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

<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">F</font> = <font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> ;<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">FUF</font> =  <a href="waybel22.html#K1" target="_self">FixedUltraFilters</a> <font color="Maroon">X</font>;<br/>
<a NAME="E1:22_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p2">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<a NAME="E2:22_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/>
<a NAME="E3:22_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_1.html#K1">RelIncl</a> <span class="p2">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font></span>)</span></span>)</span> #)
 
<b>by </b><i><a class="ref" href="yellow_1.html#D1">YELLOW_1:def 1</a></i>;<br/>
<a NAME="E4:22_1_1"/><b>then </b><i><font color="Green">E38</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> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/>
<b>let </b><font color="Maroon">Fs</font> be   <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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a><br/>

<b>assume </b><a NAME="E5:22_1_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">Fs</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>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D30">WAYBEL_0:def 30</a><br/>

<b>thus </b><a NAME="E6:22_1_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_1"/>
( <font color="Maroon">Fs</font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">Fs</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_1"><b>suppose </b></a><a NAME="E1:22_1_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Fs</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:22_1_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T149">RELAT_1:149</a></i>;<br/><a NAME="E3:22_1_1_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">Fs</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="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>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:22_1_1_1_1"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Fs</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_1_2"><b>suppose </b></a><a NAME="E1:22_1_1_1_2"/>
not <font color="Maroon">Fs</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">Fs'</font> = <font color="Maroon">Fs</font> 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> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Fs'</font>
 ;<br/><a NAME="E2:22_1_1_1_2_1"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K2">InclPoset</a> <span class="p2">(<span class="default"><a href="waybel_0.html#K8">Filt</a> <span class="p3">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 ;<br/><b>hence </b><a NAME="E3:22_1_1_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E4:22_1_1_1_2"/><b>then </b>
<font color="Maroon">Fs'</font> is <a href="setfam_1.html#V1">with_non-empty_elements</a>
 <b>by </b><i><a class="ref" href="setfam_1.html#D9">SETFAM_1:def 9</a></i>;<br/><b>then reconsider </b><font color="Maroon">K</font> =  <a href="pralg_3.html#K1">id</a> <font color="Maroon">Fs'</font> as  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">Fs'</font> <b>by </b><i/>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></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">a<sub>2</sub></font> ) </span>}</span> ,<font color="Maroon">L</font>;<br/><a NAME="E6:22_1_1_1_2"/><i><font color="Green">E61</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fs'</font> holds <br/>for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">s</font>,<font color="Olive">i</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:22_1_1_1_2_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fs'</font>
 ;<br/>

<b>let </b><font color="Maroon">s</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E2:22_1_1_1_2_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/>

<b>take </b><font color="Maroon">y</font> =  <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">s</font> ) </span>}</span> ,<font color="Maroon">L</font>;<br/>

<a NAME="E3:22_1_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E4:22_1_1_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>

