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

<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="t_0topsp.html#NM1" title="T_0TOPSP:NM.1">T_0-TopSpace</a>;<br/>

<a NAME="E1:48"/><i><font color="Green" title="E22">A</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<b>take </b><font color="Maroon" title="c2">M</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>;<br/>

<b>set </b><font color="Maroon" title="c3">J</font> = <font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="relset_1.html#M2" title="RELSET_1:mode.2">Relation</a> of the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:48"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 <b>and </b><br/><a NAME="E4:48"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> holds <br/><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E5:48"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">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" title="c5">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E4:48_1"/><i><font color="Green" title="E26">A4</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">ch</font> =  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E5:48_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="waybel18.html#E4:48"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel18.html#E3:48_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="waybel18.html#E4:48_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<i><font color="Green" title="E28">A6</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> = 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E7:48_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b2">u</font> where <font color="Olive" title="b2">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b2">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:48_1_2"/><b>then </b><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E3:48_1_2"/><b>then </b><i><font color="Green" title="E30">A8</font></i>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">R</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E5:48_1_2"/><i><font color="Green" title="E31">A9</font></i>: 
<font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <b>and </b><br/><a NAME="E6:48_1_2"/><i><font color="Green" title="E32">A10</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">R</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<i><font color="Green" title="E33">A11</font></i>: <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
<b>by </b><i><a class="txt" href="waybel18.html#E2:48_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="txt" href="waybel18.html#E5:48_1_2"><i><font color="Green" title="E31">A9</font></i></a>, <a class="txt" href="waybel18.html#E6:48_1_2"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>
;<br/>
<a NAME="E8:48_1_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:48_1_2"><i><font color="Green" title="E30">A8</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E9:48_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:48_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E7:48_1_2"><i><font color="Green" title="E33">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:48_1"/><b>then </b>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E5:48_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="waybel18.html#E6:48_1"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:48_1"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#D3" target="_self" title="WAYBEL18:def.3">Def3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c4">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">T</font>,<span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <b>by </b><i><a class="txt" href="waybel18.html#E3:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">PP</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">m</font>,<span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> where <font color="Olive" title="b1">m</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">M</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of  <a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span> <b>by </b><i><a class="ref" href="waybel18.html#T14" target="_self" title="WAYBEL18:th.14">Th14</a></i>;<br/>
<a NAME="E8:48"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">PP</font> holds <br/><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c7">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:48_2"/><i><font color="Green" title="E25">A12</font></i>: 
<font color="Maroon" title="c7">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">PP</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c7">A</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="waybel18.html#D3" target="_self" title="WAYBEL18:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">M</font><b> such that </b><br/><a NAME="E4:48_2"/><i><font color="Green" title="E26">A13</font></i>: 
<font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:48_2"><i><font color="Green" title="E25">A12</font></i></a></i>;<br/>
<a NAME="E5:48_2"/><i><font color="Green" title="E27">A14</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
;<br/>
<a NAME="E6:48_2"/><b>then </b><i><font color="Green" title="E28">A15</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">i'</font> = <font color="Maroon" title="c9">i</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<a NAME="E8:48_2"/><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E5:48_2"><i><font color="Green" title="E27">A14</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E9:48_2"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T12" title="ZFMISC_1:th.12">ZFMISC_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">V</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a>  <b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>;<br/>
