<?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>let </b><font color="Maroon">h</font> be    <a href="waybel16.html#M1">CLHomomorphism</a> of  <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>,<font color="Maroon">L</font>;<br/>







<b>assume </b><a NAME="E1:24"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">h</font> <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>
 ;<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/>
<b>set </b><font color="Maroon">cIP</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="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/>
<b>set </b><font color="Maroon">cL</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>;<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/>
<a NAME="E2:24"/><i><font color="Green">E35</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">E37</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>reconsider </b><font color="Maroon">h'</font> = <font color="Maroon">h</font> as   <a href="funct_2.html#NM1">Function</a> of 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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a>  as   <a href="funct_2.html#NM1">Function</a> of 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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Fi</font> be    <a href="subset_1.html#M1">Element</a> of 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>;<br/><a NAME="E1:24_2"/>
<font color="Maroon">Fi</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K8">Filt</a> <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>then consider </b><font color="Maroon">FF</font> being   <a href="waybel_0.html#NM4">Filter</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="E3:24_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">Fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font>
 ;<br/><a NAME="E4:24_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">FF</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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></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">FF</font> </span>}</span> ,<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/>;<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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></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">FF</font> </span>}</span> ;<br/><a NAME="E5:24_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">h</font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp; <font color="Maroon">h</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <b>by </b><i><a class="ref" href="waybel16.html#D1">WAYBEL16:def 1</a></i>;<br/><a NAME="E6:24_2"/><i><font color="Green">E61</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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></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">FF</font> </span>}</span>  <a href="tarski.html#R1">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="E1:24_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"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></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">FF</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">YY</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:24_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">YY</font> ) </span>}</span> ,<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>and </b><br/><a NAME="E4:24_2_1"/>
<font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 ;<br/>
<b>thus </b><a NAME="E5:24_2_1"/>
<font color="Maroon">p</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="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="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">YY</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:24_2_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E3:24_2_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">YY</font> ) </span>}</span> ,<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> <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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></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">FF</font> </span>}</span> 
 <b>by </b><i>, <a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>hence </b><a NAME="E4:24_2_2"/>
not <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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></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">FF</font> </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/></div><b>end;</b></div><b>then reconsider </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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<span class="p3">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p4">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p5">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span></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">FF</font> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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/>;<br/><a NAME="E9:24_2"/>
<font color="Maroon">Xs</font> is <a href="waybel_0.html#V1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3"><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 <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>, <font color="Maroon">b</font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>

<b>assume </b><a NAME="E1:24_2_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">Ya</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:24_2_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span> ,<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>and </b><br/><a NAME="E4:24_2_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">Ya</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 ;<br/>
<b>reconsider </b><font color="Maroon">Ya'</font> = <font color="Maroon">Ya</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">FF</font> <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">Yb</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E7:24_2_3"/><i><font color="Green">E69</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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span> ,<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>and </b><br/><a NAME="E8:24_2_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Yb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">Yb'</font> = <font color="Maroon">Yb</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">FF</font> <b>by </b><i><a class="txt" href="waybel22.html#E1:24"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">Xsa</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Xsb</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_2_3_1"/>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> or <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> or ( not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> &amp; not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_3_1_1"><b>suppose </b></a><a NAME="E1:24_2_3_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:24_2_3_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span> ,<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> <a href="hidden.html#R1">=</a>  <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>
 ;<br/><b>take </b>
<font color="Maroon">a</font>
;<br/><b>thus </b><a NAME="E3:24_2_3_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>thus </b><a NAME="E4:24_2_3_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 ;<br/><b>thus </b><a NAME="E5:24_2_3_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_3_1_2"><b>suppose </b></a><a NAME="E1:24_2_3_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:24_2_3_1_2"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span> ,<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> <a href="hidden.html#R1">=</a>  <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>
 ;<br/><b>take </b>
<font color="Maroon">b</font>
;<br/><b>thus </b><a NAME="E3:24_2_3_1_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/><b>thus </b><a NAME="E4:24_2_3_1_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><b>thus </b><a NAME="E5:24_2_3_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_3_1_3"><b>suppose </b></a><a NAME="E1:24_2_3_1_3"/><i><font color="Green">E73</font></i>: 
( not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> &amp; not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> )