<b>thus </b><a NAME="E5:22_1_1_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">s</font>,<font color="Maroon">i</font>]
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">FD</font> being    <a href="pboole.html#M3">ManySortedFunction</a> of <font color="Maroon">K</font>,<font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:22_1_1_1_2"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">i</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fs'</font> holds <br/> ex <font color="Olive">g</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> st <br/>( <font color="Olive">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">s</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">s</font>,<font color="Olive">s</font>,<font color="Olive">i</font>] ) )
 <b>from </b><i><a class="ref" href="mssubfam.html#S1">MSSUBFAM:sch 1</a>();<br/></i><a NAME="E9:22_1_1_1_2"/><i><font color="Green">E66</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fs'</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</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">a<sub>1</sub></font> </span>}</span> ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font>] <b>means </b> <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/><a NAME="E10:22_1_1_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_3_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_1_2_3_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>and </b><br/><a NAME="E4:22_1_1_1_2_3_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:22_1_1_1_2_3_1_1"/><i><font color="Green">E70</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E7:22_1_1_1_2_3_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E8:22_1_1_1_2_3_1_1"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>,<font color="Olive">u</font>,<font color="Maroon">s</font>]
 <b>by </b><i/>;<br/><a NAME="E9:22_1_1_1_2_3_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</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">u</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:22_1_1_1_2_3_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E11:22_1_1_1_2_3_1_1"/>
<font color="Maroon">s</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>
 ;<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="E13:22_1_1_1_2_3_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i>, , , , , <a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/><a NAME="E15:22_1_1_1_2_3_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>hence </b><a NAME="E16:22_1_1_1_2_3_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i><a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, , <a class="txt" href="waybel22.html#E1:22_1_1"><i><font color="Green">E31</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:22_1_1_1_2_3_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">s</font>)
 ;<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="E4:22_1_1_1_2_3_1"/><i><font color="Green">E76</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"><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="E5:22_1_1_1_2_3_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font>
 ;<br/><a NAME="E6:22_1_1_1_2_3_1"/><i><font color="Green">E78</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E7:22_1_1_1_2_3_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><a NAME="E8:22_1_1_1_2_3_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</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="funct_1.html#T35">FUNCT_1:35</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font><b> such that </b><br/><a NAME="E10:22_1_1_1_2_3_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font>
 <b>and </b><br/><a NAME="E11:22_1_1_1_2_3_1"/><i><font color="Green">E83</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>,<font color="Olive">u</font>,<font color="Maroon">s</font>]
 <b>by </b><i/>;<br/><a NAME="E12:22_1_1_1_2_3_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Y</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>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="E13:22_1_1_1_2_3_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="waybel22.html#E1:22_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:22_1_1_1_2_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">s</font>]
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon">FD</font> = <font color="Maroon">FD</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_4_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</font>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_1_2_4_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">s</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="E4:22_1_1_1_2_4_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fs</font>
 <b>and </b><br/><a NAME="E5:22_1_1_1_2_4_1"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T115">FUNCT_2:115</a></i>;<br/><b>reconsider </b><font color="Maroon">s'</font> = <font color="Maroon">s</font> as    <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> <b>by </b><i/>;<br/><a NAME="E7:22_1_1_1_2_4_1"/><i><font color="Green">E88</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s'</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s'</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> <b>by </b><i><a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a></i>;<br/><a NAME="E9:22_1_1_1_2_4_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">s</font>]
 <b>by </b><i/>;<br/><a NAME="E10:22_1_1_1_2_4_1"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>;<br/><b>hence </b><a NAME="E11:22_1_1_1_2_4_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 <b>by </b><i><a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:22_1_1_1_2_4"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font><b> such that </b><br/><a NAME="E4:22_1_1_1_2_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK6" target="_self">Sup</a> 
 <b>by </b><i><a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/><a NAME="E5:22_1_1_1_2_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">s</font>]
 <b>by </b><i/>;<br/><b>then </b><span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> = 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
;<br/><b>hence </b><a NAME="E7:22_1_1_1_2_4"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</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="E13:22_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E14:22_1_1_1_2"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/><a NAME="E15:22_1_1_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> holds   <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Olive">j</font></span>)</span> is <a href="waybel_0.html#V1">directed</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font>;<br/><b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">m</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D1">WAYBEL_0:def 1</a><br/>



<b>assume </b><a NAME="E1:22_1_1_1_2_5"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> )
 ;<br/>

<b>then consider </b><font color="Maroon">kd</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_1_2_5"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">kd</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:22_1_1_1_2_5"/><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">kd</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">md</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:22_1_1_1_2_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">md</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E7:22_1_1_1_2_5"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">md</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1"><i><font color="Green">Lemma27</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:22_1_1_1_2_5"/><i><font color="Green">E96</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E9:22_1_1_1_2_5"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>;<br/>
<a NAME="E10:22_1_1_1_2_5"/>
<font color="Maroon">j</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>
 