<a NAME="E11:48_2"/><i><font color="Green" title="E30">A17</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E8:48_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E12:48_2"/><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">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" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_2_1"/><i><font color="Green" title="E32">A19</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">A</font>
 ;<br/>

<a NAME="E2:48_2_1"/><b>then </b><i><font color="Green" title="E33">A20</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">a</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c13">W</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<a NAME="E3:48_2_1"/>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:48"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel18.html#E1:48_2_1"><i><font color="Green" title="E32">A19</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c14">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:48_2_1"/><i><font color="Green" title="E34">A21</font></i>: 
 <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font>
 <b>and </b><br/><a NAME="E6:48_2_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span>
 <b>and </b><br/><a NAME="E7:48_2_1"/><i><font color="Green" title="E35">A22</font></i>: 
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> holds <br/><font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:48_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="waybel18.html#E2:48_2_1"><i><font color="Green" title="E33">A20</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E8:48_2_1"/>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E8:48_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E9:48_2_1"/><b>then </b>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E7:48_2_1"><i><font color="Green" title="E35">A22</font></i></a></i>;<br/>
<a NAME="E10:48_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="waybel18.html#E11:48_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="txt" href="waybel18.html#E5:48_2_1"><i><font color="Green" title="E34">A21</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:48_2_1"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_3.html#T42" title="FUNCT_3:th.42">FUNCT_3:42</a></i>;<br/>
<a NAME="E12:48_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">uu</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">uu</font> &amp; <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">uu</font> &amp; <font color="Olive" title="b1">uu</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E13:48_2_1"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:48_2"/>
<font color="Maroon" title="c9">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">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" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:48_2_2"/><i><font color="Green" title="E32">A23</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">i</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c13">W</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<a NAME="E2:48_2_2"/>
<font color="Maroon" title="c10">i'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E6:48_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:48_2_2"/><b>then </b><i><font color="Green" title="E33">A24</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:48_2_2"><i><font color="Green" title="E32">A23</font></i></a></i>;<br/>
<a NAME="E4:48_2_2"/><i><font color="Green" title="E34">A25</font></i>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">i'</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:48_2_2"><i><font color="Green" title="E32">A23</font></i></a></i>;<br/>
<a NAME="E5:48_2_2"/><b>then </b><i><font color="Green" title="E35">A26</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:48"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<i><font color="Green" title="E36">A27</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> = 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>
;<br/>
<a NAME="E7:48_2_2"/>
for <font color="Olive" title="b1">w</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b2">u</font> where <font color="Olive" title="b2">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b2">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c14">w</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_2_2_2"/>
<font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span>
 ;<br/>

<a NAME="E2:48_2_2_2"/><b>then </b>
<font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E3:48_2_2_2"/><b>then </b><i><font color="Green" title="E37">A28</font></i>: 
<font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:48_2_2_2"/><b>then </b>
<font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:48_2_2_2"/><b>then </b><i><font color="Green" title="E38">A29</font></i>: 
<font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font> or <font color="Maroon" title="c14">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">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" title="E39">A30</font></i>: 
<font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font>
 ;<br/><div class="add"><a NAME="E2:48_2_2_2_1_1"/><b>then </b><i><font color="Green" title="E40">A31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="waybel18.html#E8:48_2"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="waybel18.html#E6:48_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="txt" href="waybel18.html#E3:48_2_2"><i><font color="Green" title="E33">A24</font></i></a>, <a class="txt" href="waybel18.html#E1:48_2_2_2_1_1"><i><font color="Green" title="E39">A30</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48_2_2_2_1_1"><i><font color="Green" title="E40">A31</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="c14">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">i</font>
