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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">LargerF</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> ) </span>}</span> ;<br/>
<a NAME="E1:30"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> ) </span>}</span> 
 
;<br/>
<a NAME="E2:30"/>
<span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="card_fil.html#K3" target="_self">Filters</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> );<br/>
<b>let </b><font color="Maroon">F2</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="E1:30_1"/>
<font color="Maroon">F2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">F2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span> 
 
;<br/>
<a NAME="E3:30_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F2</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E2:30_1"><i><font color="Green">E24</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon">F2</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K3" target="_self">Filters</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">LargerF</font> = <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_fil.html#K3" target="_self">Filters</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:30"/>
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> <font color="Maroon">LargerF</font> &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> <font color="Maroon">LargerF</font> &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="30_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><b>that </b><br/><a NAME="E1:30_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">LargerF</font>
 <b>and </b><br/><a NAME="E2:30_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X1</font> is <a href="ordinal1.html#V6">c=-linear</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2_1"/>
( <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Maroon">X1</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_1"><b>suppose </b></a><a NAME="E1:30_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">F</font>
;<br/><b>thus </b><a NAME="E2:30_2_1_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LargerF</font> &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="Maroon">X1</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_1_2"><b>suppose </b></a><a NAME="E1:30_2_1_2"/>
not <font color="Maroon">X1</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">X2</font> = <font color="Maroon">X1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="card_fil.html#K3" target="_self">Filters</a> <font color="Maroon">X</font></span>)</span> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:30_2_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="tarski.html#K3">union</a> <font color="Maroon">X2</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:30_2"><i><font color="Green">E25</font></i></a>, </i>;<br/><a NAME="E4:30_2_1_2"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">F1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:30_2_1_2_1"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">F1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> );<br/>
<a NAME="E3:30_2_1_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">F1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:30_2_1_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F1</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self">CARD_FIL:sch 1</a>();<br/></i>
<a NAME="E5:30_2_1_2_1"/>
<font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X2</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E6:30_2_1_2_1"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X2</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:30_2_1_2"/><b>then </b>
 <a href="tarski.html#K3">union</a> <font color="Maroon">X2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : ( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Olive">S</font> &amp; <font color="Olive">S</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><b>then reconsider </b><font color="Maroon">MF</font> =  <a href="tarski.html#K3">union</a> <font color="Maroon">X2</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">LargerF</font> ;<br/><b>take </b>
<font color="Maroon">MF</font>
;<br/><b>thus </b><a NAME="E7:30_2_1_2"/>
( <font color="Maroon">MF</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LargerF</font> &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="Maroon">X1</font> holds <br/><font color="Olive">X1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">MF</font> ) )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">Uf1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:30"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">Uf1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LargerF</font> &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> <font color="Maroon">LargerF</font> &amp; <font color="Olive">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Uf1</font> holds <br/> not <font color="Maroon">Uf1</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>reconsider </b><font color="Maroon">Uf1</font> = <font color="Maroon">Uf1</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">LargerF</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">Uf</font> = <font color="Maroon">Uf1</font> as    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">Uf</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> );<br/>
<a NAME="E9:30"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">Uf</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">S</font> where <font color="Olive">S</font> is   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] </span>}</span> 
 
;<br/>
<a NAME="E10:30"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">Uf</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self">CARD_FIL:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E11:30"/>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Uf</font>
 ;<br/>

<b>thus </b><a NAME="E12:30"/>
<font color="Maroon">Uf</font> is <a href="card_fil.html#V3" target="_self">being_ultrafilter</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D7" target="_self">CARD_FIL:def 7</a><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_3_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> or  not <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_1"><b>suppose </b></a><a NAME="E1:30_3_1_1"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:30_3_1_1"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> or <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_3_1_2"><b>suppose </b></a><a NAME="E1:30_3_1_2"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 ;<br/><div class="add"><a NAME="E2:30_3_1_2"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_3_1_2_1"/><i><font color="Green">E73</font></i>: 
not <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 ;<br/>

<a NAME="E2:30_3_1_2_1"/>
for <font color="Olive">Y1</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> holds <br/><font color="Olive">Y1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y1</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:30_3_1_2_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font>
 ;<br/>

<b>assume </b><a NAME="E2:30_3_1_2_1_1"/>
<font color="Maroon">Y1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E3:30_3_1_2_1_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>;<br/>
<a NAME="E4:30_3_1_2_1_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:30_3_1_2_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30_3_1_2_1"/><b>then </b><i><font color="Green">E75</font></i>: 
(  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">Uf</font>,<font color="Maroon">Z</font> is    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">Uf</font> <a href="tarski.html#R1">c=</a>  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">Uf</font>,<font color="Maroon">Z</font> &amp; <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">Uf</font>,<font color="Maroon">Z</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:30_3_1_2_1"/><b>then </b>
<font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">Uf</font>,<font color="Maroon">Z</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:30_3_1_2_1"/><b>then </b>
 <a href="card_fil.html#K2" target="_self">Extend_Filter</a> <font color="Maroon">Uf</font>,<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">LargerF</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:30_3_1_2_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:30_3_1_2"/>
( <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> or <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Uf</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
