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<b>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">X</font><b> such that </b><br/><a NAME="E2:35_1_1"/><i><font color="Green" title="E26">A1</font></i>: 
 <a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Uf</font>
 <b>and </b><br/><a NAME="E3:35_1_1"/><i><font color="Green" title="E27">A2</font></i>: 
<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#T17" target="_self" title="CARD_FIL:th.17">Th17</a></i>;<br/>
<b>take </b>
<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"> ( 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><br/>

<a NAME="E4:35_1_1"/><i><font color="Green" title="E28">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="#E2:35_1_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="card_fil.html#T20" target="_self" title="CARD_FIL:th.20">Th20</a></i>;<br/>
<a NAME="E5:35_1_1"/>
not <font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V2" title="CARD_FIL:attr.2">principal</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1_1"/>
<font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V2" title="CARD_FIL:attr.2">principal</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c3">Y</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:35_1_1_1"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Uf</font>
 <b>and </b><br/><a NAME="E4:35_1_1_1"/><i><font color="Green" title="E30">A5</font></i>: 
for <font color="Olive" title="b1">Z</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Uf</font> holds <br/><font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Z</font>
 <b>by </b><i><a class="ref" href="card_fil.html#D6" target="_self" title="CARD_FIL:def.6">Def6</a></i>;<br/>
<a NAME="E5:35_1_1_1"/><i><font color="Green" title="E31">A6</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> holds  <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="35_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Uf</font></span><br/>

<b>assume </b><a NAME="E1:35_1_1_1_1"/><i><font color="Green" title="E32">A7</font></i>: 
not <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <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="E2:35_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="subset_1.html#T55" title="SUBSET_1:th.55">SUBSET_1:55</a></i>;<br/>
<a NAME="E3:35_1_1_1_1"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Uf</font> or <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Uf</font> )
 
<b>by </b><i><a class="txt" href="#E3:35_1_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="card_fil.html#D7" target="_self" title="CARD_FIL:def.7">Def7</a></i>;<br/>
<a NAME="E4:35_1_1_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
;<br/>
<b>hence </b><a NAME="E5:35_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:35_1_1"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="#E1:35_1_1_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="#E3:35_1_1_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="card_fil.html#D5" target="_self" title="CARD_FIL:def.5">Def5</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="E6:35_1_1_1"/><i><font color="Green" title="E32">A9</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>  holds not <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>; <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> <font color="Maroon" title="c3">Y</font></span><br/>

<a NAME="E1:35_1_1_1_2"/><i><font color="Green" title="E33">A10</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E4:35_1_1_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="#E5:35_1_1_1"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:35_1_1_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</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="E3:35_1_1_1_2"/><b>then </b>
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="#E1:35_1_1_1_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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="E7:35_1_1_1"/>
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1_1_3"/>
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1_3"/><i><font color="Green" title="E33">A11</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Y</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E4:35_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:35_1_1_1"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="#E3:35_1_1_1_3"><i><font color="Green" title="E33">A11</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="E8:35_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E3:35_1_1_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="card_fil.html#D1" target="_self" title="CARD_FIL:def.1">Def1</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="E6:35_1_1"/>
( 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="txt" href="#E3:35_1_1"><i><font color="Green" title="E27">A2</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>
