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

<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   non  <a href="card_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a>  <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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">M</font>  holds <br/>(  not <font color="Olive" title="b1">F</font> is  <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a>  or  not <font color="Olive" title="b1">F</font> is  <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a>  or  not <font color="Olive" title="b1">F</font> <a href="card_fil.html#NR3" title="CARD_FIL:NR.3">is_complete_with</a> <font color="Maroon" title="c1">M</font> )</span><br/>

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E2:67"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E3:67"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:67"/>
 <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="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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> or <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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_1"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:67_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:67_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="card_fil.html#T22" target="_self" title="CARD_FIL:th.22">Th22</a></span>;<br/>
<a NAME="E5:67_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="card_fil.html#D5" target="_self" title="CARD_FIL:def.5">Def5</a></span>;<br/>
<a NAME="E6:67_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c4">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_1"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="card_fil.html#T19" target="_self" title="CARD_FIL:th.19">Th19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:67_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:67"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E7:67"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
 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>
 <span class="kw">and </span><br/><a NAME="E8:67"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:67"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="card_fil.html#T12" target="_self" title="CARD_FIL:th.12">Th12</a>, <a class="ref" href="card_fil.html#T35" target="_self" title="CARD_FIL:th.35">Th35</a></span>;<br/>
<a NAME="E9:67"/>
 not <font color="Maroon" title="c3">S</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E47">A5</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">X1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:67"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E12:67"/>
<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="67_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:67_2"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:67_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:67_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E47">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">X2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E14:67"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/>
<a NAME="E15:67"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
<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="E16:67"/>
 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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E17:67"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E18:67"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E15:67"><span class="lab"><font color="Green" title="E52">A10</font></span></a></span>;<br/>
<a NAME="E19:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E51">A9</font></span></a></span>;<br/>
<a NAME="E21:67"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
 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="67_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">X1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font> implies <font color="Maroon" title="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:67_3"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<font color="Maroon" title="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">X2</font> = <font color="Maroon" title="c8">X1</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67_3"><span class="lab"><font color="Green" title="E55">A13</font></span></a></span>;<br/>
<a NAME="E3:67_3"/>
 not <font color="Maroon" title="c9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E1:67_3"><span class="lab"><font color="Green" title="E55">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:67_3"/>
<font color="Maroon" title="c8">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:67"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="card_fil.html#T22" target="_self" title="CARD_FIL:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A14</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E50">A8</font></span></a></span>;<br/>
<a NAME="E23:67"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:67"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="txt" href="#E14:67"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E15:67"><span class="lab"><font color="Green" title="E52">A10</font></span></a></span>;<br/>
<a NAME="E24:67"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:67"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E22:67"><span class="lab"><font color="Green" title="E55">A14</font></span></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E25:67"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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