;<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="E12:22_1_1_1_2_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">kd</font> = <font color="Maroon">kd</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, , , </i>;<br/>
<b>reconsider </b><font color="Maroon">md</font> = <font color="Maroon">md</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, , , </i>;<br/>
<b>consider </b><font color="Maroon">nd</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="E16:22_1_1_1_2_5"/><i><font color="Green">E99</font></i>: 
( <font color="Maroon">nd</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FF</font> &amp; <font color="Maroon">nd</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">kd</font> &amp; <font color="Maroon">nd</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">md</font> )
 <b>by </b><i>, , , , <a class="ref" href="waybel_0.html#D2">WAYBEL_0:def 2</a></i>;<br/>
<a NAME="E17:22_1_1_1_2_5"/><i><font color="Green">E100</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">j</font>)
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">nd</font>;<br/>
<a NAME="E18:22_1_1_1_2_5"/>
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">nd</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, , , <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<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="E20:22_1_1_1_2_5"/><i><font color="Green">E102</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">nd</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="E21:22_1_1_1_2_5"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">j</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">n</font> = <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">nd</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i><a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">n</font>
;<br/>

<b>thus </b><a NAME="E23:22_1_1_1_2_5"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>

<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>,<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font><b> such that </b><br/><a NAME="E25:22_1_1_1_2_5"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E26:22_1_1_1_2_5"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font>,<font color="Olive">u</font>,<font color="Maroon">j</font>]
 <b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">kds</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">kd</font> ) </span>}</span> ;<br/>
<a NAME="E27:22_1_1_1_2_5"/><i><font color="Green">E184</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">kd</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">kd</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<b>set </b><font color="Maroon">nds</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">nd</font> ) </span>}</span> ;<br/>
<a NAME="E28:22_1_1_1_2_5"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">nd</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">nd</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<a NAME="E29:22_1_1_1_2_5"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">nd</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">kd</font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<a NAME="E30:22_1_1_1_2_5"/><i><font color="Green">E188</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">kd</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E31:22_1_1_1_2_5"/><i><font color="Green">E189</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">nd</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E32:22_1_1_1_2_5"/>
<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">nd</font> ) </span>}</span>  <a href="tarski.html#R1">c=</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">kd</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:22_1_1_1_2_5_1"/>
<font color="Maroon">w</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">nd</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="E3:22_1_1_1_2_5_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">w</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="E4:22_1_1_1_2_5_1"/><i><font color="Green">E192</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">nd</font> )
 ;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:22_1_1_1_2_5_1"/><i><font color="Green">E193</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">nd</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E7:22_1_1_1_2_5_1"/>
<font color="Maroon">w</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">kd</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E33:22_1_1_1_2_5"/>
<font color="Maroon">k</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="waybel22.html#E2"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="waybel22.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel22.html#T2" target="_self">Th2</a>, , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>

<b>set </b><font color="Maroon">mds</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">md</font> ) </span>}</span> ;<br/>
<a NAME="E34:22_1_1_1_2_5"/><i><font color="Green">E195</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">md</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">md</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E35:22_1_1_1_2_5"/><i><font color="Green">E196</font></i>: 
<font color="Maroon">nd</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">md</font>
 
<b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="yellow_1.html#T2">YELLOW_1:2</a></i>;<br/>
<a NAME="E36:22_1_1_1_2_5"/><i><font color="Green">E197</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">md</font> ) </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E37:22_1_1_1_2_5"/>
<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">nd</font> ) </span>}</span>  <a href="tarski.html#R1">c=</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">md</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w</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="E1:22_1_1_1_2_5_2"/>
<font color="Maroon">w</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">nd</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="E3:22_1_1_1_2_5_2"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">w</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="E4:22_1_1_1_2_5_2"/><i><font color="Green">E199</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">nd</font> )
 ;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:22_1_1_1_2_5_2"/><i><font color="Green">E200</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">nd</font> )
 <b>by </b><i><a class="ref" href="waybel22.html#T4" target="_self">Th4</a></i>;<br/>
<b>thus </b><a NAME="E7:22_1_1_1_2_5_2"/>
<font color="Maroon">w</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">md</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="waybel22.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel22.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E38:22_1_1_1_2_5"/>
<font color="Maroon">m</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E1:22_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="waybel22.html#T1" target="_self">Th1</a>, , , , <a class="ref" href="yellow_0.html#T35">YELLOW_0:35</a></i>;<br/>


