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

<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">T</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>;<br/>



<a NAME="E1:45"/><i><font color="Green">E27</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K5">product</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">IB</font> = <font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">IS</font> = <font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> ;<br/>
<a NAME="E2:45"/>
 <a href="lattice3.html#K3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> 1</span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="lattice3.html#D2">LATTICE3:def 2</a></i>;<br/>
<b>then </b><i><font color="Green">E28</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="yellow_1.html#D2">YELLOW_1:def 2</a></i>
<br/>.= 
 <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i><a class="ref" href="card_5.html#T1">CARD_5:1</a>, <a class="ref" href="zfmisc_1.html#T30">ZFMISC_1:30</a></i>
;<br/>
<i><font color="Green">E31</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> = 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<a NAME="E5:45"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:45_3"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">R1</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:45_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">R1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:45_3"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font>
 <b>by </b><i><a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">R2</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E7:45_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">R2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E8:45_3"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font>
 <b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R1</font> = 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
<b>by </b><i>, , , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> 
<b>by </b><i/>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R2</font>
<b>by </b><i>, <a class="txt" href="waybel18.html#E1:45"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<b>hence </b><a NAME="E10:45_3"/>
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="txt" href="waybel18.html#E3:45"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:45"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="pralg_1.html#K12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="pboole.html#R6">=</a>  <a href="pralg_1.html#K12">Carrier</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<i><font color="Green">E66</font></i>: the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> = 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>
<br/>.= 
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="waybel18.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E8:45"/><i><font color="Green">E73</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">ii</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">ii</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of  <a href="waybel18.html#K3" target="_self">product</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">T'</font> = <font color="Maroon">T</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a> ;<br/>
<a NAME="E11:45"/>
<font color="Maroon">T'</font> is <a href="waybel_8.html#V2">algebraic</a>
 
<b>by </b><i>, <a class="ref" href="waybel_8.html#T17">WAYBEL_8:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">R</font> being    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T'</font><b> such that </b><br/><a NAME="E13:45"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">yy</font></span>)</span> where <font color="Olive">yy</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T'</font> : <font color="Olive">yy</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T'</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel14.html#T42">WAYBEL14:42</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E15:45"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>and </b><br/><a NAME="E16:45"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">Y</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> holds  <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">Y</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Olive">Y</font>,<font color="Maroon">I</font>
 <b>by </b><i/>;<br/>
<a NAME="E17:45"/><i><font color="Green">E77</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:45"><i><font color="Green">E28</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E18:45"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_5"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T'</font><b> such that </b><br/><a NAME="E3:45_5"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:45_5"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T'</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:45"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E5:45_5"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Y</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <b>by </b><i>, <a class="txt" href="waybel18.html#E5:45"><i><font color="Green">E32</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E8:45_5"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">y</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i>, , <a class="ref" href="waybel_8.html#T9">WAYBEL_8:9</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:45_5"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E11:45_5"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">I</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<a NAME="E14:45_5"/>
<font color="Maroon">x</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:45"><i><font color="Green">E28</font></i></a>, , , <a class="ref" href="waybel13.html#T28">WAYBEL13:28</a></i>;<br/>
<a NAME="E15:45_5"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="waybel_8.html#T30">WAYBEL_8:30</a></i>;<br/>
<i><font color="Green">E98</font></i>:  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">Y</font> = 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="waybel_0.html#D18">WAYBEL_0:def 18</a></i>
<br/>.= 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>
<br/>.= 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="waybel_0.html#D18">WAYBEL_0:def 18</a></i>
;<br/>
<b>set </b><font color="Maroon">ZZ</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">jj</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">jj</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">jj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span> ;<br/>
<a NAME="E17:45_5"/><i><font color="Green">E99</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">jj</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">jj</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">jj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_5_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">jj</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">jj</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">jj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:45_5_2"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:45_5_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font>
 ;<br/>
<b>thus </b><a NAME="E5:45_5_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E18:45_5"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">jj</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">jj</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">jj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ZZ</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">jj</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">jj</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">jj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E20:45_5"/><i><font color="Green">E102</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">jj</font>) where <font color="Olive">jj</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">jj</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="ref" href="waybel18.html#T3" target="_self">Th3</a>);<br/></i>
<a NAME="E21:45_5"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">X'</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">ZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_5_3"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 ;<br/>