 ;<br/><div class="add"><a NAME="E2:24_2_3_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span>  <a href="tarski.html#R1">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:24_2_3_1_3_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">xz</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="E3:24_2_3_1_3_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xz</font>
 <b>and </b><br/><a NAME="E4:24_2_3_1_3_1"/>
 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">xz</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">Ya</font> )
 ;<br/>
<b>thus </b><a NAME="E5:24_2_3_1_3_1"/>
<font color="Maroon">r</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="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>, </i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Xsa</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Ya</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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/>;<br/><a NAME="E4:24_2_3_1_3"/><i><font color="Green">E75</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Xsa</font>,<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> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Xsa</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T10">WAYBEL16:10</a></i>;<br/><a NAME="E5:24_2_3_1_3"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span>  <a href="tarski.html#R1">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:24_2_3_1_3_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">xz</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="E3:24_2_3_1_3_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xz</font>
 <b>and </b><br/><a NAME="E4:24_2_3_1_3_2"/>
 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">xz</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">Yb</font> )
 ;<br/>
<b>thus </b><a NAME="E5:24_2_3_1_3_2"/>
<font color="Maroon">r</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="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>, </i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Xsb</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yb</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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/>;<br/><a NAME="E7:24_2_3_1_3"/><i><font color="Green">E77</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Xsb</font>,<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> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Xsb</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T10">WAYBEL16:10</a></i>;<br/><b>consider </b><font color="Maroon">Yab</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="E9:24_2_3_1_3"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">Yab</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>and </b><br/><a NAME="E10:24_2_3_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Yab</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Ya'</font>
 <b>and </b><br/><a NAME="E11:24_2_3_1_3"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">Yab</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Yb'</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">Yab</font> = <font color="Maroon">Yab</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">FF</font> <b>by </b><i><a class="txt" href="waybel22.html#E2:24"><i><font color="Green">E35</font></i></a></i>;<br/><b>set </b><font color="Maroon">c</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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/><b>set </b><font color="Maroon">Xsc</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ;<br/><b>thus </b><a NAME="E13:24_2_3_1_3"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of 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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font> &amp; <font color="Maroon">a</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_3"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_3_1_3_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_2_3_1_3_3_1"/>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> or  not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_3_1_3_3_1_1"><b>suppose </b></a><a NAME="E1:24_2_3_1_3_3_1_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:24_2_3_1_3_3_1_1"/><b>then </b><i><font color="Green">E82</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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> <a href="hidden.html#R1">=</a>  <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>
 ;<br/><b>take </b>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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/><b>thus </b><a NAME="E3:24_2_3_1_3_3_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:24_2_3_1_3_3_1_1"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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="txt" href="waybel22.html#E3:24_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/><b>thus </b><a NAME="E5:24_2_3_1_3_3_1_1"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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="txt" href="waybel22.html#E3:24_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_0.html#T45">YELLOW_0:45</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_2_3_1_3_3_1_2"><b>suppose </b></a><a NAME="E1:24_2_3_1_3_3_1_2"/><i><font color="Green">E83</font></i>: 
not <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:24_2_3_1_3_3_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span>  <a href="tarski.html#R1">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_3_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:24_2_3_1_3_3_1_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">xz</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="E3:24_2_3_1_3_3_1_2_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xz</font>
 <b>and </b><br/><a NAME="E4:24_2_3_1_3_3_1_2_1"/>
 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">xz</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">Yab</font> )
 ;<br/>
<b>thus </b><a NAME="E5:24_2_3_1_3_3_1_2_1"/>
<font color="Maroon">r</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="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>, </i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Xsc</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</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="txt" href="waybel22.html#E3:24_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E4:24_2_3_1_3_3_1_2"/><i><font color="Green">E86</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Xsc</font>,<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> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Xsc</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T10">WAYBEL16:10</a></i>;<br/><b>take </b>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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/><b>thus </b><a NAME="E5:24_2_3_1_3_3_1_2"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font>
 <b>by </b><i/>;<br/><a NAME="E6:24_2_3_1_3_3_1_2"/>
<font color="Maroon">a</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</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="E1:24_2_3_1_3_3_1_2_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:24_2_3_1_3_3_1_2_2"/>
<font color="Maroon">Xsc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Xsa</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_3_1_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:24_2_3_1_3_3_1_2_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xz</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="E3:24_2_3_1_3_3_1_2_2_1"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xz</font>
 <b>and </b><br/><a NAME="E4:24_2_3_1_3_3_1_2_2_1"/><i><font color="Green">E89</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">xz</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">Yab</font> )
 ;<br/>
<b>consider </b><font color="Maroon">zz</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:24_2_3_1_3_3_1_2_2_1"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">xz</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">zz</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E7:24_2_3_1_3_3_1_2_2_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Yab</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:24_2_3_1_3_3_1_2_2_1"/>
<font color="Maroon">Yab</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ya</font>
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E9:24_2_3_1_3_3_1_2_2_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsa</font>
