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<b>let </b><font color="Maroon" title="c1">X</font> be  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Uf</font> being   <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">Uf</font> is <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a> &amp; <font color="Olive" title="b1">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> holds <br/> <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="Olive" title="b1">Uf</font></span><br/><b>let </b><font color="Maroon" title="c2">Uf</font> be    <a href="card_fil.html#M1" title="CARD_FIL:mode.1">Filter</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="c2">Uf</font> is <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a> &amp; <font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> implies  <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> )</span><br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green" title="E26">A1</font></i>: 
( <font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V1" title="CARD_FIL:attr.1">uniform</a> &amp; <font color="Maroon" title="c2">Uf</font> is <a href="card_fil.html#V3" title="CARD_FIL:attr.3">being_ultrafilter</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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></span><br/>

<b>thus </b><a NAME="E2:36"/>
 <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>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">Y</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="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <font color="Maroon" title="c1">X</font> or <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> )</span><br/>

<b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_fil.html#K4" title="CARD_FIL:func.4">Frechet_Filter</a> <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="c3">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Uf</font></span><br/>

<a NAME="E2:36_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T18" target="_self" title="CARD_FIL:th.18">Th18</a></i>;<br/>
<a NAME="E3:36_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font>
 
;<br/>
<a NAME="E4:36_1"/><b>then </b>
not <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <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>by </b><i><a class="txt" href="card_fil.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="card_fil.html#D5" target="_self" title="CARD_FIL:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:36_1"/>
<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>by </b><i><a class="txt" href="card_fil.html#E1:36"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="card_fil.html#E1:36_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>; <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>


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