<?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">FUF</font> =  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">BP</font> =  <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</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/>
<a NAME="E1:23"/><i><font color="Green">E31</font></i>: 
 <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> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</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>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</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> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<i><font color="Green">E35</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> <font color="Maroon">X</font></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1">BooleLatt</a> <font color="Maroon">X</font></span>)</span></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> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="lattice3.html#D1">LATTICE3:def 1</a></i>
;<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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>now </b></a><div class="add"><a NAME="E1:23_2"/>
 <a href="relset_1.html#K4">dom</a> <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="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#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>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E2:23_2"/><b>then </b><i><font color="Green">E37</font></i>: 
 <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:23_2"/>
 <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/><b>thus </b><a NAME="E4:23_2"/>
 <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>let </b><font color="Maroon">xf</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E5:23_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">xf</font> <a href="hidden.html#R2">in</a>  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">xx</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="E7:23_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">xf</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xx</font>
 <b>and </b><br/><a NAME="E8:23_2"/><i><font color="Green">E60</font></i>: 
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">y</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E9:23_2"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xf</font> <a href="hidden.html#R1">=</a> <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_1.html#K1">.</a> <font color="Maroon">xf</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">xf</font> 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> <b>by </b><i>, , <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>set </b><font color="Maroon">Xs</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">x'</font> </span>}</span> ;<br/><a NAME="E11:23_2"/><i><font color="Green">E62</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</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">x'</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>reconsider </b><font color="Maroon">FUF'</font> =  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM3">Subset-Family</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">xf'</font> = <font color="Maroon">xf</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">FUF'</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">FUF'</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/><a NAME="E15:23_2"/>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">xf'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>
 ;<br/><b>then reconsider </b><font color="Maroon">fxf</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xf'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E17:23_2"/><i><font color="Green">E64</font></i>: 
<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">x'</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">fxf</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:23_2_1"/>
<font color="Maroon">b</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">x'</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:23_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <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">Y</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:23_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font>
 ;<br/>
<b>set </b><font color="Maroon">Xsi</font> = <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">Y</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:23_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><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#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, </i>;<br/>
<a NAME="E8:23_2_1"/>
<font color="Maroon">xx</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<a NAME="E9:23_2_1"/><b>then </b>
<font color="Maroon">xx</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<a NAME="E10:23_2_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E11:23_2_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">fxf</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">Y</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:23_2_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</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">Y</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E13:23_2_1"/><b>then </b>
<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">Y</font> ) </span>}</span>  <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D10">YELLOW_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E14:23_2_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">fxf</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:23"><i><font color="Green">E31</font></i></a>, <a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:23_2"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <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">x'</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Olive">a</font> holds <br/><font color="Maroon">fxf</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:23_2_2"/><i><font color="Green">E70</font></i>: 
<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">x'</font> </span>}</span>  <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:23_2_2"/>
<font color="Maroon">xx</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">xx</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">xx</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">x'</font> <b>by </b><i>, <a class="ref" href="waybel_0.html#T18">WAYBEL_0:18</a></i>;<br/>
<b>set </b><font color="Maroon">Xsi</font> = <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">Y</font> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:23_2_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:23_2_2_1_1"/>
<font color="Maroon">p</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">Y</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">r</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="E3:23_2_2_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</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">r</font></span>)</span>
 <b>and </b><br/><a NAME="E4:23_2_2_1_1"/><i><font color="Green">E73</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">r</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">Y</font> )
 ;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:23_2_2_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="waybel22.html#E1:23"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E7:23_2_2_1_1"/>
<font color="Maroon">xx</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E8:23_2_2_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">fxf</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:23_2_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">fxf</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:23_2_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fxf</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:23_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E5:23_2_2_1"/>
<font color="Maroon">p</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">Y</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<a NAME="E7:23_2_2"/><b>then </b>
<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">Y</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">fxf</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E8:23_2_2"/><b>then </b>
( <font color="Maroon">fxf</font> <a href="hidden.html#R1">=</a>  <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">Y</font> ) </span>}</span> ,<font color="Maroon">L</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/>
<a NAME="E9:23_2_2"/><b>then </b>
<font color="Maroon">fxf</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">x'</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:23_2_2"/>
<font color="Maroon">fxf</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:23_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xf</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">x'</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/><b>hence </b><a NAME="E20:23_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">xf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">xf</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:23"/>
<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="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div>
