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

<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2" target="_self">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>set </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> </span>}</span> ;<br/>
<a NAME="E1:45"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</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 <font color="Maroon">T</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> </span>}</span> 
 ;<br/>

<a NAME="E2:45_1"/><b>then </b>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> &amp; <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> )
 
;<br/>
<b>hence </b><a NAME="E3:45_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">O</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>reconsider </b><font color="Maroon">O</font> = <font color="Maroon">O</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<b>consider </b><font color="Maroon">R</font> being  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1" target="_self">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>;<br/>
<b>reconsider </b><font color="Maroon">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> : verum </span>}</span>  as    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E6:45"/><i><font color="Green">E30</font></i>: 
 <a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">R</font>
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">S</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E8:45"/><i><font color="Green">E34</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #) <a href="hidden.html#R1">=</a>  <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> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <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> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E9:45"/><i><font color="Green">E36</font></i>: 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<a NAME="E10:45"/><b>then </b><i><font color="Green">E38</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>
<a NAME="E11:45"/>
<font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E5:45_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="waybel19.html#E8:45"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:45_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:45"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T9">XBOOLE_1:9</a></i>;<br/>
<a NAME="E13:45"/>
<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font></span>)</span> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:45"/><b>then </b>
(  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font></span>)</span> &amp;  <a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E15:45"/><b>then </b><i><font color="Green">E63</font></i>: 
(  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> &amp;  <a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:45"/>
<font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</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">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font>
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">x</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_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> )
 ;<br/>
<a NAME="E4:45_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="waybel19.html#E9:45"><i><font color="Green">E36</font></i></a>, <a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E6:45_3"/>
 <a href="waybel_0.html#K6">uparrow</a> <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>, </i>;<br/>
<a NAME="E7:45_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">K</font> <a href="tarski.html#R1">c=</a>  <a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E6:45"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="cantor_1.html#D5">CANTOR_1:def 5</a></i>;<br/>
<a NAME="E8:45_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a>  <a href="waybel19.html#K1" target="_self">omega</a> <font color="Maroon">T</font>
 
;<br/>
<b>hence </b><a NAME="E9:45_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="txt" href="waybel19.html#E9:45"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:45"/><b>then </b><i><font color="Green">E73</font></i>: 
<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E18:45"/>
( <font color="Maroon">T</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font> &amp;  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">S</font> )
 
<b>by </b><i>, <a class="ref" href="cantor_1.html#T2">CANTOR_1:2</a>, <a class="ref" href="yellow_9.html#T44">YELLOW_9:44</a></i>;<br/>
<a NAME="E19:45"/><b>then </b>
( <font color="Maroon">T</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">R</font> &amp;  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">S</font> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E2"><i><font color="Green">Lemma28</font></i></a>, <a class="ref" href="yellow_9.html#T27">YELLOW_9:27</a></i>;<br/>
<a NAME="E20:45"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">K</font> is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E21:45"/>
<span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">W</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> : <font color="Olive">W</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> </span>}</span>  is    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>
 <b>by </b><i><a class="txt" href="waybel19.html#E8:45"><i><font color="Green">E34</font></i></a>, <a class="txt" href="waybel19.html#E9:45"><i><font color="Green">E36</font></i></a>, </i>;<br/>


</div>