 ;<br/><div class="add"><a NAME="E2:48_2_2_2_1_2"/><b>then </b><i><font color="Green" title="E39">A32</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">r</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_2_2_2_1_2"/><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font>
 <b>and </b><br/><a NAME="E5:48_2_2_2_1_2"/><i><font color="Green" title="E41">A34</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c15">r</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48_2_2_2"><i><font color="Green" title="E37">A28</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><i><font color="Green" title="E42">A35</font></i>: <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
<b>by </b><i><a class="txt" href="waybel18.html#E3:48_2_2_2"><i><font color="Green" title="E37">A28</font></i></a>, <a class="txt" href="waybel18.html#E4:48_2_2_2_1_2"><i><font color="Green" title="E40">A33</font></i></a>, <a class="txt" href="waybel18.html#E5:48_2_2_2_1_2"><i><font color="Green" title="E41">A34</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>
;<br/><a NAME="E7:48_2_2_2_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E5:48_2_2_2"><i><font color="Green" title="E38">A29</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c9">i</font>,<font color="Maroon" title="c11">V</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">w</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48_2_2_2_1_2"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="waybel18.html#E6:48_2_2_2_1_2"><i><font color="Green" title="E42">A35</font></i></a></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" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:48_2"><i><font color="Green" title="E26">A13</font></i></a>, <a class="txt" href="waybel18.html#E5:48_2_2"><i><font color="Green" title="E35">A26</font></i></a>, <a class="txt" href="waybel18.html#E6:48_2_2"><i><font color="Green" title="E36">A27</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:48_2_2"/>
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">A</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="waybel18.html#E4:48_2_2"><i><font color="Green" title="E34">A25</font></i></a>, <a class="ref" href="funct_1.html#D13" title="FUNCT_1:def.13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:48_2"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E12:48_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E15:48_2"/>
<font color="Maroon" title="c5">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E6:48_2"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:48"/><b>then </b><i><font color="Green" title="E25">A36</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T36" title="YELLOW_9:th.36">YELLOW_9:36</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
;<br/>

<a NAME="E10:48"/><i><font color="Green" title="E26">A37</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:48"/><i><font color="Green" title="E27">A38</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E12:48"/><i><font color="Green" title="E28">A39</font></i>: 
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">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" title="c7">x</font>, <font color="Maroon" title="c8">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D1" title="WAYBEL_1:def.1">WAYBEL_1:def 1</a><br/>

<b>assume </b><a NAME="E1:48_3"/><i><font color="Green" title="E29">A40</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c9">U1</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c10">U2</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E2:48_3"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:48_3_1"/><i><font color="Green" title="E30">A41</font></i>: 
<font color="Maroon" title="c11">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E4:48_3_1"/><i><font color="Green" title="E31">A42</font></i>: 
( ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> &amp; not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> ) or ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> ) )
 <b>by </b><i><a class="ref" href="t_0topsp.html#D7" title="T_0TOPSP:def.7">T_0TOPSP:def 7</a></i>;<br/>
<a NAME="E5:48_3_1"/><i><font color="Green" title="E32">A43</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:48"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E6:48_3_1"/><i><font color="Green" title="E33">A44</font></i>: 
<font color="Maroon" title="c5">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:48"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<a NAME="E7:48_3_1"/><i><font color="Green" title="E34">A45</font></i>: 
<font color="Maroon" title="c11">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:48_3_1"><i><font color="Green" title="E30">A41</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">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" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> &amp; not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> ) or ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> ) )
 </span><b>by </b><i><a class="txt" href="waybel18.html#E4:48_3_1"><i><font color="Green" title="E31">A42</font></i></a></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" title="E35">A46</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> &amp; not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c11">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E3:48_3_1_1_1"/><i><font color="Green" title="E36">A47</font></i>: 
not <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_3_1_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E3:48_3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel18.html#E1:48_3_1_1_1"><i><font color="Green" title="E35">A46</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:48_3_1_1_1"/>
