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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">B</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green" title="E4">A1</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span>
 ;<br/>

<a NAME="E2:4"/><i><font color="Green" title="E5">A2</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:4"/><b>then </b>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T95" title="ZFMISC_1:th.95">ZFMISC_1:95</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">P</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">B</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T99" title="ZFMISC_1:th.99">ZFMISC_1:99</a></i>;<br/>
<b>assume </b><a NAME="E5:4"/><i><font color="Green" title="E6">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">U</font>
 ;<br/>

<b>assume </b><a NAME="E6:4"/><i><font color="Green" title="E7">A4</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">U</font> &amp; <font color="Olive" title="b3">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/> ex <font color="Olive" title="b4">V</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">U</font> )
 ;<br/>

<b>assume </b><a NAME="E7:4"/><i><font color="Green" title="E8">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">U1</font>, <font color="Olive" title="b3">U2</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">U2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> holds <br/> ex <font color="Olive" title="b4">U</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b4">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Olive" title="b3">U2</font> )
 ;<br/>

<b>take </b>
<font color="Maroon" title="c3">P</font>
;<br/>

<b>thus </b><a NAME="E8:4"/>
<font color="Maroon" title="c3">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">T</font> be    <a href="pre_topc.html#L1" title="PRE_TOPC:struct.1">TopStruct</a> ;<br/>

<b>assume </b><a NAME="E9:4"/><i><font color="Green" title="E9">A6</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font> &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">P</font> )
 ;<br/>

<a NAME="E10:4"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c3">P</font> is <a href="topgen_3.html#V1" title="TOPGEN_3:attr.1">point-filtered</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">U1</font>, <font color="Maroon" title="c7">U2</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="topgen_3.html#D1" target="_self" title="TOPGEN_3:def.1">TOPGEN_3:def 1</a><br/>

<b>assume </b><a NAME="E1:4_1"/><i><font color="Green" title="E11">A8</font></i>: 
( <font color="Maroon" title="c6">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c7">U2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">U2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green" title="E12">A9</font></i>: 
( <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">U1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">y1</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:4_1"/><i><font color="Green" title="E13">A10</font></i>: 
( <font color="Maroon" title="c9">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">U2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">y2</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E1:4_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U1</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:4_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">V1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:4_1"/><i><font color="Green" title="E14">A11</font></i>: 
( <font color="Maroon" title="c10">V1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c10">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">U1</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="topgen_3.html#E3:4_1"><i><font color="Green" title="E12">A9</font></i></a></i>;<br/>
<a NAME="E9:4_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">U2</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E1:4_1"><i><font color="Green" title="E11">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">V2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:4_1"/><i><font color="Green" title="E15">A12</font></i>: 
( <font color="Maroon" title="c11">V2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c11">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">U2</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="topgen_3.html#E5:4_1"><i><font color="Green" title="E13">A10</font></i></a></i>;<br/>
<a NAME="E12:4_1"/><i><font color="Green" title="E16">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="c1">X</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E8:4_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">U</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:4_1"/><i><font color="Green" title="E17">A14</font></i>: 
( <font color="Maroon" title="c12">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> &amp; <font color="Maroon" title="c12">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">V1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">V2</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E7:4"><i><font color="Green" title="E8">A5</font></i></a>, <a class="txt" href="topgen_3.html#E8:4_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="topgen_3.html#E11:4_1"><i><font color="Green" title="E15">A12</font></i></a></i>;<br/>
<a NAME="E15:4_1"/>
<font color="Maroon" title="c12">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E12:4_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="topgen_3.html#E14:4_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">U</font> = <font color="Maroon" title="c12">U</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c13">U</font>
;<br/>

<b>thus </b><a NAME="E17:4_1"/>
<font color="Maroon" title="c13">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E12:4_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="topgen_3.html#E14:4_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>

