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

<b>let </b><font color="Maroon">T</font> be   <a href="t_0topsp.html#NM1">T_0-TopSpace</a>;<br/>

<b>take </b><font color="Maroon">M</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">J</font> = <font color="Maroon">M</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2">Relation</a> of the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span> =  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:48"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E4:48"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</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">M</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="48_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="E1:48_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">ch</font> =  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E5:48_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, , </i>;<br/>
<i><font color="Green">E35</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> = 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>
<br/>.= 
<font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
<br/>.= 
 <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">M</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/>
<a NAME="E7:48_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">M</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/><span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <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">M</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="48_1_2"><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="E1:48_1_2"/>
<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">M</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="E2:48_1_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:48_1_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon">R</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E5:48_1_2"/><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">M</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="E6:48_1_2"/><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">M</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="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<i><font color="Green">E66</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">M</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> = 
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="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, , <a class="txt" href="waybel18.html#E3:48_1"><i><font color="Green">E31</font></i></a>, <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/>
<a NAME="E8:48_1_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E9:48_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>thus </b><a NAME="E10:48_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <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">M</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#E4:48_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="waybel18.html#E5:48_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:48_1"/><b>then </b>
<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">M</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="txt" href="waybel18.html#E2:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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">M</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/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">M</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_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">PP</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">M</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">m</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">m</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> : verum </span>}</span>  as    <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">M</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="E7:48"/>
for <font color="Olive">A</font> being  <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">M</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PP</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <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">M</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>assume </b><a NAME="E1:48_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PP</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</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">M</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="subset_1.html#M1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:48_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">a</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">M</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="E5:48_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
;<br/>
<a NAME="E6:48_2"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E8:48_2"/><i><font color="Green">E78</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">M</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/>
<a NAME="E9:48_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="E11:48_2"/><i><font color="Green">E79</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E12:48_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_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="E1:48_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E2:48_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<a NAME="E3:48_2_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, </i>;<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="E5:48_2_1"/><i><font color="Green">E83</font></i>: 
 <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E6:48_2_1"/>
 <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">M</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>and </b><br/><a NAME="E7:48_2_1"/><i><font color="Green">E95</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</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">M</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> holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E8:48_2_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">M</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><a class="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/>
<a NAME="E9:48_2_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:48_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</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/>
<a NAME="E11:48_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_3.html#T42">FUNCT_3:42</a></i>;<br/>
<a NAME="E12:48_2_1"/><b>then </b>
 ex <font color="Olive">uu</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">uu</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">uu</font> &amp; <font color="Olive">uu</font> is <a href="pre_topc.html#V3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E13:48_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:48_2"/>
<font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_2"><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="E1:48_2_2"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i</font>
 ;<br/>

<b>set </b><font color="Maroon">W</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<a NAME="E2:48_2_2"/>
<font color="Maroon">i'</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E2:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:48_2_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:48_2_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">i'</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:48_2_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a></i>;<br/>
<i><font color="Green">E100</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> = 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>
<br/>.= 
<font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
<br/>.= 
 <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">M</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/>.= 
 <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">M</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><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>
;<br/>
<a NAME="E7:48_2_2"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</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">M</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> holds <br/><span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_2_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_2_2_2"/>
<font color="Maroon">w</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">M</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>
 ;<br/>

<a NAME="E2:48_2_2_2"/><b>then </b>
<font color="Maroon">w</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">M</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="E3:48_2_2_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:48_2_2_2"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:48_2_2_2"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_2_2_2_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> or <font color="Maroon">w</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="48_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:48_2_2_2_1_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:48_2_2_2_1_1"/><b>then </b><i><font color="Green">E105</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</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="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:48_2_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48"><i><font color="Green">E27</font></i></a>, , , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_2_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:48_2_2_2_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><a NAME="E2:48_2_2_2_1_2"/><b>then </b><i><font color="Green">E106</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">M</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">w</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_2_2_2_1_2"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</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">w</font>
 <b>and </b><br/><a NAME="E5:48_2_2_2_1_2"/><i><font color="Green">E108</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">M</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">w</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">E109</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">M</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">w</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="waybel18.html#T3" target="_self">Th3</a>, , <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/><a NAME="E7:48_2_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:48_2_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</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">M</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> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E8:48_2_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:48_2_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:48_2"/><b>then </b>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</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="E15:48_2"/>
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:48"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T36">YELLOW_9:36</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<a NAME="E9:48"/><i><font color="Green">E111</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E10:48"/><i><font color="Green">E112</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E11:48"/><i><font color="Green">E113</font></i>: 
 <a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:48_3"/>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:48_3"/><b>then </b><i><font color="Green">E114</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font>
 
