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

<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="card_1.html#V2" title="CARD_1:attr.2">limit</a> <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>;<br/>

<b>given </b><font color="Maroon" title="c2">F</font> being    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">M</font><b> such that </b><a NAME="E1:73"/><i><font color="Green" title="E46">A1</font></i>: 
( <font color="Maroon" title="c2">F</font> is <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a> &amp; <font color="Maroon" title="c2">F</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> &amp; <font color="Maroon" title="c2">F</font> <a href="card_fil.html#NR3" title="CARD_FIL:NR.3">is_complete_with</a> <font color="Maroon" title="c1">M</font> )
 ;<br/>

<a NAME="E2:73"/><i><font color="Green" title="E47">A2</font></i>: 
 <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">F</font> <a href="card_fil.html#NR4" title="CARD_FIL:NR.4">is_complete_with</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:73"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="card_fil.html#T12" target="_self" title="CARD_FIL:th.12">Th12</a></i>;<br/>
<a NAME="E3:73"/>
 <a href="card_fil.html#K5" title="CARD_FIL:func.5">Frechet_Ideal</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">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" title="c3">X</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><br/>

<b>assume </b><a NAME="E1:73_1"/><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K5" title="CARD_FIL:func.5">Frechet_Ideal</a> <font color="Maroon" title="c1">M</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">X1</font> = <font color="Maroon" title="c3">X</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E1:73_1"><i><font color="Green" title="E48">A3</font></i></a></i>;<br/>
<a NAME="E3:73_1"/><i><font color="Green" title="E49">A4</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">X1</font> <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:73_1"><i><font color="Green" title="E48">A3</font></i></a>, <a class="ref" href="card_fil.html#T19" target="_self" title="CARD_FIL:th.19">Th19</a></i>;<br/>
<b>assume </b><a NAME="E4:73_1"/>
not <font color="Maroon" title="c3">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">F</font>
 ;<br/>

<a NAME="E5:73_1"/><b>then </b>
<font color="Maroon" title="c4">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:73"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="card_fil.html#T22" target="_self" title="CARD_FIL:th.22">Th22</a></i>;<br/>
<a NAME="E6:73_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E1:73"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="card_fil.html#D5" target="_self" title="CARD_FIL:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:73_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E3:73_1"><i><font color="Green" title="E49">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c3">S</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:73"/><i><font color="Green" title="E48">A5</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E6:73"/><i><font color="Green" title="E49">A6</font></i>: 
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="c3">S</font> holds <br/> not <font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">F</font>
 <b>and </b><br/><a NAME="E7:73"/><i><font color="Green" title="E50">A7</font></i>: 
for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</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="c3">S</font> &amp; <font color="Olive" title="b2">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Olive" title="b1">X1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">X2</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">X2</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E2:73"><i><font color="Green" title="E47">A2</font></i></a>, <a class="ref" href="card_fil.html#T35" target="_self" title="CARD_FIL:th.35">Th35</a></i>;<br/>
<a NAME="E8:73"/><i><font color="Green" title="E51">A8</font></i>: 
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="c3">S</font> holds <br/><font color="Olive" title="b1">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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" title="c4">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:73_2"/><i><font color="Green" title="E52">A9</font></i>: 
<font color="Maroon" title="c4">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">X2</font> = <font color="Maroon" title="c4">X1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E1:73_2"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>
<a NAME="E3:73_2"/>
not <font color="Maroon" title="c5">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E6:73"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="card_fil.html#E1:73_2"><i><font color="Green" title="E52">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:73_2"/>
<font color="Maroon" title="c4">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E1:73"><i><font color="Green" title="E46">A1</font></i></a>, <a class="ref" href="card_fil.html#T22" target="_self" title="CARD_FIL:th.22">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:73"/><i><font color="Green" title="E52">A10</font></i>: 
not <font color="Maroon" title="c3">S</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="txt" href="card_fil.html#E5:73"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="card_4.html#T1" title="CARD_4:th.1">CARD_4:1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:73"/><i><font color="Green" title="E53">A11</font></i>: 
<font color="Maroon" title="c4">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</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/>
<a NAME="E12:73"/>
<font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<a NAME="E2:73_3"/><b>then </b>
<font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E3:73_3"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E9:73"><i><font color="Green" title="E52">A10</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c5">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E14:73"/><i><font color="Green" title="E54">A12</font></i>: 
<font color="Maroon" title="c5">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E15:73"/><i><font color="Green" title="E55">A13</font></i>: 
<font color="Maroon" title="c3">S</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c3">S</font>
 
;<br/>
<a NAME="E16:73"/>
not <font color="Maroon" title="c5">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">X1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:73"><i><font color="Green" title="E54">A12</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E17:73"/><b>then </b><i><font color="Green" title="E56">A14</font></i>: 
( <font color="Maroon" title="c5">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> &amp; <font color="Maroon" title="c5">X2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">X1</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:73"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="card_fil.html#E15:73"><i><font color="Green" title="E55">A13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:73"/><b>then </b>
<font color="Maroon" title="c4">X1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">X2</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E7:73"><i><font color="Green" title="E50">A7</font></i></a>, <a class="txt" href="card_fil.html#E11:73"><i><font color="Green" title="E53">A11</font></i></a></i>;<br/>
<a NAME="E19:73"/><b>then </b><i><font color="Green" title="E57">A15</font></i>: 
<font color="Maroon" title="c4">X1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c5">X2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">X1</font> = <font color="Maroon" title="c4">X1</font>, <font color="Maroon" title="c7">X2</font> = <font color="Maroon" title="c5">X2</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E11:73"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="card_fil.html#E17:73"><i><font color="Green" title="E56">A14</font></i></a></i>;<br/>
<a NAME="E21:73"/>
( <font color="Maroon" title="c6">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c7">X2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )
 
<b>by </b><i><a class="txt" href="card_fil.html#E8:73"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="card_fil.html#E11:73"><i><font color="Green" title="E53">A11</font></i></a>, <a class="txt" href="card_fil.html#E14:73"><i><font color="Green" title="E54">A12</font></i></a>, <a class="txt" href="card_fil.html#E15:73"><i><font color="Green" title="E55">A13</font></i></a></i>;<br/>
<a NAME="E22:73"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:73"><i><font color="Green" title="E57">A15</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E23:73"/>
contradiction
 <b>by </b><i><a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></i>;<br/>


</div>
