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

<b>let </b><font color="Maroon">M</font> be  non <a href="card_1.html#V2">limit</a> <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>given </b><font color="Maroon">F</font> being    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">M</font><b> such that </b><a NAME="E1:73"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">F</font> is <a href="card_fil.html#V1" target="_self">uniform</a> &amp; <font color="Maroon">F</font> is <a href="card_fil.html#V3" target="_self">being_ultrafilter</a> &amp; <font color="Maroon">F</font> <a href="card_fil.html#NR3" target="_self">is_complete_with</a> <font color="Maroon">M</font> )
 ;<br/>

<a NAME="E2:73"/><i><font color="Green">E24</font></i>: 
 <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">F</font> <a href="card_fil.html#NR4" target="_self">is_complete_with</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:73"/>
 <a href="card_fil.html#K5" target="_self">Frechet_Ideal</a> <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_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="E1:73_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K5" target="_self">Frechet_Ideal</a> <font color="Maroon">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="card_fil.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E3:73_1"/><i><font color="Green">E47</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X1</font> <a href="card_1.html#NR2" target="_self">&lt;`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T9" target="_self">Th9</a>, <a class="txt" href="card_fil.html#E2:73"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:73_1"/>
not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E5:73_1"/><b>then </b>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:73_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:73_1"/>
contradiction
 <b>by </b><i><a class="ref" href="card_fil.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">S</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E5:73"/><i><font color="Green">E58</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E6:73"/><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> <font color="Maroon">S</font> holds <br/> not <font color="Olive">X1</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E7:73"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">X1</font>, <font color="Olive">X2</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">X1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">X2</font> holds <br/><font color="Olive">X1</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">X2</font>
 <b>by </b><i>, <a class="ref" href="card_fil.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E8:73"/><i><font color="Green">E67</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> <font color="Maroon">S</font> holds <br/><font color="Olive">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_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="E1:73_2"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">X2</font> = <font color="Maroon">X1</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="card_fil.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E3:73_2"/>
not <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#D2" target="_self">Def2</a>, <a class="ref" href="card_fil.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E4:73_2"/>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:73"/><i><font color="Green">E69</font></i>: 
not <font color="Maroon">S</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_fil.html#T9" target="_self">Th9</a>, <a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<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="E11:73"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E12:73"/>
<font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X1</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_3"/>
<font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X1</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:73_3"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E3:73_3"/>
contradiction
 <b>by </b><i><a class="ref" href="card_fil.html#D3" target="_self">Def3</a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">X2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:73"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X1</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E15:73"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">S</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X1</font></span><a href="tarski.html#K1">}</a></span> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">S</font>
 
;<br/>
<a NAME="E16:73"/>
not <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">X1</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="card_fil.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E17:73"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">X1</font> )
 
<b>by </b><i><a class="ref" href="card_fil.html#D4" target="_self">Def4</a>, <a class="ref" href="card_fil.html#T12" target="_self">Th12</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:73"/><b>then </b>
<font color="Maroon">X1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">X2</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T10" target="_self">Th10</a>, </i>;<br/>
<a NAME="E19:73"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">X1</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">X2</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X1</font>, <font color="Maroon">X2</font> = <font color="Maroon">X2</font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i>, </i>;<br/>
<a NAME="E21:73"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="card_fil.html#T11" target="_self">Th11</a>, , <a class="ref" href="card_fil.html#D4" target="_self">Def4</a>, <a class="ref" href="card_fil.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E22:73"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E23:73"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
