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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:62"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="card_fil.html#V7" target="_self">measurable</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:62"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="card_fil.html#NR3" target="_self">is_complete_with</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:62"/><i><font color="Green">E41</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> is <a href="card_fil.html#V2" target="_self">principal</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="card_fil.html#V3" target="_self">being_ultrafilter</a> )
 <b>by </b><i><a class="ref" href="card_fil.html#D16" target="_self">Def16</a></i>;<br/>
<a NAME="E5:62"/>
<font color="Maroon">c<sub>2</sub></font> <a href="card_fil.html#NR3" target="_self">is_complete_with</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:62"><i><font color="Green">E40</font></i></a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="card_fil.html#V1" target="_self">uniform</a>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:62"><i><font color="Green">E41</font></i></a>, <a class="ref" href="card_fil.html#T23" target="_self">Th23</a></i>;<br/>
<b>assume </b><a NAME="E7:62"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="card_5.html#V1">regular</a>
 ;<br/>

<a NAME="E8:62"/><b>then </b>
(  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="card_5.html#D2">CARD_5:def 2</a>, <a class="ref" href="card_5.html#D4">CARD_5:def 4</a></i>;<br/>
<a NAME="E9:62"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#T13">CARD_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E11:62"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E12:62"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E13:62"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="ordinal2.html#V2">increasing</a>
 <b>and </b><br/><a NAME="E14:62"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E15:62"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="card_3.html#NM1">Cardinal-Function</a>
 <b>and </b><br/><a NAME="E16:62"/>
not 0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="card_5.html#T41">CARD_5:41</a></i>;<br/>
<a NAME="E17:62"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:62"><i><font color="Green">E46</font></i></a>, <a class="ref" href="ordinal2.html#D9">ORDINAL2:def 9</a></i>;<br/>
<a NAME="E18:62"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T32">CARD_4:32</a></i>;<br/>
<a NAME="E19:62"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E12:62"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_fil.html#E17:62"><i><font color="Green">E47</font></i></a>, <a class="ref" href="card_fil.html#T32" target="_self">Th32</a></i>;<br/>
<a NAME="E20:62"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T80">CARD_2:80</a></i>;<br/>
<a NAME="E21:62"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_5.html#K1">cf</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E11:62"><i><font color="Green">E44</font></i></a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E22:62"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E9:62"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E23:62"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:62_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:62_1"/><i><font color="Green">E52</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>4</sub></font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E12:62"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_fil.html#E1:62_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="card_1.html#T25">CARD_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="card_fil.html#E12:62"><i><font color="Green">E45</font></i></a>, <a class="txt" href="card_fil.html#E1:62_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:62_1"/>
not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:62_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:62_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E6:62"><i><font color="Green">E42</font></i></a>, <a class="ref" href="card_fil.html#D5" target="_self">Def5</a></i>;<br/>
<a NAME="E3:62_1_1"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:62_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E2:62_1"><i><font color="Green">E52</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:62_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="card_fil.html#E4:62"><i><font color="Green">E41</font></i></a>, <a class="ref" href="card_fil.html#T22" target="_self">Th22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E24:62"/>
 <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font> <a href="card_fil.html#NR4" target="_self">is_complete_with</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E3:62"><i><font color="Green">E40</font></i></a>, <a class="ref" href="card_fil.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E25:62"/><b>then </b>
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="card_fil.html#K1" target="_self">dual</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:62"><i><font color="Green">E48</font></i></a>, <a class="txt" href="card_fil.html#E22:62"><i><font color="Green">E49</font></i></a>, <a class="txt" href="card_fil.html#E23:62"><i><font color="Green">E50</font></i></a>, <a class="ref" href="card_fil.html#D4" target="_self">Def4</a>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>
<b>hence </b><a NAME="E26:62"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E19:62"><i><font color="Green">E48</font></i></a>, <a class="ref" href="card_fil.html#D2" target="_self">Def2</a></i>;<br/>


</div>
