26_5_1_1"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="pcomps_1.html#E4:26_5_1_1"><i><font color="Green" title="E34">A36</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:26_5_1"/>
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:26_5"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>


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

<b>thus </b><a NAME="E24:26"/>
<font color="Maroon" title="c8">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Z</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_6"><b>proof </b></a><div class="add">

<a NAME="E1:26_6"/>
<font color="Maroon" title="c9">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">Y</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T20" target="_self" title="PCOMPS_1:th.20">Th20</a>, <a class="ref" href="setfam_1.html#T18" title="SETFAM_1:th.18">SETFAM_1:18</a></i>;<br/>
<b>hence </b><a NAME="E2:26_6"/>
<font color="Maroon" title="c8">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c9">Z</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>


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


</div>
