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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>  ex <font color="Olive" title="b2">Uf</font> being   <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Uf</font> &amp; <font color="Olive" title="b2">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> )</span><br/><b>let </b><font color="Maroon" title="c2">F</font> be    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Uf</font> being   <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Uf</font> &amp; <font color="Olive" title="b1">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> )</span><br/>



<b>set </b><font color="Maroon" title="c3">LargerF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> ) </span>}</span> ;<br/>
<a NAME="E1:30"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> ) </span>}</span> 
 
;<br/>
<a NAME="E2:30"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">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" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 &amp; $1 is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> );<br/>
<b>let </b><font color="Maroon" title="c4">F2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> ) </span>}</span>  or <font color="Maroon" title="c4">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon" title="c4">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">X</font></span><br/>

<a NAME="E2:30_1"/><b>then </b><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c4">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] </span>}</span> 
 
;<br/>
<a NAME="E3:30_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">F2</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E2:30_1"><i><font color="Green" title="E24">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon" title="c4">F2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c4">LargerF</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> ) </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="card_fil.html#E1:30"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E4:30"/>
for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">LargerF</font> &amp; <font color="Olive" title="b1">Z</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> holds <br/> ex <font color="Olive" title="b2">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b3">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Z</font> holds <br/><font color="Olive" title="b3">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">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" title="c5">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">LargerF</font> &amp; <font color="Maroon" title="c5">X1</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> implies  ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b2">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) ) )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:30_2"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c5">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">LargerF</font>
 <b>and </b><br/><a NAME="E2:30_2"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c5">X1</font> is <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b2">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) )</span><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" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not <font color="Maroon" title="c5">X1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">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" title="E26">A5</font></i>: 
<font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b2">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) )</span><br/><div class="add"><b>take </b>
<font color="Maroon" title="c2">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> ) )</span><br/><b>thus </b><a NAME="E2:30_2_1_1"/>
( <font color="Maroon" title="c2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> ) )
 <b>by </b><i><a class="txt" href="card_fil.html#E1:30_2_1_1"><i><font color="Green" title="E26">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c5">X1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">Y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b2">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b2">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> ) )</span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">X2</font> = <font color="Maroon" title="c5">X1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="card_fil.html#K3" title="CARD_FIL:func.3">Filters</a> <font color="Maroon" title="c1">X</font></span>)</span> <b>by </b><i><a class="txt" href="card_fil.html#E1:30_2"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:30_2_1_2"/><i><font color="Green" title="E26">A6</font></i>: 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">X2</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E2:30_2"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="card_fil.html#T16" target="_self" title="CARD_FIL:th.16">Th16</a></i>;<br/><a NAME="E4:30_2_1_2"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">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" title="c7">F1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:30_2_1_2_1"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 &amp; $1 is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> );<br/>
<a NAME="E3:30_2_1_2_1"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c7">F1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] </span>}</span> 
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:30_2"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="card_fil.html#E2:30_2_1_2_1"><i><font color="Green" title="E27">A7</font></i></a></i>;<br/>
<a NAME="E4:30_2_1_2_1"/><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">F1</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E3:30_2_1_2_1"><i><font color="Green" title="E28">A8</font></i></a>);<br/></i>
<a NAME="E5:30_2_1_2_1"/>
<font color="Maroon" title="c7">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">X2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E2:30_2_1_2_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E6:30_2_1_2_1"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">X2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E4:30_2_1_2_1"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:30_2_1_2"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> &amp; <font color="Olive" title="b1">S</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="card_fil.html#E3:30_2_1_2"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">MF</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">X2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">LargerF</font> ;<br/><b>take </b>
<font color="Maroon" title="c7">MF</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">MF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">MF</font> ) )</span><br/><b>thus </b><a NAME="E7:30_2_1_2"/>
( <font color="Maroon" title="c7">MF</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b1">X1</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">MF</font> ) )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">Uf1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:30"/><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c5">Uf1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; ( for <font color="Olive" title="b1">Z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font> &amp; <font color="Olive" title="b1">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">Uf1</font> holds <br/> not <font color="Maroon" title="c5">Uf1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font> ) )
 <b>by </b><i><a class="ref" href="orders_1.html#T175" title="ORDERS_1:th.175">ORDERS_1:175</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">Uf1</font> = <font color="Maroon" title="c5">Uf1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c4">LargerF</font> <b>by </b><i><a class="txt" href="card_fil.html#E6:30"><i><font color="Green" title="E24">A10</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">Uf</font> = <font color="Maroon" title="c6">Uf1</font> as    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="card_fil.html#T15" target="_self" title="CARD_FIL:th.15">Th15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">Uf</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Uf</font> &amp; <font color="Maroon" title="c7">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> $1 &amp; $1 is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> );<br/>
<a NAME="E9:30"/><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c7">Uf</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">S</font> where <font color="Olive" title="b1">S</font> is   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">X</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">S</font>] </span>}</span> 
 