;<br/>
<b>set </b><font color="Maroon">U1</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">U2</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E3:48_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:48_3_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E4:48_3_1"/><i><font color="Green">E116</font></i>: 
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) )
 <b>by </b><i><a class="ref" href="t_0topsp.html#D7">T_0TOPSP:def 7</a></i>;<br/>
<a NAME="E5:48_3_1"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E6:48_3_1"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E7:48_3_1"/><i><font color="Green">E119</font></i>: 
<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">T</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_3_1_1"/>
( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) or ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_1_1"><b>suppose </b></a><a NAME="E1:48_3_1_1_1"/><i><font color="Green">E120</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:48_3_1_1_1"/><i><font color="Green">E121</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_3_1_1_1_1"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E3:48_3_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:48_3_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:48_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>hence </b><a NAME="E6:48_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, , , , , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_1_2"><b>suppose </b></a><a NAME="E1:48_3_1_1_2"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">V</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:48_3_1_1_2"/><i><font color="Green">E123</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_3_1_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_3_1_1_2_1"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E3:48_3_1_1_2_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E4:48_3_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E5:48_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>hence </b><a NAME="E6:48_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, , , , , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div><b>end;</b></div>
<a NAME="E12:48"/><i><font color="Green">E124</font></i>: 
 <a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:48"/><i><font color="Green">E125</font></i>: 
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">V</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">M</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">V</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:48_4"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</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="E1:48_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:48_4_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:48_4_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E5:48_4_1"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>set </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/><a NAME="E6:48_4_1"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E7:48_4_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, , </i>;<br/><i><font color="Green">E133</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> = 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>
<br/>.= 
<font color="Maroon">M</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
<br/>.= 
 <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">M</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/>.= 
 <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">M</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">V</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="E9:48_4_1"/>
for <font color="Olive">W</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">W</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">M</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">V</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/><span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">W</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">M</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">V</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">W</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:48_4_1_2"/>
<font color="Maroon">W</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">M</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">V</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="E2:48_4_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
<font color="Maroon">W</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">M</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="E3:48_4_1_2"/><b>then </b><i><font color="Green">E135</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:48_4_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:48_4_1_2"/><b>then </b><i><font color="Green">E137</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_4_1_2_1"/>
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> or <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">V</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_2_1_1"><b>suppose </b></a><a NAME="E1:48_4_1_2_1_1"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font>
 ;<br/><div class="add"><a NAME="E2:48_4_1_2_1_1"/><b>then </b><i><font color="Green">E139</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">M</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">V</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">W</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="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:48_4_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>hence </b><a NAME="E4:48_4_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">W</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">M</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">V</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">W</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_2_1_2"><b>suppose </b></a><a NAME="E1:48_4_1_2_1_2"/>
<font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">V</font>
 ;<br/><div class="add"><a NAME="E2:48_4_1_2_1_2"/><b>then </b><i><font color="Green">E140</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">M</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">V</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">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">M</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">W</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_4_1_2_1_2"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M</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">W</font>
 <b>and </b><br/><a NAME="E5:48_4_1_2_1_2"/><i><font color="Green">E142</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">M</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">W</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">E143</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">M</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">W</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/><a NAME="E7:48_4_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:48_4_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">W</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">M</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">V</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">W</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:48_4_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<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">M</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">V</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="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E11:48_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:48_4_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:48_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">M</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">V</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<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="E3:48_4"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon">M</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">V</font>,<span class="p3"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3">/\</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E4:48_4"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:48_4"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:48_4"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E8:48_4"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">Q</font> = <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ;<br/>
<a NAME="E9:48_4"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, , </i>;<br/>
<a NAME="E10:48_4"/>
<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">M</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">V</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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<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="E12:48_4"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E13:48_4"/>
 <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">M</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">V</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="E14:48_4"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">W</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">W</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">M</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">V</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">W</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">M</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">V</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">W</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E15:48_4"/>
<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">T</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E16:48_4"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E17:48_4"/><b>then </b><i><font color="Green">E151</font></i>: 
<font color="Maroon">V</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">M</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/>
<a NAME="E18:48_4"/><b>then </b>
<font color="Maroon">V</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">M</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">V</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="E19:48_4"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">V</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">M</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">V</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">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E20:48_4"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</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="funct_7.html#T33">FUNCT_7:33</a></i>;<br/>
<a NAME="E21:48_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <span class="p2">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">V</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/>
<a NAME="E22:48_4"/><b>then </b>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">u</font> where <font color="Olive">u</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_3.html#T42">FUNCT_3:42</a></i>;<br/>
<a NAME="E23:48_4"/><b>then </b>
 ex <font color="Olive">u</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">u</font> &amp; <font color="Olive">u</font> is <a href="pre_topc.html#V3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E24:48_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:48"/>
for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:48_5"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:48_5"/><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 <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">V</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:48_5"/><i><font color="Green">E153</font></i>: 
 <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">M</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">W</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="hidden.html#R2">in</a> <font color="Maroon">PP</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">Q</font> =  <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">M</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">V</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> 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">M</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="E6:48_5"/><i><font color="Green">E154</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:48_5"/>
<font color="Maroon">PP</font> <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">M</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/>
<a NAME="E8:48_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" target="_self">Image</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E9:48_5"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K10">.:</a> <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/>
<b>hence </b><a NAME="E10:48_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="relset_1.html#K11">"</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, <a class="txt" href="waybel18.html#E2:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tops_2.html#T67">TOPS_2:67</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:48"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font></span>)</span> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E16:48"/>
 <a href="waybel18.html#K8" target="_self">corestr</a> <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , <a class="txt" href="waybel18.html#E2:48"><i><font color="Green">E27</font></i></a>, <a class="txt" href="waybel18.html#E3:48"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>


</div>
