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

<b>let </b><font color="Maroon">L</font> be  <a href="waybel_1.html#V2">distributive</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_0.html#NM4">Filter</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">I</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> ;<br/>
<a NAME="E2:37"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 ;<br/><a NAME="E2:37_1"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> )
 ;<br/><b>hence </b><a NAME="E3:37_1"/>
( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font> )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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><b>that </b><br/><a NAME="E1:37_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span>  )
 <b>and </b><br/><a NAME="E2:37_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/><a NAME="E3:37_2"/>
<font color="Maroon">Z</font> <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="37_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:37_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:37_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:37_2_1"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> )
 
;<br/>
<b>hence </b><a NAME="E4:37_2_1"/>
<font color="Maroon">x</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><b>then reconsider </b><font color="Maroon">ZI</font> = <font color="Maroon">Z</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">L</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:37_2_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZI</font>
 ;<br/><a NAME="E2:37_2_2"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:37_2_2"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> )
 ;<br/><b>hence </b><a NAME="E4:37_2_2"/>
<font color="Maroon">A</font> is <a href="waybel_0.html#V12">lower</a>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">J</font> =  <a href="setfam_1.html#K5">union</a> <font color="Maroon">ZI</font> as  <a href="waybel_0.html#V12">lower</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel_0.html#T26">WAYBEL_0:26</a></i>;<br/><div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_3"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">Z</font>;<br/><a NAME="E2:37_2_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i/>;<br/><a NAME="E3:37_2_3"/><b>then </b>
( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font> )
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><b>hence </b><a NAME="E4:37_2_3"/>
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">J</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E5:37_2_3"/>
not <font color="Maroon">J</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E6:37_2_3"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZI</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZI</font> )
 ;<br/><a NAME="E7:37_2_3"/><b>then </b>
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="xboole_0.html#R3">are_c=-comparable</a> 
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/><a NAME="E8:37_2_3"/><b>then </b>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/><a NAME="E9:37_2_3"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> or <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/><b>hence </b><a NAME="E10:37_2_3"/>
 ex <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZI</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:37_2_4"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">ZI</font>
 ;<br/><a NAME="E2:37_2_4"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:37_2_4"/><b>then </b>
 ex <font color="Olive">P</font> being  <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">P</font> &amp; <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> )
 ;<br/><b>hence </b><a NAME="E4:37_2_4"/>
<font color="Maroon">A</font> is <a href="waybel_0.html#V1">directed</a>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">J</font> = <font color="Maroon">J</font> as   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="waybel_7.html#T3" target="_self">Th3</a>, <a class="ref" href="waybel_0.html#T46">WAYBEL_0:46</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:37_2_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_2_5"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:37_2_5"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">F</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:37_2_5"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E11:37_2"/><b>then </b>
<font color="Maroon">J</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E12:37_2"/>
 <a href="tarski.html#K3">union</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel_7.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:37"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span>  &amp; ( for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span>  &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font> holds <br/> not <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Z</font> ) )
 <b>by </b><i>, <a class="ref" href="orders_1.html#T177">ORDERS_1:177</a></i>;<br/>
<b>consider </b><font color="Maroon">P</font> being   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:37"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D1" target="_self">WAYBEL_7:def 1</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:37_3"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/><b>set </b><font color="Maroon">Px</font> =  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">Py</font> =  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span>;<br/><a NAME="E2:37_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:37_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="waybel_0.html#T61">WAYBEL_0:61</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E4:37_3"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E5:37_3"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp;  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> &amp;  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_3_1"/>
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:37_3_1"/><b>then </b>
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:37_3_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_7.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:37_3"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_3_2"/>
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:37_3_2"/><b>then </b>
 <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">P</font> where <font color="Olive">P</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> : ( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Olive">P</font> &amp; <font color="Olive">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:37_3_2"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="waybel_7.html#T3" target="_self">Th3</a>, </i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">v</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:37_3"/><i><font color="Green">E93</font></i>: 
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K3">downarrow</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K11">finsups</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span>)</span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font>, <font color="Maroon">v</font> = <font color="Maroon">v</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="waybel_7.html#T4" target="_self">Th4</a></i>;<br/><b>consider </b><font color="Maroon">u'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:37_3"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">u'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">u</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">v'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E16:37_3"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">v'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">v</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">v'</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span></span>)</span> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E17:37_3"/><i><font color="Green">E98</font></i>: 
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">y</font></span><a href="domain_1.html#K6">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T39">YELLOW_0:39</a></i>;<br/><b>set </b><font color="Maroon">w</font> = <font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font>;<br/><a NAME="E18:37_3"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="waybel_7.html#T5" target="_self">Th5</a>, <a class="txt" href="waybel_7.html#E1:37"><i><font color="Green">E33</font></i></a>, <a class="ref" href="waybel_0.html#T40">WAYBEL_0:40</a></i>;<br/><a NAME="E19:37_3"/>
( <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v'</font> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font></span>)</span> &amp; <font color="Maroon">v'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">u'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T14">LATTICE3:14</a></i>;<br/><a NAME="E20:37_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">v'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/><a NAME="E21:37_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">u</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">v</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T5" target="_self">Th5</a>, <a class="txt" href="waybel_7.html#E1:37"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E22:37_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="waybel_7.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/><a NAME="E23:37_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T41">WAYBEL_0:41</a></i>;<br/><a NAME="E24:37_3"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">u'</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">v'</font></span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K12">"/\"</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T3" target="_self">Th3</a>, , <a class="ref" href="waybel_0.html#T40">WAYBEL_0:40</a>, <a class="ref" href="waybel_1.html#T6">WAYBEL_1:6</a></i>;<br/><b>hence </b><a NAME="E25:37_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E10:37"/>
( <font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>


</div>