<b>thus </b><a NAME="E18:4_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">U</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E5:4"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="topgen_3.html#E12:4_1"><i><font color="Green" title="E16">A13</font></i></a>, <a class="txt" href="topgen_3.html#E14:4_1"><i><font color="Green" title="E17">A14</font></i></a></i>;<br/>

<a NAME="E19:4_1"/>
<font color="Maroon" title="c10">V1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c11">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">U2</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E8:4_1"><i><font color="Green" title="E14">A11</font></i></a>, <a class="txt" href="topgen_3.html#E11:4_1"><i><font color="Green" title="E15">A12</font></i></a>, <a class="ref" href="xboole_1.html#T27" title="XBOOLE_1:th.27">XBOOLE_1:27</a></i>;<br/>
<b>hence </b><a NAME="E20:4_1"/>
<font color="Maroon" title="c13">U</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">U1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c7">U2</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E14:4_1"><i><font color="Green" title="E17">A14</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>now </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> ;<br/><b>consider </b><font color="Maroon" title="c6">U</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>;<br/><b>assume </b><a NAME="E2:4_2"/><i><font color="Green" title="E11">A15</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><a NAME="E3:4_2"/><b>then </b>
not <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="pboole.html#D16" title="PBOOLE:def.16">PBOOLE:def 16</a></i>;<br/><a NAME="E4:4_2"/><b>then </b>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">U</font> &amp; <font color="Maroon" title="c6">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E5:4"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="topgen_3.html#E2:4_2"><i><font color="Green" title="E11">A15</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/><b>hence </b><a NAME="E5:4_2"/>
 ex <font color="Olive" title="b1">U</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">U</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U</font> )
 ;<br/></div><b>end;</b></div>
<a NAME="E12:4"/><b>then </b>
<font color="Maroon" title="c3">P</font> is <a href="abian.html#V2" title="ABIAN:attr.2">covering</a>
 
<b>by </b><i><a class="ref" href="topgen_3.html#T1" target="_self" title="TOPGEN_3:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E13:4"/>
<font color="Maroon" title="c4">T</font> is   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>
 <b>by </b><i><a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="topgen_3.html#E10:4"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="topgen_3.html#T2" target="_self" title="TOPGEN_3:th.2">Th2</a></i>;<br/>

<b>let </b><font color="Maroon" title="c5">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>assume </b><a NAME="E14:4"/><i><font color="Green" title="E11">A16</font></i>: 
<font color="Maroon" title="c5">T'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">B'</font> = <font color="Maroon" title="c2">B</font> as   <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c5">T'</font> <b>by </b><i><a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a></i>;<br/>
<a NAME="E16:4"/>
<font color="Maroon" title="c6">B'</font> is    <a href="topgen_2.html#M1" title="TOPGEN_2:mode.1">Neighborhood_System</a> of <font color="Maroon" title="c5">T'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c5">T'</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="topgen_2.html#D3" title="TOPGEN_2:def.3">TOPGEN_2:def 3</a><br/>

<a NAME="E1:4_3"/><i><font color="Green" title="E12">A17</font></i>: 
<font color="Maroon" title="c6">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">B</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="topgen_3.html#E14:4"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">Bx</font> = <font color="Maroon" title="c6">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c5">T'</font> <b>by </b><i><a class="txt" href="topgen_3.html#E1:4"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="topgen_3.html#E14:4"><i><font color="Green" title="E11">A16</font></i></a></i>;<br/>
<a NAME="E3:4_3"/>
<font color="Maroon" title="c8">Bx</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c7">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:4_3_1"/>
( <font color="Maroon" title="c8">Bx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E1:4_3"><i><font color="Green" title="E12">A17</font></i></a>, <a class="ref" href="cantor_1.html#T1" title="CANTOR_1:th.1">CANTOR_1:1</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E2:4_3_1"/>
<font color="Maroon" title="c8">Bx</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">T'</font>
 <b>by </b><i><a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="topgen_3.html#E14:4"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2" title="YELLOW_8:def.2">YELLOW_8:def 2</a><br/>