;<br/>
<a NAME="E10:30"/><i><font color="Green" title="E26">A12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">Uf</font>]
 
<b>from </b><i><a class="ref" href="card_fil.html#S1" target="_self" title="CARD_FIL:sch.1">CARD_FIL:sch 1</a>(<a class="txt" href="card_fil.html#E9:30"><i><font color="Green" title="E25">A11</font></i></a>);<br/></i>
<b>hence </b><a NAME="E11:30"/>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Uf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a></span><br/>

<b>thus </b><a NAME="E12:30"/>
<font color="Maroon" title="c7">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c8">Z</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="card_fil.html#D7" target="_self" title="CARD_FIL:def.7">CARD_FIL:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> or <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> )</span><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" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> or  not <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> or <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> )</span><br/><div class="add"><b>hence </b><a NAME="E2:30_3_1_1"/>
( <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> or <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="E27">A13</font></i>: 
not <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> or <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> )</span><br/><div class="add"><a NAME="E2:30_3_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">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" title="E28">A14</font></i>: 
not <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:30_3_1_2_1"/>
for <font color="Olive" title="b1">Y1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> holds <br/><font color="Olive" title="b1">Y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">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" title="c9">Y1</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> implies <font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">Z</font> )</span><br/>

<b>assume </b><a NAME="E1:30_3_1_2_1_1"/><i><font color="Green" title="E29">A15</font></i>: 
<font color="Maroon" title="c9">Y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c8">Z</font></span><br/>

<b>assume </b><a NAME="E2:30_3_1_2_1_1"/>
<font color="Maroon" title="c9">Y1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">Z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:30_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Y1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c8">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T83" title="XBOOLE_1:th.83">XBOOLE_1:83</a></i>;<br/>
<a NAME="E4:30_3_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">Y1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:30_3_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E1:30_3_1_2_1"><i><font color="Green" title="E28">A14</font></i></a>, <a class="txt" href="card_fil.html#E1:30_3_1_2_1_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:30_3_1_2_1"/><b>then </b><i><font color="Green" title="E29">A16</font></i>: 
(  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c7">Uf</font>,<font color="Maroon" title="c8">Z</font> is    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c7">Uf</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c7">Uf</font>,<font color="Maroon" title="c8">Z</font> &amp; <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c7">Uf</font>,<font color="Maroon" title="c8">Z</font> )
 
<b>by </b><i><a class="ref" href="card_fil.html#T14" target="_self" title="CARD_FIL:th.14">Th14</a></i>;<br/>
<a NAME="E4:30_3_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c7">Uf</font>,<font color="Maroon" title="c8">Z</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E10:30"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:30_3_1_2_1"/><b>then </b>
 <a href="card_fil.html#K2" title="CARD_FIL:func.2">Extend_Filter</a> <font color="Maroon" title="c7">Uf</font>,<font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">LargerF</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:30_3_1_2_1"><i><font color="Green" title="E29">A16</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:30_3_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E6:30"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="card_fil.html#E1:30_3_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="card_fil.html#E3:30_3_1_2_1"><i><font color="Green" title="E29">A16</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:30_3_1_2"/>
( <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> or <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c8">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Uf</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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


</div>