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48_3_1"><i><font color="Green" title="E30">A41</font></i></a>, <a class="txt" href="waybel18.html#E1:48_3_1_1_1"><i><font color="Green" title="E35">A46</font></i></a></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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="waybel18.html#E7:48_3_1"><i><font color="Green" title="E34">A45</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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#E1:48_3"><i><font color="Green" title="E29">A40</font></i></a>, <a class="txt" href="waybel18.html#E5:48_3_1"><i><font color="Green" title="E32">A43</font></i></a>, <a class="txt" href="waybel18.html#E6:48_3_1"><i><font color="Green" title="E33">A44</font></i></a>, <a class="txt" href="waybel18.html#E7:48_3_1"><i><font color="Green" title="E34">A45</font></i></a>, <a class="txt" href="waybel18.html#E3:48_3_1_1_1"><i><font color="Green" title="E36">A47</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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" title="E35">A48</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> &amp; not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">V</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c12">v</font> = <font color="Maroon" title="c11">V</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/><a NAME="E3:48_3_1_1_2"/><i><font color="Green" title="E36">A49</font></i>: 
not <font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 ;<br/>

<a NAME="E2:48_3_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">v</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E3:48_3_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="waybel18.html#E1:48_3_1_1_2"><i><font color="Green" title="E35">A48</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:48_3_1_1_2"/>
<font color="Maroon" title="c12">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48_3_1"><i><font color="Green" title="E30">A41</font></i></a>, <a class="txt" href="waybel18.html#E1:48_3_1_1_2"><i><font color="Green" title="E35">A48</font></i></a></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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="waybel18.html#E7:48_3_1"><i><font color="Green" title="E34">A45</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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#E1:48_3"><i><font color="Green" title="E29">A40</font></i></a>, <a class="txt" href="waybel18.html#E5:48_3_1"><i><font color="Green" title="E32">A43</font></i></a>, <a class="txt" href="waybel18.html#E6:48_3_1"><i><font color="Green" title="E33">A44</font></i></a>, <a class="txt" href="waybel18.html#E7:48_3_1"><i><font color="Green" title="E34">A45</font></i></a>, <a class="txt" href="waybel18.html#E3:48_3_1_1_2"><i><font color="Green" title="E36">A49</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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="E13:48"/><i><font color="Green" title="E29">A50</font></i>: 
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E9:48"><i><font color="Green" title="E25">A36</font></i></a>, <a class="ref" href="waybel18.html#T11" target="_self" title="WAYBEL18:th.11">Th11</a></i>;<br/>
<a NAME="E14:48"/><i><font color="Green" title="E30">A51</font></i>: 
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Olive" title="b1">V</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">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" title="c7">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:48_4"/><i><font color="Green" title="E31">A52</font></i>: 
<font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:48_4_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">V</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:48_4_1"/><i><font color="Green" title="E32">A53</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E4:48_4_1"/><i><font color="Green" title="E33">A54</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">V</font>
 <b>and </b><br/><a NAME="E5:48_4_1"/><i><font color="Green" title="E34">A55</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/><b>set </b><font color="Maroon" title="c10">Q</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/><a NAME="E6:48_4_1"/><i><font color="Green" title="E35">A56</font></i>: 
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel18.html#E1:48_4"><i><font color="Green" title="E31">A52</font></i></a>, <a class="txt" href="waybel18.html#E4:48_4_1"><i><font color="Green" title="E33">A54</font></i></a></i>;<br/><a NAME="E7:48_4_1"/><i><font color="Green" title="E36">A57</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E4:48"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel18.html#E3:48_4_1"><i><font color="Green" title="E32">A53</font></i></a>, <a class="txt" href="waybel18.html#E5:48_4_1"><i><font color="Green" title="E34">A55</font></i></a></i>;<br/><i><font color="Green" title="E37">A58</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> = 
the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">FUNCT_3:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>
;<br/><a NAME="E9:48_4_1"/>
for <font color="Olive" title="b1">W</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b2">u</font> where <font color="Olive" title="b2">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">u</font> &amp; <font color="Olive" title="b2">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">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" title="c11">W</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:48_4_1_2"/>
