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

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="card_5.html#NM1" title="CARD_5:NM.1">Aleph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">M</font> is <a href="card_fil.html#V7" title="CARD_FIL:attr.7">measurable</a> implies <font color="Maroon" title="c1">M</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a> )</span><br/>

<b>assume </b><a NAME="E1:62"/>
<font color="Maroon" title="c1">M</font> is <a href="card_fil.html#V7" title="CARD_FIL:attr.7">measurable</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">M</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a></span><br/>

<b>then consider </b><font color="Maroon" title="c2">Uf</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><br/><a NAME="E3:62"/><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c2">Uf</font> <a href="card_fil.html#NR3" title="CARD_FIL:NR.3">is_complete_with</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E4:62"/><i><font color="Green" title="E41">A2</font></i>: 
( not <font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V2" title="CARD_FIL:attr.2">principal</a> &amp; <font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> )
 <b>by </b><i><a class="ref" href="card_fil.html#D16" target="_self" title="CARD_FIL:def.16">Def16</a></i>;<br/>
<a NAME="E5:62"/>
<font color="Maroon" title="c2">Uf</font> <a href="card_fil.html#NR3" title="CARD_FIL:NR.3">is_complete_with</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#E3:62"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green" title="E42">A3</font></i>: 
<font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a>
 
<b>by </b><i><a class="txt" href="card_fil.html#E4:62"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="card_fil.html#T23" target="_self" title="CARD_FIL:th.23">Th23</a></i>;<br/>
<b>assume </b><a NAME="E7:62"/>
not <font color="Maroon" title="c1">M</font> is <a href="card_5.html#V1" title="CARD_5:attr.1">regular</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E8:62"/><b>then </b>
(  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">M</font> &amp;  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">M</font> )
 
<b>by </b><i><a class="ref" href="card_5.html#D2" title="CARD_5:def.2">CARD_5:def 2</a>, <a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></i>;<br/>
<a NAME="E9:62"/><b>then </b><i><font color="Green" title="E43">A4</font></i>: 
 <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">xi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E11:62"/><i><font color="Green" title="E44">A5</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">xi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E12:62"/><i><font color="Green" title="E45">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">M</font>
 <b>and </b><br/><a NAME="E13:62"/>
<font color="Maroon" title="c3">xi</font> is <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a>
 <b>and </b><br/><a NAME="E14:62"/><i><font color="Green" title="E46">A7</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">xi</font>
 <b>and </b><br/><a NAME="E15:62"/>
<font color="Maroon" title="c3">xi</font> is   <a href="card_3.html#NM1" title="CARD_3:NM.1">Cardinal-Function</a>
 <b>and </b><br/><a NAME="E16:62"/>
not  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font>
 <b>by </b><i><a class="ref" href="card_5.html#T41" title="CARD_5:th.41">CARD_5:41</a></i>;<br/>
<a NAME="E17:62"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E14:62"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="ordinal2.html#D9" title="ORDINAL2:def.9">ORDINAL2:def 9</a></i>;<br/>
<a NAME="E18:62"/>
<font color="Maroon" title="c1">M</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 
<b>by </b><i><a class="ref" href="card_4.html#T32" title="CARD_4:th.32">CARD_4:32</a></i>;<br/>
<a NAME="E19:62"/><b>then </b><i><font color="Green" title="E48">A9</font></i>: 
<font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font></span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E12:62"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="card_fil.html#E17:62"><i><font color="Green" title="E47">A8</font></i></a>, <a class="ref" href="card_fil.html#T32" target="_self" title="CARD_FIL:th.32">Th32</a></i>;<br/>
<a NAME="E20:62"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">xi</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T80" title="CARD_2:th.80">CARD_2:80</a></i>;<br/>
<a NAME="E21:62"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="card_5.html#K1" title="CARD_5:func.1">cf</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E11:62"><i><font color="Green" title="E44">A5</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<a NAME="E22:62"/><b>then </b><i><font color="Green" title="E49">A10</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E9:62"><i><font color="Green" title="E43">A4</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/>
<a NAME="E23:62"/><i><font color="Green" title="E50">A11</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</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">Uf</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" title="c4">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> <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">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font> or <font color="Maroon" title="c4">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">Uf</font> )</span><br/>

<b>assume </b><a NAME="E1:62_1"/><i><font color="Green" title="E51">A12</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">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">Uf</font></span><br/>

<a NAME="E2:62_1"/><i><font color="Green" title="E52">A13</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c4">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E12:62"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="card_fil.html#E1:62_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="card_1.html#T25" title="CARD_1:th.25">CARD_1:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">X1</font> = <font color="Maroon" title="c4">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#E12:62"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="card_fil.html#E1:62_1"><i><font color="Green" title="E51">A12</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E4:62_1"/>
not <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Uf</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" title="c5">X1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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="E2:62_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">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#E6:62"><i><font color="Green" title="E42">A3</font></i></a>, <a class="ref" href="card_fil.html#D5" target="_self" title="CARD_FIL:def.5">Def5</a></i>;<br/>
<a NAME="E3:62_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c5">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">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" title="E52">A13</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="E5:62_1"/>
<font color="Maroon" title="c4">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">Uf</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E4:62"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="card_fil.html#T22" target="_self" title="CARD_FIL:th.22">Th22</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="E24:62"/>
 <a href="card_fil.html#K1" title="CARD_FIL:func.1">dual</a> <font color="Maroon" title="c2">Uf</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#E3:62"><i><font color="Green" title="E40">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="E25:62"/><b>then </b>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">xi</font></span>)</span> <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">Uf</font>
 
<b>by </b><i><a class="txt" href="card_fil.html#E19:62"><i><font color="Green" title="E48">A9</font></i></a>, <a class="txt" href="card_fil.html#E22:62"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="card_fil.html#E23:62"><i><font color="Green" title="E50">A11</font></i></a>, <a class="ref" href="card_fil.html#D4" target="_self" title="CARD_FIL:def.4">Def4</a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">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" title="E48">A9</font></i></a>, <a class="ref" href="card_fil.html#D2" target="_self" title="CARD_FIL:def.2">Def2</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>
