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

<b>consider </b><font color="Maroon">Uf</font> being    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E2:37_1_1"/><i><font color="Green">E22</font></i>: 
 <a href="card_fil.html#K4" target="_self">Frechet_Filter</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Uf</font>
 <b>and </b><br/><a NAME="E3:37_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">Uf</font> is <a href="card_fil.html#V3" target="_self">being_ultrafilter</a>
 <b>by </b><i><a class="txt" href="card_fil.html#E2:37_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">Uf</font>
;<br/>

<a NAME="E4:37_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">Uf</font> is <a href="card_fil.html#V1" target="_self">uniform</a>
 
<b>by </b><i>, <a class="txt" href="card_fil.html#E4:37_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E5:37_1_1"/>
not <font color="Maroon">Uf</font> is <a href="card_fil.html#V2" target="_self">principal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_1"/>
<font color="Maroon">Uf</font> is <a href="card_fil.html#V2" target="_self">principal</a>
 ;<br/>

<b>then consider </b><font color="Maroon">Y</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:37_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 <b>and </b><br/><a NAME="E4:37_1_1_1"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">Z</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> holds <br/><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="card_fil.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E5:37_1_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> holds  <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:37_1_1_1_1"/><i><font color="Green">E65</font></i>: 
not <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 ;<br/>

<a NAME="E2:37_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T55">SUBSET_1:55</a></i>;<br/>
<a NAME="E3:37_1_1_1_1"/><b>then </b><i><font color="Green">E67</font></i>: 
( <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> or <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:37_1_1_1_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
;<br/>
<b>hence </b><a NAME="E5:37_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:37_1_1_1"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  holds not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>

<a NAME="E1:37_1_1_1_2"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:37_1_1_1_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:37_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E4:37_1_1_1_2"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:37_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1_1_1_3"/>
<font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:37_1_1_1_3"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E4:37_1_1_1_3"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:37_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:37_1_1"/>
( not <font color="Maroon">Uf</font> is <a href="card_fil.html#V2" target="_self">principal</a> &amp; <font color="Maroon">Uf</font> is <a href="card_fil.html#V3" target="_self">being_ultrafilter</a> )
 <b>by </b><i/>;<br/>


</div>
