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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<br/>





<b>set </b><font color="Maroon">IP</font> =  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">F</font> = <font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> ;<br/>
<b>reconsider </b><font color="Maroon">T</font> =  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> ;<br/>
<a NAME="E2:20"/><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="waybel16.html#T15">WAYBEL16:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">E</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span> ;<br/>
<a NAME="E5:20"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</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">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> ) </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E6:20"/><i><font color="Green">E38</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</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">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/>
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</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">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</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">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <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">X</font></span>)</span><b> such that </b><br/><a NAME="E5:20_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span>
 <b>and </b><br/><a NAME="E6:20_1"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:20_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:20"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T32">YELLOW_0:32</a></i>;<br/>
<a NAME="E8:20"/><b>then </b>
 <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <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">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">T</font> </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<b>hence </b><a NAME="E9:20"/>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K4">Top</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p4">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="waybel15.html#T5">WAYBEL15:5</a></i>;<br/>


</div>