<a NAME="E2:45_5_3"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i>, , <a class="txt" href="waybel18.html#E6:45"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E4:45_5_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">gg</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:45_5_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font>
 <b>and </b><br/><a NAME="E7:45_5_3"/><i><font color="Green">E107</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E8:45_5_3"/><i><font color="Green">E108</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E9:45_5_3"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i>, , <a class="txt" href="waybel18.html#E6:45"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">h1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">h2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:45_5_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E12:45_5_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E13:45_5_3"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 <b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<i><font color="Green">E112</font></i>: <font color="Maroon">h1</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font>
<b>by </b><i/>
;<br/>
<a NAME="E15:45_5_3"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font> holds <br/><font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_3_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_5_3_2"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:45_5_3_2"/><i><font color="Green">E113</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:45_5_3_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/>
<i><font color="Green">E115</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon">RS</font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">xj</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font>, <font color="Maroon">yj</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font><b> such that </b><br/><a NAME="E7:45_5_3_2"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="waybel_3.html#K6">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E8:45_5_3_2"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">xj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E9:45_5_3_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">yj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E10:45_5_3_2"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">xj</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yj</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:45_5_3_2"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">xj</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">xj</font>, <font color="Maroon">b</font> = <font color="Maroon">yj</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E13:45_5_3_2"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E14:45_5_3_2"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span>
 
;<br/>
<a NAME="E15:45_5_3_2"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">yj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_3_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_5_3_2_2"/>
<font color="Maroon">yj</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:45_5_3_2_2"/><b>then </b><i><font color="Green">E124</font></i>: 
1 <a href="tarski.html#R1">c=</a> 0
 
<b>by </b><i><a class="ref" href="waybel18.html#T4" target="_self">Th4</a>, , <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<a NAME="E3:45_5_3_2_2"/>
0 <a href="tarski.html#R1">c=</a> 1
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:45_5_3_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:45_5_3_2"/><i><font color="Green">E125</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, , , , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:45_5_3_2"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> holds <br/><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_3_2_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_5_3_2_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:45_5_3_2_3"/><b>then </b><i><font color="Green">E127</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_3_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_5_3_2_3_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_3_2_3_1_1"><b>suppose </b></a><a NAME="E1:45_5_3_2_3_1_1"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:45_5_3_2_3_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:45_5_3_2_3_1_1"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , <a class="ref" href="waybel18.html#D3" target="_self">Def3</a>, <a class="txt" href="waybel18.html#E7:45"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:45_5_3_2_3_1_1"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_3_2_3_1_2"><b>suppose </b></a><a NAME="E1:45_5_3_2_3_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:45_5_3_2_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:45_5_3_2_3_1_2"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E18:45_5_3_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="waybel18.html#D2" target="_self">Def2</a>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:45_5_3"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">ZZ</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E6:45"><i><font color="Green">E44</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E22:45_5"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">ZZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_5_4"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">ZZ</font>
 ;<br/>

<a NAME="E2:45_5_4"/><b>then </b><i><font color="Green">E131</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i>, , <a class="txt" href="waybel18.html#E6:45"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">h1</font> =  <a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font>;<br/>
<i><font color="Green">E132</font></i>:  <a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i/>
;<br/>
<a NAME="E5:45_5_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">h2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:45_5_4"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font>
 <b>and </b><br/><a NAME="E8:45_5_4"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:45_5_4"/><i><font color="Green">E134</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E10:45_5_4"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_4_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_5_4_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>take </b><font color="Maroon">RS</font> =  <a href="yellow_1.html#K3">BoolePoset</a> 1;<br/>

