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

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

<b>given </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:23"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">r</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Olive">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon">F</font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E2:23"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span> ;<br/>
<a NAME="E3:23"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E4:23"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span>  holds <br/>( <font color="Olive">x</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> &amp; <font color="Olive">x</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </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="E1:23_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:23_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 
;<br/>
<b>hence </b><a NAME="E3:23_1"/>
( <font color="Maroon">x</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">x</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:23"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span>  holds <br/>( <font color="Maroon">F</font>,<font color="Olive">X1</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  &amp; <font color="Olive">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:23_2"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 
;<br/>
<b>hence </b><a NAME="E3:23_2"/>
( <font color="Maroon">F</font>,<font color="Maroon">X1</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  &amp; <font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:23"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">H</font> being   <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>  st <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Olive">H</font> holds <br/><font color="Olive">H</font> <a href="hidden.html#R1">=</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="23_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">H</font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:23_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 ;<br/>

<b>thus </b><a NAME="E2:23_3"/>
<font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<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="E3:23_3"/>
<font color="Maroon">x</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>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E5:23_3"/>
<font color="Maroon">r</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:23_3"/><b>then </b>
<font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>
<b>hence </b><a NAME="E7:23_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:23"/>
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Olive">Z</font> is <a href="ordinal1.html#V6">c=-linear</a> 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="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span>  &amp; ( for <font color="Olive">X1</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Z</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">Y</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>proof </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="E1:23_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:23_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">Z</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<b>take </b><font color="Maroon">Y</font> =  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font>;<br/>
<a NAME="E4:23_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
;<br/>
<a NAME="E5:23_4"/>
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</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="E6:23_4"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">Y</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E8:23_4"/>
<font color="Maroon">V</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="23_4_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="E1:23_4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_4_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E4:23_4_1"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E5:23_4_1"/><b>then </b>
<font color="Maroon">X1</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:23_4_1"/>
<font color="Maroon">x</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/>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E10:23_4"/>
<font color="Maroon">V</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</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="filter_0.html#D1" target="_self">FILTER_0:def 1</a><br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<b>thus </b><a NAME="E1:23_4_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> implies <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:23_4_2_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_4_2_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>assume </b><a NAME="E4:23_4_2_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:23_4_2_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:23_4_2_1"/>
( ( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> ) )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E8:23_4_2_1"/><b>then </b>
( ( ( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) or <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> ) &amp; ( ( <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span>  &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) or <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> ) )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E9:23_4_2_1"/><b>then </b>
( <font color="Maroon">X1</font>,<font color="Maroon">X2</font> <a href="xboole_0.html#R3">are_c=-comparable</a>  &amp; <font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">X1</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">X2</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> )
 
<b>by </b><i>, , , , <a class="ref" href="ordinal1.html#D9">ORDINAL1:def 9</a></i>;<br/>
<a NAME="E10:23_4_2_1"/><b>then </b>
( ( <font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X2</font> or <font color="Maroon">X2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X1</font> ) &amp; <font color="Maroon">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">X1</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">X2</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D9">XBOOLE_0:def 9</a></i>;<br/>
<a NAME="E11:23_4_2_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> or <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font> )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E12:23_4_2_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="filter_0.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


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

<b>assume </b><a NAME="E2:23_4_2"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:23_4_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E5:23_4_2"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:23_4_2"/><b>then </b>
<font color="Maroon">X1</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E7:23_4_2"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:23_4_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">V</font> = <font color="Maroon">V</font> as    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E12:23_4"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_3"><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>thus </b><a NAME="E1:23_4_3"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_4_4"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/><b>then consider </b><font color="Maroon">X1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:23_4_4"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> &amp; <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E4:23_4_4"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> or <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="filter_0.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E5:23_4_4"/><b>then </b>
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span>  or <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:23_4_4"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E7:23_4_4"/>
contradiction
 <b>by </b><i>, <a class="ref" href="filter_0.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:23_4"/><b>then </b>
( <font color="Maroon">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> )
 
;<br/>
<b>hence </b><a NAME="E15:23_4"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">X1</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E16:23_4"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E17:23_4"/><b>then </b>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">F</font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E18:23_4"/>
<font color="Maroon">X1</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/>


</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="E9:23"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</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#T175">ORDERS_1:175</a></i>;<br/>
<b>consider </b><font color="Maroon">H</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:23"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E13:23"/>
( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> &amp; <font color="Maroon">H</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D4" target="_self">FILTER_0:def 4</a><br/>

<b>let </b><font color="Maroon">G</font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E14:23"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> )
 ;<br/>

<a NAME="E15:23"/><b>then </b>
( <font color="Maroon">G</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> &amp; <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E16:23"/><b>then </b>
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:23"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font>
 <b>by </b><i>, , </i>;<br/>


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