<?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">Z</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<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="E2:46"/><i><font color="Green">E30</font></i>: 
( the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <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#T51">YELLOW_9:51</a>, <a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B1</font> =  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">T</font> as    <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></i>;<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/>
<a NAME="E5:46"/><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> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Olive">F</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> where <font color="Olive">F</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> : <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E7:46"/>
<font color="Maroon">T</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44">YELLOW_9:44</a></i>;<br/>
<a NAME="E8:46"/><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>
 
<b>by </b><i><a class="txt" href="waybel19.html#E2"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E9:46"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font></span>)</span> is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T59">YELLOW_9:59</a></i>;<br/>
<a NAME="E10:46"/><i><font color="Green">E38</font></i>: 
 <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font> <a href="hidden.html#R1">=</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:46_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font>
 ;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:46_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">z</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i><a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a></i>;<br/><b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:46_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><a NAME="E8:46_1_1"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp;  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="waybel19.html#E2:46"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/><a NAME="E9:46_1_1"/><b>then </b>
( <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#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <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#K9">/\</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E10:46_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>;<br/><b>hence </b><a NAME="E11:46_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a>, </i>;<br/></div>
<b>end;</b></div>

<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="E2:46_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">W</font> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 ;<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">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E4:46_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</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#K4">uparrow</a> <font color="Maroon">F</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> &amp; <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<a NAME="E5:46_1"/>
<font color="Maroon">W</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E6:46_1"/><b>then </b>
<font color="Maroon">x</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="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>;<br/>
<a NAME="E7:46_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">x</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#K4">uparrow</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E9:46_1"/>
( <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font> )
 
<b>by </b><i>, <a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/>
<b>hence </b><a NAME="E10:46_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">B1</font>,<font color="Maroon">B2</font>
 <b>by </b><i><a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:46"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">B1</font> <a href="tarski.html#R1">c=</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</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 <font color="Maroon">T</font> ;<br/>
<b>set </b><font color="Maroon">F'</font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">R</font>;<br/>
<b>reconsider </b><font color="Maroon">G</font> =  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">R</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E4:46_2"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/>
<a NAME="E5:46_2"/><b>then </b>
( <font color="Maroon">x'</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">G</font> is <a href="finset_1.html#V1">finite</a> )
 
;<br/>
<b>hence </b><a NAME="E6:46_2"/>
<font color="Maroon">x</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:46"/>
<font color="Maroon">B2</font> <a href="tarski.html#R1">c=</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:46_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B2</font>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:46_3"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">F</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E5:46_3"/>
(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B1</font> &amp;  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E2:46"><i><font color="Green">E30</font></i></a>, , <a class="ref" href="pre_topc.html#D1">PRE_TOPC:def 1</a>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/>
<a NAME="E6:46_3"/><b>then </b>
( <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#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:46_3"/>
<font color="Maroon">x</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:46"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">B2</font> <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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E14:46"/>
<font color="Maroon">B1</font> <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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="waybel19.html#E5:46"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<b>hence </b><a NAME="E15:46"/>
<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#K4">uparrow</a> <font color="Olive">F</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">F</font> is   <a href="struct_0.html#NM2">Subset</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> &amp; <font color="Olive">F</font> is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  is    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>


</div>