<a NAME="E3:4_3_1"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Bx</font> holds <br/><font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">a</font>
 
<b>by </b><i><a class="txt" href="topgen_3.html#E5:4"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="topgen_3.html#E14:4"><i><font color="Green" title="E11">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:4_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="setfam_1.html#K8" title="SETFAM_1:func.8">Intersect</a> <font color="Maroon" title="c8">Bx</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58" title="SETFAM_1:th.58">SETFAM_1:58</a></i>;<br/>

<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="c5">T'</font>;<br/>

<b>assume </b><a NAME="E5:4_3_1"/>
<font color="Maroon" title="c9">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c5">T'</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a><br/>

<b>then consider </b><font color="Maroon" title="c10">Y</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c5">T'</font><b> such that </b><br/><a NAME="E7:4_3_1"/><i><font color="Green" title="E13">A18</font></i>: 
( <font color="Maroon" title="c10">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c9">A</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="c10">Y</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="topgen_3.html#E14:4"><i><font color="Green" title="E11">A16</font></i></a>, <a class="ref" href="cantor_1.html#D1" title="CANTOR_1:def.1">CANTOR_1:def 1</a></i>;<br/>
<b>assume </b><a NAME="E8:4_3_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:4_3_1"/><i><font color="Green" title="E14">A19</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c10">Y</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E7:4_3_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<a NAME="E11:4_3_1"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c11">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">b</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E7:4_3_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="topgen_3.html#E10:4_3_1"><i><font color="Green" title="E14">A19</font></i></a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">V</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E13:4_3_1"/><i><font color="Green" title="E15">A20</font></i>: 
( <font color="Maroon" title="c12">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">a</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E6:4"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="topgen_3.html#E10:4_3_1"><i><font color="Green" title="E14">A19</font></i></a></i>;<br/>
<a NAME="E14:4_3_1"/>
( <font color="Maroon" title="c11">a</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c3">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">P</font> &amp; <font color="Maroon" title="c12">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E2:4"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topgen_3.html#E9:4"><i><font color="Green" title="E9">A6</font></i></a>, <a class="txt" href="topgen_3.html#E14:4"><i><font color="Green" title="E11">A16</font></i></a>, <a class="txt" href="topgen_3.html#E7:4_3_1"><i><font color="Green" title="E13">A18</font></i></a>, <a class="txt" href="topgen_3.html#E10:4_3_1"><i><font color="Green" title="E14">A19</font></i></a>, <a class="txt" href="topgen_3.html#E13:4_3_1"><i><font color="Green" title="E15">A20</font></i></a>, <a class="ref" href="cantor_1.html#T1" title="CANTOR_1:th.1">CANTOR_1:1</a>, <a class="ref" href="card_5.html#T10" title="CARD_5:th.10">CARD_5:10</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<a NAME="E15:4_3_1"/><b>then </b>
( <font color="Maroon" title="c12">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A</font> &amp; <font color="Maroon" title="c12">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cantor_1.html#K1" title="CANTOR_1:func.1">UniCl</a> <font color="Maroon" title="c3">P</font> )
 
<b>by </b><i><a class="txt" href="topgen_3.html#E13:4_3_1"><i><font color="Green" title="E15">A20</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E16:4_3_1"/>
 ex <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c5">T'</font> st <br/>( <font color="Olive" title="b1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Bx</font> &amp; <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">A</font> )
 <b>by </b><i><a class="txt" href="topgen_3.html#E13:4_3_1"><i><font color="Green" title="E15">A20</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:4_3"/>
<font color="Maroon" title="c6">B'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> is    <a href="yellow_8.html#M1" title="YELLOW_8:mode.1">Basis</a> of <font color="Maroon" title="c7">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:4"/>
<font color="Maroon" title="c2">B</font> is    <a href="topgen_2.html#M1" title="TOPGEN_2:mode.1">Neighborhood_System</a> of <font color="Maroon" title="c5">T'</font>
 ;<br/>


</div>