</div><b>end;</b></div><b>then </b><i><font color="Green">E201</font></i>:  <a href="yellow_2.html#NK7" target="_self">Inf</a>  = 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="waybel_5.html#T19">WAYBEL_5:19</a></i>
<br/>.= 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span></span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_2.html#D5">YELLOW_2:def 5</a></i>
;<br/><a NAME="E17:22_1_1_1_2"/>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Fs'</font> is   <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>by </b><i><a class="ref" href="waybel16.html#T9">WAYBEL16:9</a></i>;<br/><a NAME="E18:22_1_1_1_2"/><b>then </b>
 <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Fs'</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>
 ;<br/><b>then reconsider </b><font color="Maroon">mFs</font> =  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Fs'</font> as    <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> <b>by </b><i/>;<br/><a NAME="E20:22_1_1_1_2"/><i><font color="Green">E203</font></i>: 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Fs'</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <font color="Maroon">Fs'</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T10">WAYBEL16:10</a></i>;<br/><b>set </b><font color="Maroon">smFs</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">mFs</font> </span>}</span> ;<br/><a NAME="E21:22_1_1_1_2"/><i><font color="Green">E205</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <font color="Maroon">Fs'</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">mFs</font> </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>reconsider </b><font color="Maroon">pdFD</font> =  <a href="card_3.html#K4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">FD</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="fraenkel.html#V1">functional</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>reconsider </b><font color="Maroon">dFFD</font> = <span class="p1">(<span class="default"><a href="card_3.html#K4">product</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">FD</font></span>)</span></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Fs'</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">pdFD</font> ;<br/><b>reconsider </b><font color="Maroon">FFD</font> =  <a href="waybel_5.html#K2">Frege</a> <font color="Maroon">FD</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">dFFD</font>,<font color="Maroon">L</font> ;<br/><a NAME="E4:22_1_1_1_2_7"/><i><font color="Green">E209</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FFD</font> <a href="hidden.html#R1">=</a> <font color="Maroon">pdFD</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">pdFD</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</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="Maroon">a<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> </span>}</span> ;<br/><div><i><font color="Green">E211</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">pdFD</font>;<br/><a NAME="E1:22_1_1_1_2_7_1"/><i><font color="Green">E212</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_7_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_1_2_7_1_1_1"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>and </b><br/><a NAME="E4:22_1_1_1_2_7_1_1_1"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:22_1_1_1_2_7_1_1_1"/><i><font color="Green">E215</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> <b>by </b><i>, , <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font><b> such that </b><br/><a NAME="E8:22_1_1_1_2_7_1_1_1"/><i><font color="Green">E216</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E9:22_1_1_1_2_7_1_1_1"/><i><font color="Green">E217</font></i>: 
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>,<font color="Olive">v</font>,<font color="Maroon">u</font>]
 <b>by </b><i/>;<br/><a NAME="E10:22_1_1_1_2_7_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:22_1_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/><a NAME="E12:22_1_1_1_2_7_1_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font></span>)</span> <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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a></i>;<br/><b>hence </b><a NAME="E13:22_1_1_1_2_7_1_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i>, , <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:22_1_1_1_2_7_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font><b> such that </b><br/><a NAME="E4:22_1_1_1_2_7_1_1"/><i><font color="Green">E219</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"><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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E5:22_1_1_1_2_7_1_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> ;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font><b> such that </b><br/><a NAME="E8:22_1_1_1_2_7_1_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">u</font>
 <b>and </b><br/><a NAME="E9:22_1_1_1_2_7_1_1"/><i><font color="Green">E222</font></i>: 
for <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font>,<font color="Olive">v</font>,<font color="Maroon">u</font>]