<b>consider </b><font color="Maroon">RB</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:45_5_4_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">RB</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:45_5_4_2"/><i><font color="Green">E137</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RB</font>
 <b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E6:45_5_4_2"/><i><font color="Green">E138</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, , , , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:45_5_4_2"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:45_5_4_2"/><b>then </b><i><font color="Green">E140</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_4_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_5_4_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_4_2_1_1"><b>suppose </b></a><a NAME="E1:45_5_4_2_1_1"/><i><font color="Green">E141</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:45_5_4_2_1_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/><a NAME="E3:45_5_4_2_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="txt" href="waybel18.html#E6:45"><i><font color="Green">E44</font></i></a>, <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>then consider </b><font color="Maroon">h2'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:45_5_4_2_1_1"/><i><font color="Green">E142</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2'</font>
 <b>and </b><br/><a NAME="E6:45_5_4_2_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E7:45_5_4_2_1_1"/><i><font color="Green">E143</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> holds <br/><font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E8:45_5_4_2_1_1"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel18.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/><a NAME="E9:45_5_4_2_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>by </b><i/>;<br/><a NAME="E10:45_5_4_2_1_1"/><i><font color="Green">E145</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="waybel18.html#D2" target="_self">Def2</a>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>reconsider </b><font color="Maroon">xi</font> = 1, <font color="Maroon">yi</font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">xi</font>
;<br/><b>take </b>
<font color="Maroon">yi</font>
;<br/><b>thus </b><a NAME="E12:45_5_4_2_1_1"/>
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>thus </b><a NAME="E13:45_5_4_2_1_1"/>
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>thus </b><a NAME="E14:45_5_4_2_1_1"/>
<font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E15:45_5_4_2_1_1"/>
<font color="Maroon">xi</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yi</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_4_2_1_2"><b>suppose </b></a><a NAME="E1:45_5_4_2_1_2"/><i><font color="Green">E146</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:45_5_4_2_1_2"/>
 ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> st <br/>( <font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_4_2_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_4_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_5_4_2_1_2_1_1"/>
( <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_4_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:45_5_4_2_1_2_1_1_1"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">xi</font> = 0, <font color="Maroon">yi</font> = 0 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">xi</font>
;<br/><b>take </b>
<font color="Maroon">yi</font>
;<br/><b>thus </b><a NAME="E3:45_5_4_2_1_2_1_1_1"/>
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>thus </b><a NAME="E4:45_5_4_2_1_2_1_1_1"/>
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>thus </b><a NAME="E5:45_5_4_2_1_2_1_1_1"/>
<font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:45_5_4_2_1_2_1_1_1"/>
<font color="Maroon">xi</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yi</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_5_4_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:45_5_4_2_1_2_1_1_2"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">xi</font> = 0, <font color="Maroon">yi</font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">xi</font>
;<br/><b>take </b>
<font color="Maroon">yi</font>
;<br/><b>thus </b><a NAME="E3:45_5_4_2_1_2_1_1_2"/>
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>thus </b><a NAME="E4:45_5_4_2_1_2_1_1_2"/>
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>thus </b><a NAME="E5:45_5_4_2_1_2_1_1_2"/>
<font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><a NAME="E6:45_5_4_2_1_2_1_1_2"/>
<font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">yi</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E7:45_5_4_2_1_2_1_1_2"/>
<font color="Maroon">xi</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yi</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<a NAME="E11:45_5_4"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i>, <a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, , <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:45_5_4"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:45_5"/><b>then </b>
<font color="Maroon">X'</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">ZZ</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E24:45_5"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:45"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> ;<br/>
<b>consider </b><font color="Maroon">ZZ</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:45_6"/><i><font color="Green">E150</font></i>: 
<font color="Maroon">ZZ</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E5:45_6"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">ZZ</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E6:45_6"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">ZZ</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E5:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">a<sub>1</sub></font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E8:45_6"/><i><font color="Green">E153</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>and </b><br/><a NAME="E9:45_6"/><i><font color="Green">E154</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">e</font>)
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<a NAME="E10:45_6"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:45_6_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">e</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>
<a NAME="E4:45_6_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:45_6_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:45_6"/><b>then </b>
<font color="Maroon">ZZ</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then consider </b><font color="Maroon">x'</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E13:45_6"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">x'</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E14:45_6"/><i><font color="Green">E157</font></i>: 
<font color="Maroon">x'</font> is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E15:45_6"/><i><font color="Green">E158</font></i>: 
<font color="Maroon">F</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ZZ</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#T195">ORDERS_1:195</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x'</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">PP</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> </span>}</span> ;<br/>
<a NAME="E17:45_6"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">ZZ</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_6_2"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E4:45_6_2"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font>
 <b>and </b><br/><a NAME="E5:45_6_2"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> <b>by </b><i>, </i>;<br/>
<a NAME="E7:45_6_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">e</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="waybel18.html#E6:45"><i><font color="Green">E44</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:45_6_2"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E18:45_6"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">ZZ</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:45_6_3"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">e</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:45_6_3"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font>
 ;<br/>
<a NAME="E5:45_6_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:45_6_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E19:45_6"/><b>then </b><i><font color="Green">E165</font></i>: 
<font color="Maroon">ZZ</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p5">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Olive">i</font>,<span class="p4"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x'</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">I</font></span>)</span> <b>by </b><i><a class="ref" href="waybel_8.html#T28">WAYBEL_8:28</a></i>;<br/>
<a NAME="E21:45_6"/><i><font color="Green">E217</font></i>: 
<font color="Maroon">x</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="ref" href="waybel18.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel_8.html#T30">WAYBEL_8:30</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> ;<br/>
<a NAME="E23:45_6"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">y</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i><a class="txt" href="waybel18.html#E3:45"><i><font color="Green">E28</font></i></a>, , <a class="ref" href="waybel13.html#T28">WAYBEL13:28</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T'</font> <b>by </b><i/>;<br/>
<i><font color="Green">E219</font></i>:  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">Y</font> = 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">Y</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i><a class="ref" href="waybel_0.html#D18">WAYBEL_0:def 18</a></i>
<br/>.= 
 <a href="waybel_0.html#K4">uparrow</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span>
<b>by </b><i>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>
<br/>.= 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="waybel_0.html#D18">WAYBEL_0:def 18</a></i>
;<br/>
<a NAME="E26:45_6"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">X'</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_5"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 ;<br/>

