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

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>set </b><font color="Maroon">T</font> =  <a href="yellow_1.html#K3">BoolePoset</a> 1;<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">S</font>;<br/>
<b>reconsider </b><font color="Maroon">T'</font> =  <a href="waybel25.html#K1">Omega</a> <a href="waybel18.html#K9">Sierpinski_Space</a>  as  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_1.html#K3">BoolePoset</a> 1 <b>by </b><i><a class="ref" href="waybel27.html#D2" target="_self">Def2</a>, <a class="ref" href="waybel26.html#T4">WAYBEL26:4</a></i>;<br/>
<a NAME="E3:51"/>
(  <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">S'</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S'</font> #) &amp;  <a href="yellow_1.html#K3">BoolePoset</a> 1 <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>then </b><i><font color="Green">E36</font></i>:  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span> = 
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S'</font>,<font color="Maroon">T'</font>
<b>by </b><i/>
<br/>.= 
 <a href="waybel17.html#K2">SCMaps</a> <font color="Maroon">S'</font>,<font color="Maroon">T'</font>
<b>by </b><i/>
<br/>.= 
 <a href="waybel24.html#K3">ContMaps</a> <font color="Maroon">S'</font>,<font color="Maroon">T'</font>
<b>by </b><i><a class="ref" href="waybel24.html#T38">WAYBEL24:38</a></i>
<br/>.= 
 <a href="waybel26.html#K1">oContMaps</a> <font color="Maroon">S'</font>,<a href="waybel18.html#K9">Sierpinski_Space</a> 
<b>by </b><i><a class="ref" href="waybel26.html#D1">WAYBEL26:def 1</a></i>
;<br/>
<a NAME="E5:51"/><i><font color="Green">E38</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">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T51">YELLOW_9:51</a></i>;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:51"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">G</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>and </b><br/><a NAME="E8:51"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">V</font> being <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S'</font> holds  <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">V</font> <a href="hidden.html#R1">=</a>  <a href="funct_3.html#K5">chi</a> <font color="Olive">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E4:51"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel26.html#T5">WAYBEL26:5</a></i>;<br/>
<b>take </b><font color="Maroon">F</font> = <font color="Maroon">G</font> <a href="tops_2.html#NK3" target="_self">"</a> ;<br/>

<a NAME="E9:51"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">G</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<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="txt" href="waybel27.html#E5:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/>
<a NAME="E10:51"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:51"/>
<font color="Maroon">F</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel27.html#E5:51"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel_0.html#T68">WAYBEL_0:68</a></i>;<br/>

<b>let </b><font color="Maroon">f</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span>;<br/>

<a NAME="E12:51"/>
<font color="Maroon">f</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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> 1</span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">V</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:51"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="waybel27.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">G</font> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="yellow_1.html#T1">YELLOW_1:1</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S'</font> <b>by </b><i>, , <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>thus </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font> = 
<font color="Maroon">V</font>
<b>by </b><i><a class="ref" href="waybel27.html#T1" target="_self">Th1</a>, , , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K5">chi</a> <font color="Maroon">V</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S'</font></span>)</span> <a href="relset_1.html#K11">"</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/>
<br/>.= 
<font color="Maroon">f</font> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="waybel27.html#E7:51"><i><font color="Green">E39</font></i></a>, </i>
;<br/>


</div>