 <b>by </b><i/>;<br/><a NAME="E10:22_1_1_1_2_7_1_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">u</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E11:22_1_1_1_2_7_1_1"/><b>then </b>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i>, , <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/><a NAME="E12:22_1_1_1_2_7_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font></span>)</span> <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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E13:22_1_1_1_2_7_1_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, , , , <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:22_1_1_1_2_7_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_7_2"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">s</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">pdFD</font><b> such that </b><br/><a NAME="E3:22_1_1_1_2_7_2"/><i><font color="Green">E223</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="yellow_2.html#NK7" target="_self">Inf</a> 
 <b>by </b><i><a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/><a NAME="E4:22_1_1_1_2_7_2"/><i><font color="Green">E224</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/><a NAME="E5:22_1_1_1_2_7_2"/><i><font color="Green">E225</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font>
 <b>by </b><i>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/><a NAME="E6:22_1_1_1_2_7_2"/>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:22_1_1_1_2_7_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">te</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_1_2_7_2_1"/><i><font color="Green">E227</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">te</font> &amp; <font color="Maroon">te</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:22_1_1_1_2_7_2_1"/><i><font color="Green">E228</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">te</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E6:22_1_1_1_2_7_2_1"/>
<font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel_5.html#T6">WAYBEL_5:6</a></i>;<br/>
<b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font></span>)</span> = 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
;<br/>
<a NAME="E8:22_1_1_1_2_7_2_1"/><b>then </b><i><font color="Green">E229</font></i>: 
<font color="Maroon">te</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/>
<a NAME="E9:22_1_1_1_2_7_2_1"/>
<font color="Maroon">u</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/>
<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="E11:22_1_1_1_2_7_2_1"/><i><font color="Green">E230</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E12:22_1_1_1_2_7_2_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Y</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font></span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/><b>set </b><font color="Maroon">Ys</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="22_1_1_1_2_7_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1_1_1_2_7_2_2"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fs'</font>
 ;<br/><a NAME="E2:22_1_1_1_2_7_2_2"/><b>then </b>
<font color="Maroon">Z</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>
 ;<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="E4:22_1_1_1_2_7_2_2"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FF</font>
 <b>by </b><i/>;<br/><a NAME="E5:22_1_1_1_2_7_2_2"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, , <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E6:22_1_1_1_2_7_2_2"/><b>then </b><i><font color="Green">E234</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E7:22_1_1_1_2_7_2_2"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_5.html#T9">WAYBEL_5:9</a></i>;<br/><a NAME="E8:22_1_1_1_2_7_2_2"/>
<font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel_5.html#T6">WAYBEL_5:6</a></i>;<br/><b>then </b><i><font color="Green">E236</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font></span>)</span> = 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">Z</font>
<b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
;<br/><b>reconsider </b><font color="Maroon">sZ</font> = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, , <a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">Y'</font> = <font color="Maroon">Y</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="yellow_1.html#T4">YELLOW_1:4</a></i>;<br/><a NAME="E12:22_1_1_1_2_7_2_2"/>
<font color="Maroon">sZ</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">Y'</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E3:22_1_1"><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="E13:22_1_1_1_2_7_2_2"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/></div><b>end;</b></div><a NAME="E9:22_1_1_1_2_7_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">mFs</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><a NAME="E10:22_1_1_1_2_7_2"/><b>then </b><i><font color="Green">E238</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> ) </span>}</span> ,<font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">Y</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> : <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">mFs</font> </span>}</span> 
 ;<br/><b>set </b><font color="Maroon">idFFDs</font> = <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> ;<br/><a NAME="E11:22_1_1_1_2_7_2"/><i><font color="Green">E240</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">t</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="E1:22_1_1_1_2_7_2_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font><b> such that </b><br/><a NAME="E3:22_1_1_1_2_7_2_3"/><i><font color="Green">E241</font></i>: 
<font color="Maroon">t</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:22_1_1_1_2_7_2_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/>
<a NAME="E5:22_1_1_1_2_7_2_3"/>