<a NAME="E2:45_6_5"/><b>then </b><i><font color="Green">E222</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i>, , </i>;<br/>
<b>set </b><font color="Maroon">h1</font> =  <a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font>;<br/>
<a NAME="E4:45_6_5"/><i><font color="Green">E223</font></i>: 
 <a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:45_6_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">h2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E7:45_6_5"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font>
 <b>and </b><br/><a NAME="E8:45_6_5"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:45_6_5"/><i><font color="Green">E225</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E10:45_6_5"/>
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_5_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_5_1"/><i><font color="Green">E226</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>take </b><font color="Maroon">RS</font> =  <a href="yellow_1.html#K3">BoolePoset</a> 1;<br/>

<b>consider </b><font color="Maroon">RB</font> being    <a href="struct_0.html#L1">1-sorted</a> <b> such that </b><br/><a NAME="E4:45_6_5_1"/><i><font color="Green">E227</font></i>: 
<font color="Maroon">RB</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:45_6_5_1"/><i><font color="Green">E228</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RB</font>
 <b>by </b><i>, <a class="ref" href="pralg_1.html#D13">PRALG_1:def 13</a></i>;<br/>
<a NAME="E6:45_6_5_1"/><i><font color="Green">E229</font></i>: 
<span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">0,1</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, , <a class="ref" href="waybel18.html#D2" target="_self">Def2</a>, , <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E7:45_6_5_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:45_6_5_1"/><b>then </b><i><font color="Green">E231</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_6_5_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> or  not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_5_1_1_1"><b>suppose </b></a><a NAME="E1:45_6_5_1_1_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><a NAME="E2:45_6_5_1_1_1"/><b>then </b>
 <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 <b>by </b><i/>;<br/><a NAME="E3:45_6_5_1_1_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/><b>then consider </b><font color="Maroon">h2'</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:45_6_5_1_1_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2'</font>
 <b>and </b><br/><a NAME="E6:45_6_5_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2'</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E7:45_6_5_1_1_1"/><i><font color="Green">E234</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> holds <br/><font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/><a NAME="E8:45_6_5_1_1_1"/>
<font color="Maroon">i'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>;<br/><a NAME="E9:45_6_5_1_1_1"/><b>then </b><i><font color="Green">E235</font></i>: 
<font color="Maroon">h2'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font>
 <b>by </b><i><a class="ref" href="waybel18.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E10:45_6_5_1_1_1"/><i><font color="Green">E236</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">i'</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><b>reconsider </b><font color="Maroon">xi</font> = 1, <font color="Maroon">yi</font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">xi</font>
;<br/><b>take </b>
<font color="Maroon">yi</font>
;<br/><b>thus </b><a NAME="E12:45_6_5_1_1_1"/>
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>thus </b><a NAME="E13:45_6_5_1_1_1"/>
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>thus </b><a NAME="E14:45_6_5_1_1_1"/>
<font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , , , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E15:45_6_5_1_1_1"/>
<font color="Maroon">xi</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yi</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_5_1_1_2"><b>suppose </b></a><a NAME="E1:45_6_5_1_1_2"/><i><font color="Green">E237</font></i>: 
not <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:45_6_5_1_1_2"/>
 ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> st <br/>( <font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_5_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_5_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_6_5_1_1_2_1_1"/>
( <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_5_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:45_6_5_1_1_2_1_1_1"/><i><font color="Green">E238</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">xi</font> = 0, <font color="Maroon">yi</font> = 0 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">xi</font>
;<br/><b>take </b>
<font color="Maroon">yi</font>
;<br/><b>thus </b><a NAME="E3:45_6_5_1_1_2_1_1_1"/>
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>thus </b><a NAME="E4:45_6_5_1_1_2_1_1_1"/>
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>thus </b><a NAME="E5:45_6_5_1_1_2_1_1_1"/>
<font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:45_6_5_1_1_2_1_1_1"/>
