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<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_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a> )</span><br/>

<b>assume </b><a NAME="E1:68"/>
<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_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a></span><br/>

<b>then consider </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><br/><a NAME="E3:68"/><i><font color="Green" title="E47">A1</font></i>: 
<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>
 <b>and </b><br/><a NAME="E4:68"/><i><font color="Green" title="E48">A2</font></i>: 
( not <font color="Maroon" title="c2">F</font> is <a href="card_fil.html#V2" title="CARD_FIL:attr.2">principal</a> &amp; <font color="Maroon" title="c2">F</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="E5:68"/><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c2">F</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="#E3:68"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="card_1.html#D5" title="CARD_1:def.5">CARD_1:def 5</a></i>;<br/>
<b>assume </b><a NAME="E6:68"/>
not <font color="Maroon" title="c1">M</font> is <a href="card_1.html#V2" title="CARD_1:attr.2">limit_cardinal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>hence </b><a NAME="E7:68"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:68"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="#E4:68"><i><font color="Green" title="E48">A2</font></i></a>, <a class="txt" href="#E5:68"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="card_fil.html#T23" target="_self" title="CARD_FIL:th.23">Th23</a>, <a class="ref" href="card_fil.html#T37" target="_self" title="CARD_FIL:th.37">Th37</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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