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:48_4_1_2"/><b>then </b><i><font color="Green" title="E38">A59</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E3:48_4_1_2"/><b>then </b><i><font color="Green" title="E39">A60</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:48_4_1_2"/><b>then </b><i><font color="Green" title="E40">A61</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E5:48_4_1_2"/><b>then </b><i><font color="Green" title="E41">A62</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">V</font> or <font color="Maroon" title="c11">W</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">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" title="E42">A63</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">V</font>
 ;<br/><div class="add"><a NAME="E2:48_4_1_2_1_1"/><b>then </b><i><font color="Green" title="E43">A64</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48_4_1_2"><i><font color="Green" title="E38">A59</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="waybel18.html#E6:48_4_1"><i><font color="Green" title="E35">A56</font></i></a>, <a class="txt" href="waybel18.html#E4:48_4_1_2"><i><font color="Green" title="E40">A61</font></i></a>, <a class="txt" href="waybel18.html#E1:48_4_1_2_1_1"><i><font color="Green" title="E42">A63</font></i></a>, <a class="ref" href="funct_3.html#D3" title="FUNCT_3:def.3">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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48_4_1_2_1_1"><i><font color="Green" title="E43">A64</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">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" title="c11">W</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">V</font>
 ;<br/><div class="add"><a NAME="E2:48_4_1_2_1_2"/><b>then </b><i><font color="Green" title="E42">A65</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">r</font> being    <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> <b> such that </b><br/><a NAME="E4:48_4_1_2_1_2"/><i><font color="Green" title="E43">A66</font></i>: 
<font color="Maroon" title="c12">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font>
 <b>and </b><br/><a NAME="E5:48_4_1_2_1_2"/><i><font color="Green" title="E44">A67</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">r</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48_4_1_2"><i><font color="Green" title="E39">A60</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/><i><font color="Green" title="E45">A68</font></i>: <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> 
<b>by </b><i><a class="txt" href="waybel18.html#E3:48_4_1_2"><i><font color="Green" title="E39">A60</font></i></a>, <a class="txt" href="waybel18.html#E4:48_4_1_2_1_2"><i><font color="Green" title="E43">A66</font></i></a>, <a class="txt" href="waybel18.html#E5:48_4_1_2_1_2"><i><font color="Green" title="E44">A67</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="waybel18.html#D9" target="_self" title="WAYBEL18:def.9">Def9</a></i>
;<br/><a NAME="E7:48_4_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E5:48_4_1_2"><i><font color="Green" title="E41">A62</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">W</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E2:48_4_1_2_1_2"><i><font color="Green" title="E42">A65</font></i></a>, <a class="txt" href="waybel18.html#E6:48_4_1_2_1_2"><i><font color="Green" title="E45">A68</font></i></a></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" title="E38">A69</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E7:48_4_1"><i><font color="Green" title="E36">A57</font></i></a>, <a class="txt" href="waybel18.html#E8:48_4_1"><i><font color="Green" title="E37">A58</font></i></a>, <a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/><a NAME="E11:48_4_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E3:48_4_1"><i><font color="Green" title="E32">A53</font></i></a>, <a class="txt" href="waybel18.html#E5:48_4_1"><i><font color="Green" title="E34">A55</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E12:48_4_1"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/><b>hence </b><a NAME="E13:48_4_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E10:48_4_1"><i><font color="Green" title="E38">A69</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:48_4"/><i><font color="Green" title="E32">A70</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 ;<br/>

<a NAME="E4:48_4"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#T10" target="_self" title="WAYBEL18:th.10">Th10</a></i>;<br/>
<a NAME="E5:48_4"/><b>then </b>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:48_4"><i><font color="Green" title="E32">A70</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:48_4"/><i><font color="Green" title="E33">A71</font></i>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font>
 <b>and </b><br/><a NAME="E8:48_4"/><i><font color="Green" title="E34">A72</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">Q</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ;<br/>
<a NAME="E9:48_4"/><i><font color="Green" title="E35">A73</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_3.html#K5" title="FUNCT_3:func.5">chi</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:48"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel18.html#E7:48_4"><i><font color="Green" title="E33">A71</font></i></a>, <a class="txt" href="waybel18.html#E8:48_4"><i><font color="Green" title="E34">A72</font></i></a></i>;<br/>
<a NAME="E10:48_4"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:48_4"><i><font color="Green" title="E32">A70</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E12:48_4"/><i><font color="Green" title="E36">A74</font></i>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">g</font>
 <b>and </b><br/><a NAME="E13:48_4"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E14:48_4"/><i><font color="Green" title="E37">A75</font></i>: 
for <font color="Olive" title="b1">W</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> holds <br/><font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">W</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">W</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E15:48_4"/>