<font color="Maroon">t</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</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="E1:22_1_1_1_2_7_2_3_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">t</font>
 ;<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="E3:22_1_1_1_2_7_2_3_1"/><i><font color="Green">E242</font></i>: 
<font color="Maroon">v</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="E4:22_1_1_1_2_7_2_3_1"/><i><font color="Green">E243</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</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="E6:22_1_1_1_2_7_2_3_1"/><i><font color="Green">E244</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>
 <b>and </b><br/><a NAME="E7:22_1_1_1_2_7_2_3_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:22_1_1_1_2_7_2_3_1"/>
 <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#E2"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:22_1_1_1_2_7_2_3_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:22_1_1_1_2_7_2_3"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_7_2_4_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font><b> such that </b><br/><a NAME="E3:22_1_1_1_2_7_2_4_1"/><i><font color="Green">E245</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"><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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:22_1_1_1_2_7_2_4_1"/><i><font color="Green">E246</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><a NAME="E5:22_1_1_1_2_7_2_4_1"/>
<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_2_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</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="E1:22_1_1_1_2_7_2_4_1_1"/>
<font color="Maroon">v</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</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="E3:22_1_1_1_2_7_2_4_1_1"/><i><font color="Green">E247</font></i>: 
<font color="Maroon">v</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="E4:22_1_1_1_2_7_2_4_1_1"/><i><font color="Green">E248</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 ;<br/>
<b>consider </b><font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E6:22_1_1_1_2_7_2_4_1_1"/><i><font color="Green">E249</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>
 <b>and </b><br/><a NAME="E7:22_1_1_1_2_7_2_4_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E8:22_1_1_1_2_7_2_4_1_1"/>
 <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="ref" href="waybel22.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E9:22_1_1_1_2_7_2_4_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">Y1</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/><a NAME="E7:22_1_1_1_2_7_2_4_1"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>hence </b><a NAME="E8:22_1_1_1_2_7_2_4_1"/>
<font color="Maroon">t</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#NK4" target="_self">inf</a> <font color="Olive">YY</font></span>)</span> where <font color="Olive">YY</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:22_1_1_1_2_7_2_4"/>
<font color="Maroon">t</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#NK4" target="_self">inf</a> <font color="Olive">YY</font></span>)</span> where <font color="Olive">YY</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span>  </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">YY</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:22_1_1_1_2_7_2_4"/><i><font color="Green">E250</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">YY</font>
 <b>and </b><br/><a NAME="E5:22_1_1_1_2_7_2_4"/><i><font color="Green">E251</font></i>: 
<font color="Maroon">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> 
 ;<br/><b>consider </b><font color="Maroon">u1</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font><b> such that </b><br/><a NAME="E7:22_1_1_1_2_7_2_4"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">YY</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u1</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E8:22_1_1_1_2_7_2_4"/><i><font color="Green">E254</font></i>: 
<font color="Maroon">u1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a></i>;<br/><b>thus </b><a NAME="E9:22_1_1_1_2_7_2_4"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i><a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, , <a class="txt" href="waybel22.html#E2"><i><font color="Green">Lemma40</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E13:22_1_1_1_2_7_2"/><b>then </b><i><font color="Green">E255</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>) <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">YY</font></span>)</span> where <font color="Olive">YY</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">YY</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><i><font color="Green">E256</font></i>:  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">L</font> = 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>),<font color="Maroon">L</font>
<b>by </b><i><a class="txt" href="waybel22.html#E1:22_1_1"><i><font color="Green">E31</font></i></a></i>
<br/>.= 
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <span class="p2">{<span class="default"> <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="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> </span>)</span>,<font color="Maroon">L</font>
<b>by </b><i>, <a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, </i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_7_2_6_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">te</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_1_1_1_2_7_2_6_1"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">te</font>
