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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<a NAME="E1:56"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#D2" title="CARD_5:def.2">CARD_5:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:56"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:56"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:56"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
( 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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_fil.html#D16" target="_self" title="CARD_FIL:def.16">Def16</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:56"/>
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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:56"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#D4" title="CARD_5:def.4">CARD_5:def 4</a></span>;<br/>
<a NAME="E8:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:56"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="card_1.html#T13" title="CARD_1:th.13">CARD_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">xi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><span class="kw"> such that </span><br/><a NAME="E10:56"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E11:56"/><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E12:56"/>
<font color="Maroon" title="c3">xi</font> is  <a href="ordinal2.html#V2" title="ORDINAL2:attr.2">increasing</a> 
 <span class="kw">and </span><br/><a NAME="E13:56"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E14:56"/>
<font color="Maroon" title="c3">xi</font> is   <a href="card_3.html#NM1" title="CARD_3:NM.1">Cardinal-Function</a>
 <span class="kw">and </span><br/><a NAME="E15:56"/>
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T41" title="CARD_5:th.41">CARD_5:41</a></span>;<br/>
<a NAME="E16:56"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:56"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="ordinal2.html#D9" title="ORDINAL2:def.9">ORDINAL2:def 9</a></span>;<br/>
<a NAME="E17:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="card_fil.html#T32" target="_self" title="CARD_FIL:th.32">Th32</a></span>;<br/>
<a NAME="E18:56"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/>
<a NAME="E19:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A9</font></span>: 
<font color="Maroon" title="c2">Uf</font> is  <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="card_fil.html#T23" target="_self" title="CARD_FIL:th.23">Th23</a></span>;<br/>
<a NAME="E20:56"/><span class="lab"><font color="Green" title="E48">A10</font></span>: 
 <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="56_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: 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="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:56_1"/><span class="lab"><font color="Green" title="E49">A11</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></span>;<br/>
<a NAME="E3:56_1"/><span class="lab"><font color="Green" title="E50">A12</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:56"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="txt" href="#E1:56_1"><span class="lab"><font color="Green" title="E49">A11</font></span></a>, <a class="ref" href="card_1.html#T25" title="CARD_1:th.25">CARD_1:25</a></span>;<br/>
<a NAME="E4:56_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="56_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:56_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:56_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:56"><span class="lab"><font color="Green" title="E47">A9</font></span></a>, <a class="ref" href="card_fil.html#D5" target="_self" title="CARD_FIL:def.5">Def5</a></span>;<br/>
<a NAME="E3:56_1_1"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:56_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:56_1"><span class="lab"><font color="Green" title="E50">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:56_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:56"><span class="lab"><font color="Green" title="E41">A3</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E21:56"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_2.html#T80" title="CARD_2:th.80">CARD_2:80</a></span>;<br/>
<a NAME="E22:56"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:56"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></span>;<br/>
<a NAME="E23:56"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A13</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:56"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></span>;<br/>
<a NAME="E24:56"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:56"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="card_fil.html#T12" target="_self" title="CARD_FIL:th.12">Th12</a></span>;<br/>
<a NAME="E25:56"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="txt" href="#E23:56"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="txt" href="#E20:56"><span class="lab"><font color="Green" title="E48">A10</font></span></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></span>;<br/>
<span class="kw">hence </span><a NAME="E26:56"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:56"><span class="lab"><font color="Green" title="E46">A8</font></span></a>, <a class="ref" href="card_fil.html#D2" target="_self" title="CARD_FIL:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