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:48_4"><i><font color="Green" title="E31">A52</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E16:48_4"/><b>then </b>
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E17:48_4"/><b>then </b><i><font color="Green" title="E38">A76</font></i>: 
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E18:48_4"/><b>then </b>
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T32" title="FUNCT_7:th.32">FUNCT_7:32</a></i>;<br/>
<a NAME="E19:48_4"/><b>then </b>
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">V</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E14:48_4"><i><font color="Green" title="E37">A75</font></i></a></i>;<br/>
<a NAME="E20:48_4"/><b>then </b>
<font color="Maroon" title="c11">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E17:48_4"><i><font color="Green" title="E38">A76</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">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" title="FUNCT_3:func.5">chi</a> <span class="p2">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> ,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="waybel18.html#E9:48_4"><i><font color="Green" title="E35">A73</font></i></a>, <a class="txt" href="waybel18.html#E12:48_4"><i><font color="Green" title="E36">A74</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E22:48_4"/><b>then </b>
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">u</font> where <font color="Olive" title="b1">u</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> : ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_3.html#T42" title="FUNCT_3:th.42">FUNCT_3:42</a></i>;<br/>
<a NAME="E23:48_4"/><b>then </b>
 ex <font color="Olive" title="b1">u</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">V</font> &amp; <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">u</font> &amp; <font color="Olive" title="b1">u</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
;<br/>
<b>hence </b><a NAME="E24:48_4"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">V</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E7:48_4"><i><font color="Green" title="E33">A71</font></i></a>, <a class="txt" href="waybel18.html#E8:48_4"><i><font color="Green" title="E34">A72</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:48"/>
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">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" title="c7">V</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>assume </b><a NAME="E1:48_5"/><i><font color="Green" title="E31">A77</font></i>: 
<font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:48_5"/><b>then </b>
<font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">W</font> = <font color="Maroon" title="c7">V</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">M</font> <b>by </b><i><a class="ref" href="yellow_1.html#T1" title="YELLOW_1:th.1">YELLOW_1:1</a></i>;<br/>
<a NAME="E4:48_5"/><i><font color="Green" title="E32">A78</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c8">W</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">PP</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">Q</font> =  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> <font color="Maroon" title="c7">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E6:48_5"/><i><font color="Green" title="E33">A79</font></i>: 
<span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Q</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E14:48"><i><font color="Green" title="E30">A51</font></i></a>, <a class="txt" href="waybel18.html#E1:48_5"><i><font color="Green" title="E31">A77</font></i></a></i>;<br/>
<a NAME="E7:48_5"/>
<font color="Maroon" title="c6">PP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" title="WAYBEL18:func.3">product</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <a href="waybel18.html#K9" title="WAYBEL18:func.9">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5" title="CANTOR_1:def.5">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" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K7" title="WAYBEL18:func.7">Image</a> <font color="Maroon" title="c5">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:48_5"><i><font color="Green" title="E32">A78</font></i></a>, <a class="txt" href="waybel18.html#E6:48_5"><i><font color="Green" title="E33">A79</font></i></a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">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" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">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" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c7">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="waybel18.html#E11:48"><i><font color="Green" title="E27">A38</font></i></a>, <a class="txt" href="waybel18.html#E12:48"><i><font color="Green" title="E28">A39</font></i></a>, <a class="ref" href="tops_2.html#T67" title="TOPS_2:th.67">TOPS_2:67</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:48"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E1:48"><i><font color="Green" title="E22">A</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>
<b>hence </b><a NAME="E17:48"/>
 <a href="waybel18.html#K8" title="WAYBEL18:func.8">corestr</a> <font color="Maroon" title="c5">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="waybel18.html#E10:48"><i><font color="Green" title="E26">A37</font></i></a>, <a class="txt" href="waybel18.html#E11:48"><i><font color="Green" title="E27">A38</font></i></a>, <a class="txt" href="waybel18.html#E12:48"><i><font color="Green" title="E28">A39</font></i></a>, <a class="txt" href="waybel18.html#E13:48"><i><font color="Green" title="E29">A50</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>


</div>