<font color="Maroon">xi</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yi</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_5_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:45_6_5_1_1_2_1_1_2"/><i><font color="Green">E239</font></i>: 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">xi</font> = 0, <font color="Maroon">yi</font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font> <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">xi</font>
;<br/><b>take </b>
<font color="Maroon">yi</font>
;<br/><b>thus </b><a NAME="E3:45_6_5_1_1_2_1_1_2"/>
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>thus </b><a NAME="E4:45_6_5_1_1_2_1_1_2"/>
<font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/><b>thus </b><a NAME="E5:45_6_5_1_1_2_1_1_2"/>
<font color="Maroon">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i/>;<br/><a NAME="E6:45_6_5_1_1_2_1_1_2"/>
<font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">yi</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>hence </b><a NAME="E7:45_6_5_1_1_2_1_1_2"/>
<font color="Maroon">xi</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">yi</font>
 <b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<a NAME="E11:45_6_5"/><b>then </b>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i>, , , <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:45_6_5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E27:45_6"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_6"/><i><font color="Green">E240</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:45_6_6"/><b>then </b><i><font color="Green">E241</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:45_6_6"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">gg</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:45_6_6"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">gg</font>
 <b>and </b><br/><a NAME="E7:45_6_6"/><i><font color="Green">E243</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">gg</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E8:45_6_6"/><i><font color="Green">E244</font></i>: 
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">gg</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<a NAME="E9:45_6_6"/>
<font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">h1</font> being   <a href="funct_1.html#NM1">Function</a>, <font color="Maroon">h2</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E11:45_6_6"/><i><font color="Green">E245</font></i>: 
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E12:45_6_6"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">h2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font>
 <b>and </b><br/><a NAME="E13:45_6_6"/><i><font color="Green">E247</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> holds <br/> ex <font color="Olive">R</font> being   <a href="orders_2.html#L1">RelStr</a>  ex <font color="Olive">xi</font>, <font color="Olive">yi</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Olive">R</font> st <br/>( <font color="Olive">R</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">yi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Olive">xi</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">yi</font> )
 <b>by </b><i>, <a class="ref" href="yellow_1.html#D4">YELLOW_1:def 4</a></i>;<br/>
<a NAME="E14:45_6_6"/><i><font color="Green">E248</font></i>: 
<font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Maroon">x</font>,<font color="Maroon">I</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E15:45_6_6"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font> holds <br/><font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_6_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_6_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZZ</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:45_6_6_1"/><i><font color="Green">E249</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>and </b><br/><a NAME="E4:45_6_6_1"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>
<i><font color="Green">E251</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_7.html#T32">FUNCT_7:32</a></i>
<br/>.= 
<font color="Maroon">I</font>
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon">RS</font> being    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">xj</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font>, <font color="Maroon">yj</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">RS</font><b> such that </b><br/><a NAME="E7:45_6_6_1"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span> <a href="waybel_3.html#K6">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E8:45_6_6_1"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">xj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E9:45_6_6_1"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">yj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E10:45_6_6_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">xj</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">yj</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:45_6_6_1"/><i><font color="Green">E256</font></i>: 
<font color="Maroon">xj</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , <a class="ref" href="waybel18.html#T4" target="_self">Th4</a>, <a class="ref" href="funct_3.html#D3">FUNCT_3:def 3</a></i>;<br/>
<a NAME="E12:45_6_6_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">RS</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K3">BoolePoset</a> 1
 
