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

<span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </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><br/><a NAME="E3:68"/><span class="lab"><font color="Green" title="E46">A1</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>
 <span class="kw">and </span><br/><a NAME="E4:68"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
(  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>  )
 <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="E5:68"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:68"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="card_1.html#D2" title="CARD_1:def.2">CARD_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:68"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:68"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E4:68"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E5:68"><span class="lab"><font color="Green" title="E48">A3</font></span></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></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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