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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">IS</font> = <font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> ;<br/>
<b>set </b><font color="Maroon">pB</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">P</font> =  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span>;<br/>
<a NAME="E1:41"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:41_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:41_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>
<a NAME="E4:41_1"/>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:41_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:41_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:41_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E5:41_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:41_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E7:41_1_1"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:41_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E3:41_1_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:41_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">R</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E6:41_1_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>and </b><br/><a NAME="E7:41_1_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<i><font color="Green">E66</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> 
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i/>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_2_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:41_1_1_1_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:41_1_1_1_2_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:41_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel18.html#E3:41_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_2_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1_2_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:41_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:41_1"/><b>then </b>
<font color="Maroon">x</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:41_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E4:41"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:41_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:41_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E5:41_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T12">ZFMISC_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a>  <b>by </b><i/>;<br/>
<a NAME="E7:41_2"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/>
<a NAME="E8:41_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:41_2"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E10:41_2"/><i><font color="Green">E77</font></i>: 
 <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <a href="waybel18.html#K4" target="_self">.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E11:41_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">V</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E12:41_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:41"/><i><font color="Green">E78</font></i>: 
 <a href="waybel18.html#K2" target="_self">product_prebasis</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> is    <a href="cantor_1.html#M2">prebasis</a> of  <a href="waybel18.html#K3" target="_self">product</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:41"/><b>then </b><i><font color="Green">E79</font></i>: 
 <a href="waybel18.html#K2" target="_self">product_prebasis</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> =  <a href="waybel18.html#K2" target="_self">product_prebasis</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E9:41"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font> is    <a href="cantor_1.html#M1">Basis</a> of  <a href="waybel18.html#K3" target="_self">product</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">(<span class="default"><a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> as    <a href="cantor_1.html#M1">Basis</a> of  <a href="waybel18.html#K3" target="_self">product</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <b>by </b><i/>;<br/>
<a NAME="E11:41"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:41_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E3:41_3"/><i><font color="Green">E82</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:41_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y1</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:41_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E7:41_3"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">Y1</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E8:41_3"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/>
<a NAME="E9:41_3"/><i><font color="Green">E97</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Y2</font> = <font color="Maroon">Y1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E11:41_3"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E12:41_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E13:41_3"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">Y2</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E14:41_3"/><i><font color="Green">E102</font></i>: 
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font>
 
<b>by </b><i>, <a class="ref" href="setfam_1.html#T59">SETFAM_1:59</a></i>;<br/>
<a NAME="E15:41_3"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:41_3_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font>
 ;<br/>

<a NAME="E2:41_3_1"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:41_3_1"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> holds <br/><font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_3_1_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">S</font> being    <a href="pre_topc.html#L1">TopStruct</a> , <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:41_3_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E5:41_3_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:41_3_1_1"/><i><font color="Green">E108</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E7:41_3_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">Y'</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<font color="Maroon">V</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<b>consider </b><font color="Maroon">RR</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E9:41_3_1_1"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">RR</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E10:41_3_1_1"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RR</font>
 <b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E11:41_3_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">V'</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a>  <b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E13:41_3_1_1"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E14:41_3_1_1"/><b>then </b>
<font color="Maroon">V'</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> 
 
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E15:41_3_1_1"/><b>then </b><i><font color="Green">E113</font></i>: 
<font color="Maroon">V'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="enumset1.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span></span><a href="enumset1.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E16:41_3_1_1"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">V'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_3_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">V'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:41_3_1_1_1"/><i><font color="Green">E116</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E3:41_3_1_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E4:41_3_1_1_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i</font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:41_3_1_1_1"/><b>then </b>
<font color="Maroon">Y'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="card_3.html#T37">CARD_3:37</a></i>;<br/>
<a NAME="E6:41_3_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, , <a class="ref" href="mssubfam.html#T3">MSSUBFAM:3</a></i>;<br/>
<b>hence </b><a NAME="E7:41_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_3_1_1_2"/>
( <font color="Maroon">V'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">V'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="enumset1.html#D1">ENUMSET1:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_1_2_1"><b>suppose </b></a><a NAME="E1:41_3_1_1_2_1"/>
<font color="Maroon">V'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><a NAME="E2:41_3_1_1_2_1"/><b>then </b>
<font color="Maroon">Y'</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:41_3_1_1_2_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E4:41_3_1_1_2_1"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E5:41_3_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_3_1_1_2_2"><b>suppose </b></a><a NAME="E1:41_3_1_1_2_2"/>
<font color="Maroon">V'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 ;<br/><div class="add"><b>then </b><font color="Maroon">V'</font> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/><b>hence </b><a NAME="E3:41_3_1_1_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T37">FUNCT_7:37</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:41_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y1</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:41_3"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E17:41_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:41"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E13:41"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span>  is    <a href="cantor_1.html#M2">prebasis</a> of  <a href="waybel18.html#K3" target="_self">product</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>


</div>
