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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  implies ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ; <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>hence </b><a NAME="E2:61_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="filter_0.html#NM2">I_Lattice</a> <b>by </b><i><a class="ref" href="filter_0.html#T40" target="_self">Th40</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E5:61_1"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<a NAME="E7:61_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E8:61_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:61_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span>
 
<b>by </b><i><a class="txt" href="filter_0.html#E5:61_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E10:61_1"/><b>then </b>
<span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E1:61_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="filter_0.html#E8:61_1"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E11:61_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K6">}</a></span></span><a href="filter_0.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="filter_0.html#K2" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E12:61_1"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="filter_0.html#K3" target="_self">.)</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T26" target="_self">Th26</a>, <a class="ref" href="filter_0.html#T47" target="_self">Th47</a></i>;<br/>
<a NAME="E13:61_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="filter_0.html#K4" target="_self">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="filter_0.html#K4" target="_self">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T53" target="_self">Th53</a>, <a class="ref" href="filter_0.html#T55" target="_self">Th55</a></i>;<br/>
<b>hence </b><a NAME="E14:61_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="lattices.html#T17">LATTICES:17</a></i>;<br/>


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

<b>assume </b><b>that </b><br/><a NAME="E2:61"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:61"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>thus </b><a NAME="E4:61"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="filter_0.html#E2:61"><i><font color="Green">E44</font></i></a></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">c<sub>3</sub></font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E5:61"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E6:61"/><b>then </b>
not for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:61"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="filter_0.html#E5:61"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="filter_0.html#E8:61"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E10:61"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E3:61"><i><font color="Green">E45</font></i></a>, <a class="txt" href="filter_0.html#E8:61"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E11:61"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM4">0_Lattice</a> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E5:61"><i><font color="Green">E46</font></i></a>, <a class="ref" href="filter_0.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:61"/><b>then </b>
(  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K1" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="filter_0.html#K1" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_0.html#K1" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="filter_0.html#K1" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K2" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="filter_0.html#K2" target="_self">.)</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> is   <a href="lattices.html#NM4">0_Lattice</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_0.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="filter_0.html#K3" target="_self">.)</a></span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T20" target="_self">Th20</a>, <a class="ref" href="filter_0.html#T26" target="_self">Th26</a>, <a class="ref" href="lattices.html#T47">LATTICES:47</a></i>;<br/>
<b>hence </b><a NAME="E13:61"/>
contradiction
 <b>by </b><i><a class="txt" href="filter_0.html#E5:61"><i><font color="Green">E46</font></i></a>, <a class="ref" href="filter_0.html#T31" target="_self">Th31</a></i>;<br/>


</div>
