<?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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a> implies for <font color="Olive" title="b1">Bn</font> being  <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font>  st  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Olive" title="b1">Bn</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> holds <br/>for <font color="Olive" title="b2">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">p</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">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> holds <br/> ex <font color="Olive" title="b4">O</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="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b4">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp; <font color="Olive" title="b4">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Olive" title="b1">Bn</font> ) )</span><br/>

<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c1">T</font> is <a href="pre_topc.html#V9" title="PRE_TOPC:attr.9">regular</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Bn</font> being  <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font>  st  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Olive" title="b1">Bn</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> holds <br/>for <font color="Olive" title="b2">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">p</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">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font> holds <br/> ex <font color="Olive" title="b4">O</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="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b4">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U</font> &amp; <font color="Olive" title="b4">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Olive" title="b1">Bn</font> )</span><br/>

<b>let </b><font color="Maroon" title="c2">Bn</font> be   <a href="nagata_1.html#NM1" title="NAGATA_1:NM.1">FamilySequence</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font> implies for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</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">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b3">O</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> ) )</span><br/>

<b>assume </b><a NAME="E2:39"/><i><font color="Green" title="E22">A2</font></i>: 
 <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> is    <a href="cantor_1.html#M1" title="CANTOR_1:mode.1">Basis</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</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">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b3">O</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )</span><br/>

<a NAME="E3:39"/>
for <font color="Olive" title="b1">U</font> being <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</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">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b3">O</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">U</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</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="Maroon" title="c3">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> holds <br/> ex <font color="Olive" title="b2">O</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="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b2">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )</span><br/><b>let </b><font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> implies  ex <font color="Olive" title="b1">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> ) )</span><br/>



<b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c3">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1_1"/>
( <font color="Maroon" title="c3">U</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> or <font color="Maroon" title="c3">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_1"><b>suppose </b></a><a NAME="E1:39_1_1_1_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c3">U</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">O</font>, <font color="Olive" title="b2">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b2">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )</span><br/><div class="add"><b>consider </b><font color="Maroon" title="c5">O</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="E3:39_1_1_1_1"/><i><font color="Green" title="E25">A5</font></i>: 
( <font color="Maroon" title="c5">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">O</font> &amp; <font color="Maroon" title="c5">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> )
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E1:39_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></i>;<br/><b>take </b><font color="Maroon" title="c6">O</font> = <font color="Maroon" title="c5">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )</span><br/><a NAME="E4:39_1_1_1_1"/>
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c6">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Maroon" title="c6">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )
 <b>by </b><i><a class="txt" href="#E1:39_1_1_1_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="#E3:39_1_1_1_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/><b>hence </b><a NAME="E5:39_1_1_1_1"/>
 ex <font color="Olive" title="b1">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1_2"><b>suppose </b></a><a NAME="E1:39_1_1_1_2"/>
<font color="Maroon" title="c3">U</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">O</font>, <font color="Olive" title="b2">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b2">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b2">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )</span><br/><div class="add"><a NAME="E2:39_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c3">U</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</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="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:39_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c3">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c3">U</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="#E1:39_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xboole_1.html#T105" title="XBOOLE_1:th.105">XBOOLE_1:105</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">W</font>, <font color="Maroon" title="c6">V</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="E5:39_1_1_1_2"/><i><font color="Green" title="E24">A6</font></i>: 
( <font color="Maroon" title="c5">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c6">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font> &amp; <font color="Maroon" title="c3">U</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> &amp; <font color="Maroon" title="c5">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">V</font> )
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="compts_1.html#D5" title="COMPTS_1:def.5">COMPTS_1:def 5</a></i>;<br/><a NAME="E6:39_1_1_1_2"/><i><font color="Green" title="E25">A7</font></i>: 
( <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Maroon" title="c6">V</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="#E5:39_1_1_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="subset_1.html#T36" title="SUBSET_1:th.36">SUBSET_1:36</a>, <a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">O</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="E8:39_1_1_1_2"/><i><font color="Green" title="E26">A8</font></i>: 
( <font color="Maroon" title="c7">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">O</font> &amp; <font color="Maroon" title="c7">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">W</font> )
 <b>by </b><i><a class="txt" href="#E2:39"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E5:39_1_1_1_2"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="yellow_9.html#T31" title="YELLOW_9:th.31">YELLOW_9:31</a></i>;<br/><b>take </b><font color="Maroon" title="c8">O</font> = <font color="Maroon" title="c7">O</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )</span><br/><a NAME="E9:39_1_1_1_2"/>
<font color="Maroon" title="c8">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E6:39_1_1_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="#E8:39_1_1_1_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E10:39_1_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">O</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">V</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#T49" title="PRE_TOPC:th.49">PRE_TOPC:49</a></i>;<br/><a NAME="E11:39_1_1_1_2"/><b>then </b>
 <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c8">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="#E6:39_1_1_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="pre_topc.html#T52" title="PRE_TOPC:th.52">PRE_TOPC:52</a></i>;<br/><b>hence </b><a NAME="E12:39_1_1_1_2"/>
 ex <font color="Olive" title="b1">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )
 <b>by </b><i><a class="txt" href="#E6:39_1_1_1_2"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="#E8:39_1_1_1_2"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_1"/>
 ex <font color="Olive" title="b1">O</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="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b1">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U</font> &amp; <font color="Olive" title="b1">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
for <font color="Olive" title="b1">U</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">p</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">U</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> holds <br/> ex <font color="Olive" title="b3">O</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">O</font> &amp;  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Olive" title="b3">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U</font> &amp; <font color="Olive" title="b3">O</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="nagata_1.html#K2" title="NAGATA_1:func.2">Union</a> <font color="Maroon" title="c2">Bn</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