 <b>and </b><br/><a NAME="E4:22_1_1_1_2_7_2_6_1"/><i><font color="Green">E258</font></i>: 
<font color="Maroon">te</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font><b> such that </b><br/><a NAME="E6:22_1_1_1_2_7_2_6_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">te</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E7:22_1_1_1_2_7_2_6_1"/><i><font color="Green">E260</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E2"><i><font color="Green">Lemma40</font></i></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="E9:22_1_1_1_2_7_2_6_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">t</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="E10:22_1_1_1_2_7_2_6_1"/><i><font color="Green">E262</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i>, <a class="ref" href="waybel22.html#T1" target="_self">Th1</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="E12:22_1_1_1_2_7_2_6_1"/><i><font color="Green">E263</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i/>;<br/><a NAME="E13:22_1_1_1_2_7_2_6_1"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E14:22_1_1_1_2_7_2_6_1"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>hence </b><a NAME="E15:22_1_1_1_2_7_2_6_1"/>
<font color="Maroon">t</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="ref" href="waybel22.html#T2" target="_self">Th2</a>, </i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:22_1_1_1_2_7_2_6"/>
<font color="Maroon">t</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="E4:22_1_1_1_2_7_2_6"/><i><font color="Green">E264</font></i>: 
<font color="Maroon">t</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="E5:22_1_1_1_2_7_2_6"/><i><font color="Green">E265</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="E7:22_1_1_1_2_7_2_6"/><i><font color="Green">E266</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"><i><font color="Green">Lemma40</font></i></a></i>;<br/><b>consider </b><font color="Maroon">ze</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:22_1_1_1_2_7_2_6"/><i><font color="Green">E268</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ze</font> &amp; <font color="Maroon">ze</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">s</font> )
 <b>by </b><i><a class="ref" href="waybel22.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:22_1_1_1_2_7_2_6"/><i><font color="Green">E269</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> &amp; <font color="Maroon">ze</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> <b>by </b><i>, <a class="ref" href="waybel22.html#T2" target="_self">Th2</a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E13:22_1_1_1_2_7_2_6"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">t</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">x1</font></span>)</span></span>)</span> where <font color="Olive">x1</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">x1</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> 
 <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/><a NAME="E14:22_1_1_1_2_7_2_6"/>
<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">x1</font></span>)</span></span>)</span> where <font color="Olive">x1</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">x1</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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="waybel22.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E15:22_1_1_1_2_7_2_6"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E16:22_1_1_1_2_7_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p4">(<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">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span>  where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span>  <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="E17:22_1_1_1_2_7_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="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">mFs</font> </span>}</span> 
 <b>by </b><i>, <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E7:22_1_1_1_2_7"/>
<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="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">mFs</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="E9:22_1_1_1_2_7"/><i><font color="Green">E271</font></i>: 
<font color="Maroon">r</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="E10:22_1_1_1_2_7"/><i><font color="Green">E272</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">mFs</font>
 ;<br/><b>set </b><font color="Maroon">s</font> = <font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font>;<br/><b>set </b><font color="Maroon">s'</font> = <font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font>;<br/><i><font color="Green">E273</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">FD</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font>
<b>by </b><i><a class="ref" href="funct_6.html#T89">FUNCT_6:89</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">w</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1_1_1_2_7_4"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">FD</font></span>)</span>
 ;<br/><a NAME="E2:22_1_1_1_2_7_4"/><b>then </b><i><font color="Green">E274</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Fs'</font>