 <b>by </b><i><a class="ref" href="waybel22.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel22.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_2_3_1_3_3_1_2_2"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Xsa</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Xsc</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T7">SETFAM_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:24_2_3_1_3_3_1_2_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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="txt" href="waybel22.html#E2"><i><font color="Green">Lemma40</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:24_2_3_1_3_3_1_2"/>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/><a NAME="E8:24_2_3_1_3_3_1_2"/>
<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_3_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</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="E1:24_2_3_1_3_3_1_2_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:24_2_3_1_3_3_1_2_3"/>
<font color="Maroon">Xsc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Xsb</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_3_1_3_3_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:24_2_3_1_3_3_1_2_3_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsc</font>
 ;<br/>

<b>then consider </b><font color="Maroon">xz</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="E3:24_2_3_1_3_3_1_2_3_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xz</font>
 <b>and </b><br/><a NAME="E4:24_2_3_1_3_3_1_2_3_1"/><i><font color="Green">E94</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">xz</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">Yab</font> )
 ;<br/>
<b>consider </b><font color="Maroon">zz</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:24_2_3_1_3_3_1_2_3_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">xz</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">zz</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E7:24_2_3_1_3_3_1_2_3_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">zz</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Yab</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:24_2_3_1_3_3_1_2_3_1"/>
<font color="Maroon">Yab</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Yb</font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E3:24"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<b>hence </b><a NAME="E9:24_2_3_1_3_3_1_2_3_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsb</font>
 <b>by </b><i><a class="ref" href="waybel22.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel22.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:24_2_3_1_3_3_1_2_3"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Xsb</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Xsc</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T7">SETFAM_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:24_2_3_1_3_3_1_2_3"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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="txt" href="waybel22.html#E2"><i><font color="Green">Lemma40</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E9:24_2_3_1_3_3_1_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">Yab</font> ) </span>}</span> ,<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="yellow_1.html#T3">YELLOW_1:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E10:24_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">h</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">Xs</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/><a NAME="E11:24_2"/><i><font color="Green">E98</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">Xs</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>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>set </b><font color="Maroon">Xsf</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">FF</font> </span>}</span> ;<br/><b>set </b><font color="Maroon">FUF</font> =  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_2_4_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xs</font>
 ;<br/><b>then consider </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_2_4_1"/>
<font color="Maroon">t</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="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>and </b><br/><a NAME="E4:24_2_4_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font>
 <b>and </b><br/><a NAME="E5:24_2_4_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>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="E7:24_2_4_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">t</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"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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> ,<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>and </b><br/><a NAME="E8:24_2_4_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">Xsi</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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/><a NAME="E9:24_2_4_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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="tarski.html#R1">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:24_2_4_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">xz</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="E3:24_2_4_1_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xz</font>
 <b>and </b><br/><a NAME="E4:24_2_4_1_1"/>
 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">xz</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>thus </b><a NAME="E5:24_2_4_1_1"/>
<font color="Maroon">r</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="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>, </i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Xsi</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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>  as   <a href="struct_0.html#NM2">Subset</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="E11:24_2_4_1"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">h</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">Xsi</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/><a NAME="E12:24_2_4_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Xsi</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>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E13:24_2_4_1"/><b>then </b><i><font color="Green">E186</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Xsi</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/><b>set </b><font color="Maroon">Xsif</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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_4_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_2_4_1_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font>
 ;<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="E3:24_2_4_1_2_1"/>
<font color="Maroon">v</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="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>and </b><br/><a NAME="E4:24_2_4_1_2_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsi</font>
 <b>and </b><br/><a NAME="E5:24_2_4_1_2_1"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</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="E7:24_2_4_1_2_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E8:24_2_4_1_2_1"/><i><font color="Green">E191</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">Y</font> )
 <b>by </b><i><a class="txt" href="waybel22.html#E2:24"><i><font color="Green">E35</font></i></a></i>;<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="E10:24_2_4_1_2_1"/><i><font color="Green">E192</font></i>: 
( <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="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#E3:24_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E11:24_2_4_1_2_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:24"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E12:24_2_4_1_2_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E13:24_2_4_1_2_1"/>
<font color="Maroon">u</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>, <a class="txt" href="waybel22.html#E3:24"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel22.html#E3:24_2"><i><font color="Green">E38</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:24_2_4_1_2"/>
<font color="Maroon">u</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">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="E4:24_2_4_1_2"/><i><font color="Green">E193</font></i>: 
