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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<b>consider </b><font color="Maroon">B</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E2:19"/><i><font color="Green">E29</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="waybel23.html#T75">WAYBEL23:75</a></i>;<br/>
<b>set </b><font color="Maroon">B1</font> = <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span> ;<br/>
<a NAME="E3:19"/><i><font color="Green">E31</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span> 
 ;<br/>

<a NAME="E2:19_1"/><b>then </b>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) )
 
;<br/>
<b>hence </b><a NAME="E3:19_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span> 
 ;<br/><a NAME="E2:19_2"/><b>then </b>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) )
 ;<br/><b>hence </b><a NAME="E3:19_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>] )
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:19"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span>  &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>and </b><br/><a NAME="E7:19"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1">WELLORD2:sch 1</a>();<br/></i>
<b>set </b><font color="Maroon">G</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>;<br/>
<a NAME="E8:19"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">G</font> =  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> as  <a href="tops_2.html#V1">open</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="tops_2.html#T18">TOPS_2:18</a></i>;<br/>
<b>take </b>
<font color="Maroon">G</font>
;<br/>

<b>thus </b><a NAME="E10:19"/>
( <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T95">ZFMISC_1:95</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E11:19"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i>, <a class="ref" href="tops_2.html#D1">TOPS_2:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">W0</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:19_3"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">W0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W0</font> &amp; <font color="Maroon">W0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<a NAME="E8:19_3"/>
<font color="Maroon">W0</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:19_3"/><b>then </b>
( <font color="Maroon">W0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">W0</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">W0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E10:19_3"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">W0</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">W0</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:19_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">G</font>
 ;<br/>


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

<a NAME="E12:19"/>
(  <a href="card_1.html#K1">Card</a> <font color="Maroon">G</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span>  &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Olive">W</font> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp;  ex <font color="Olive">U</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U</font> ) ) </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T27">CARD_1:27</a>, <a class="ref" href="card_1.html#T28">CARD_1:28</a></i>;<br/>
<b>hence </b><a NAME="E13:19"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">G</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="waybel23.html#K2">weight</a> <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div>