<b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">yj</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E14:45_6_6_1"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span>
 
;<br/>
<a NAME="E15:45_6_6_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">yj</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_6_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:45_6_6_1_2"/>
<font color="Maroon">yj</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:45_6_6_1_2"/><b>then </b><i><font color="Green">E260</font></i>: 
1 <a href="tarski.html#R1">c=</a> 0
 
<b>by </b><i>, <a class="ref" href="waybel18.html#D3" target="_self">Def3</a>, , <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<a NAME="E3:45_6_6_1_2"/>
0 <a href="tarski.html#R1">c=</a> 1
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>hence </b><a NAME="E4:45_6_6_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:45_6_6_1"/><i><font color="Green">E261</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="waybel18.html#D2" target="_self">Def2</a>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E17:45_6_6_1"/>
for <font color="Olive">w</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> holds <br/><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_6_6_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_6_6_1_3"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span>
 ;<br/>

<a NAME="E2:45_6_6_1_3"/><b>then </b><i><font color="Green">E262</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_6_1_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_6_6_1_3_1"/>
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> or <font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_6_1_3_1_1"><b>suppose </b></a><a NAME="E1:45_6_6_1_3_1_1"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:45_6_6_1_3_1_1"/><b>then </b><i><font color="Green">E264</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="funct_7.html#T33">FUNCT_7:33</a></i>;<br/><a NAME="E3:45_6_6_1_3_1_1"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, , , <a class="txt" href="waybel18.html#E7:45"><i><font color="Green">E66</font></i></a>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:45_6_6_1_3_1_1"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_6_6_1_3_1_2"><b>suppose </b></a><a NAME="E1:45_6_6_1_3_1_2"/>
<font color="Maroon">w</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:45_6_6_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="ref" href="funct_7.html#T34">FUNCT_7:34</a></i>;<br/><b>hence </b><a NAME="E3:45_6_6_1_3_1_2"/>
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pralg_1.html#K12">Carrier</a> <span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span> <a href="funct_7.html#K2">+*</a> <font color="Maroon">j</font>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, , , <a class="ref" href="waybel18.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E18:45_6_6_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="waybel18.html#D2" target="_self">Def2</a>, , , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:45_6_6"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X'</font>
 <b>by </b><i>, , , , <a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:45_6"/><b>then </b><i><font color="Green">E265</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="waybel18.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E29:45_6"/>
<font color="Maroon">Y</font> is <a href="waybel_3.html#V1">compact</a>
 
<b>by </b><i>, <a class="txt" href="waybel18.html#E6:45"><i><font color="Green">E44</font></i></a>, <a class="ref" href="waybel_8.html#T9">WAYBEL_8:9</a></i>;<br/>
<a NAME="E30:45_6"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">T'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/>
<b>hence </b><a NAME="E31:45_6"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="txt" href="waybel18.html#E1:45"><i><font color="Green">E27</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:45"/><b>then </b><i><font color="Green">E266</font></i>: 
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="txt" href="waybel18.html#E4:45"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P'</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E22:45"/>
<font color="Maroon">P'</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="txt" href="waybel18.html#E5:45"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_9.html#T23">YELLOW_9:23</a></i>;<br/>
<b>hence </b><a NAME="E23:45"/>
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><a href="waybel18.html#K3" target="_self">product</a> <span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="pre_circ.html#K2">--&gt;</a> <a href="waybel18.html#K9" target="_self">Sierpinski_Space</a> </span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T26">YELLOW_9:26</a></i>;<br/>


</div>