<font color="Maroon">u</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="E5:24_2_4_1_2"/><i><font color="Green">E195</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">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="E7:24_2_4_1_2"/><i><font color="Green">E196</font></i>: 
( <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="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/>;<br/><a NAME="E8:24_2_4_1_2"/><i><font color="Green">E197</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsi</font>
 <b>by </b><i/>;<br/><a NAME="E9:24_2_4_1_2"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:24"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E10:24_2_4_1_2"/><b>then </b>
<font color="Maroon">h</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> <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>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E11:24_2_4_1_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:24"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel22.html#E3:24_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div><a NAME="E15:24_2_4_1"/><b>then </b>
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font> <a href="hidden.html#R1">=</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><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E16:24_2_4_1"/>
<font color="Maroon">s</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">FF</font> </span>}</span> 
 <b>by </b><i>, <a class="txt" href="waybel22.html#E1:24"><i><font color="Green">E31</font></i></a>, , </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:24_2_4"/>
<font color="Maroon">s</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">FF</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="E4:24_2_4"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">s</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="E5:24_2_4"/><i><font color="Green">E199</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font>
 ;<br/><b>set </b><font color="Maroon">Xsi</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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/><a NAME="E6:24_2_4"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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="tarski.html#R1">c=</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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="E1:24_2_4_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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">xz</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="E3:24_2_4_2"/><i><font color="Green">E200</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">xz</font>
 <b>and </b><br/><a NAME="E4:24_2_4_2"/>
 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">xz</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>thus </b><a NAME="E5:24_2_4_2"/>
<font color="Maroon">r</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="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="txt" href="waybel22.html#E1:24"><i><font color="Green">E31</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Xsi</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></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>  as   <a href="struct_0.html#NM2">Subset</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="E8:24_2_4"/><i><font color="Green">E201</font></i>: 
<font color="Maroon">h</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <font color="Maroon">Xsi</font>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/><a NAME="E9:24_2_4"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Xsi</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>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E10:24_2_4"/><b>then </b><i><font color="Green">E203</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Xsi</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:24"><i><font color="Green">E31</font></i></a>, <a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a></i>;<br/><b>set </b><font color="Maroon">Xsif</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/><a NAME="E11:24_2_4"/><i><font color="Green">E205</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Xsi</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xs</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_2_4_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">u</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_2_4_3_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font>
 ;<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="E3:24_2_4_3_1"/>
<font color="Maroon">v</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="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>and </b><br/><a NAME="E4:24_2_4_3_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsi</font>
 <b>and </b><br/><a NAME="E5:24_2_4_3_1"/><i><font color="Green">E211</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</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="E7:24_2_4_3_1"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E8:24_2_4_3_1"/><i><font color="Green">E213</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">Y</font> )
 <b>by </b><i/>;<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="E10:24_2_4_3_1"/><i><font color="Green">E214</font></i>: 
( <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="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#E3:24"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E11:24_2_4_3_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E3:24_2"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E12:24_2_4_3_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E13:24_2_4_3_1"/>
<font color="Maroon">u</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><a class="txt" href="waybel22.html#E2:24"><i><font color="Green">E35</font></i></a>, , <a class="txt" href="waybel22.html#E3:24"><i><font color="Green">E37</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:24_2_4_3"/>
<font color="Maroon">u</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">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="E4:24_2_4_3"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">u</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="E5:24_2_4_3"/><i><font color="Green">E216</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">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="E7:24_2_4_3"/><i><font color="Green">E217</font></i>: 
( <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="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#E2:24"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E8:24_2_4_3"/><i><font color="Green">E219</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Xsi</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:24"><i><font color="Green">E35</font></i></a></i>;<br/><a NAME="E9:24_2_4_3"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/><a NAME="E10:24_2_4_3"/><b>then </b>
<font color="Maroon">h</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> <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>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E11:24_2_4_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E3:24"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div><a NAME="E13:24_2_4"/><b>then </b>
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xsi</font> <a href="hidden.html#R1">=</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><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>hence </b><a NAME="E14:24_2_4"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xs</font>
 <b>by </b><i>, , , <a class="ref" href="funct_2.html#T43">FUNCT_2:43</a></i>;<br/></div><b>end;</b></div><a NAME="E13:24_2"/><b>then </b><i><font color="Green">E220</font></i>: 
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xs</font> <a href="hidden.html#R1">=</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">FF</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><b>thus </b><font color="Maroon">h'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fi</font> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Xs</font></span>)</span>
<b>by </b><i>, , <a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
<font color="Maroon">F'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Fi</font>
<b>by </b><i>, , </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:24"/>
<font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