 <b>by </b><i>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E3:22_1_1_1_2_7_4"/><b>then </b><i><font color="Green">E275</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_6.html#T31">FUNCT_6:31</a></i>;<br/><a NAME="E4:22_1_1_1_2_7_4"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>,<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="pboole.html#D18">PBOOLE:def 18</a></i>;<br/><a NAME="E5:22_1_1_1_2_7_4"/>
<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>then </b><i><font color="Green">E277</font></i>:  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FD</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font></span>)</span> = 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
<b>by </b><i><a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
<font color="Maroon">w</font>
<b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
;<br/><a NAME="E7:22_1_1_1_2_7_4"/>
<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E8:22_1_1_1_2_7_4"/>
<span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">FD</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, , <a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">pdFD</font> <b>by </b><i>, <a class="ref" href="card_3.html#T18">CARD_3:18</a></i>;<br/><a NAME="E14:22_1_1_1_2_7"/><i><font color="Green">E278</font></i>: 
 <a href="yellow_2.html#NK7" target="_self">Inf</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span>)</span>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="yellow_2.html#D6">YELLOW_2:def 6</a></i>;<br/><a NAME="E15:22_1_1_1_2_7"/><i><font color="Green">E279</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i><a class="txt" href="waybel22.html#E1:22_1_1"><i><font color="Green">E31</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1_1_2_7_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">t</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:22_1_1_1_2_7_5_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 ;<br/><b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font><b> such that </b><br/><a NAME="E3:22_1_1_1_2_7_5_1"/><i><font color="Green">E280</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"><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">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> ) </span>}</span> ,<font color="Maroon">L</font>
 <b>and </b><br/><a NAME="E4:22_1_1_1_2_7_5_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 ;<br/><a NAME="E5:22_1_1_1_2_7_5_1"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E6:22_1_1_1_2_7_5_1"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:22_1_1_1_2_7_5"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E3:22_1_1_1_2_7_5"/><b>then </b><i><font color="Green">E281</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font>;<br/><a NAME="E5:22_1_1_1_2_7_5"/><i><font color="Green">E282</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/><a NAME="E6:22_1_1_1_2_7_5"/><i><font color="Green">E283</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">FD</font>
 <b>by </b><i>, <a class="ref" href="waybel_5.html#T8">WAYBEL_5:8</a></i>;<br/><a NAME="E7:22_1_1_1_2_7_5"/><i><font color="Green">E284</font></i>: 
( <font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">Fs'</font> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">Y</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fs'</font> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/><a NAME="E8:22_1_1_1_2_7_5"/>
<font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E9:22_1_1_1_2_7_5"/>
<font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">s</font>)
 <b>by </b><i>, <a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="waybel22.html#E4:22_1_1"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="waybel22.html#E2"><i><font color="Green">Lemma40</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E17:22_1_1_1_2_7"/><b>then </b><i><font color="Green">E285</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#K2">"/\"</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p5">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Olive">x</font></span>)</span></span>)</span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <font color="Maroon">X</font></span>)</span> :  ex <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">z</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Olive">u</font> ) </span>}</span> ,<font color="Maroon">L</font></span>)</span> where <font color="Olive">u</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">Fs'</font> : <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">FFD</font> <a href="waybel_5.html#K1">.</a> <font color="Maroon">s</font></span>)</span> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E18:22_1_1_1_2_7"/>
 <a href="yellow_0.html#K2">"/\"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">r</font></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/><b>hence </b><a NAME="E19:22_1_1_1_2_7"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>
 <b>by </b><i><a class="txt" href="waybel22.html#E2:22_1_1"><i><font color="Green">E35</font></i></a>, , <a class="txt" href="waybel22.html#E3:22_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="waybel_5.html#T14">WAYBEL_5:14</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E23:22_1_1_1_2"/>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">Fs</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="waybel22.html#K2" target="_self">-extension_to_hom</a> </span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">Fs</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="tarski.html#T2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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