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

<b>let </b><font color="Maroon">M</font> be   <a href="card_5.html#NM1">Aleph</a>;<br/>

<b>assume </b><a NAME="E1:76"/>
<font color="Maroon">M</font> is <a href="card_fil.html#V7" target="_self">measurable</a>
 ;<br/>

<b>then consider </b><font color="Maroon">F</font> being    <a href="card_fil.html#M1" target="_self">Filter</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E3:76"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">F</font> <a href="card_fil.html#NR3" target="_self">is_complete_with</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:76"/><i><font color="Green">E24</font></i>: 
( not <font color="Maroon">F</font> is <a href="card_fil.html#V2" target="_self">principal</a> &amp; <font color="Maroon">F</font> is <a href="card_fil.html#V3" target="_self">being_ultrafilter</a> )
 <b>by </b><i><a class="ref" href="card_fil.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:76"/>
<font color="Maroon">F</font> <a href="card_fil.html#NR3" target="_self">is_complete_with</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T9" target="_self">Th9</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E6:76"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">F</font> is <a href="card_fil.html#V1" target="_self">uniform</a>
 
<b>by </b><i><a class="ref" href="card_fil.html#D2" target="_self">Def2</a>, </i>;<br/>
<b>assume </b><a NAME="E7:76"/>
not <font color="Maroon">M</font> is <a href="card_fil.html#V5" target="_self">strong_limit</a>
 ;<br/>

<b>then consider </b><font color="Maroon">N</font> being   <a href="card_1.html#NM1">Cardinal</a><b> such that </b><br/><a NAME="E9:76"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">N</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E10:76"/><i><font color="Green">E58</font></i>: 
not  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">N</font> <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>
<a NAME="E11:76"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> 2,<font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#D3" target="_self">Def3</a>, <a class="ref" href="card_1.html#T14">CARD_1:14</a></i>;<br/>
<a NAME="E12:76"/><b>then </b>
<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="funct_2.html#K1">Funcs</a> <font color="Maroon">N</font>,2</span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#D3">CARD_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E14:76"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="funct_2.html#K1">Funcs</a> <font color="Maroon">N</font>,2
 <b>and </b><br/><a NAME="E15:76"/><i><font color="Green">E67</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="card_fil.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E16:76"/>
not <font color="Maroon">N</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_1"/>
<font color="Maroon">N</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:76_1"/><b>then </b>
 <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">N</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="finset_1.html#T24">FINSET_1:24</a></i>;<br/>
<a NAME="E3:76_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">N</font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/>
<a NAME="E4:76_1"/>
<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_2.html#K3">exp</a> 2,<span class="p1">(<span class="default"><a href="card_1.html#K1">Card</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E5:76_1"/><b>then </b>
<font color="Maroon">M</font> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">N</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T44">CARD_2:44</a></i>;<br/>
<b>hence </b><a NAME="E6:76_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">N1</font> = <font color="Maroon">N</font> as   <a href="card_5.html#NM1">Aleph</a> ;<br/>
<a NAME="E18:76"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">Y</font>,<font color="Maroon">M</font> <a href="tarski.html#R2">are_equipotent</a> 
 
<b>by </b><i><a class="ref" href="card_fil.html#T12" target="_self">Th12</a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E20:76"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E21:76"/><i><font color="Green">E71</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E22:76"/><i><font color="Green">E72</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">M</font>,<font color="Maroon">Y</font> <b>by </b><i><a class="txt" href="card_fil.html#E10:76"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_fil.html#E11:76"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_2.html#T3">FUNCT_2:3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>;<br/>
<a NAME="E24:76"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">A</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font>  ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Olive">i</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font>;<br/>

<b>set </b><font color="Maroon">Y0</font> = <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y1</font> = <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> ;<br/>
<b>reconsider </b><font color="Maroon">Inv0</font> = <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E10:76"><i><font color="Green">E58</font></i></a>, <a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<b>reconsider </b><font color="Maroon">Inv1</font> = <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E10:76"><i><font color="Green">E58</font></i></a>, <a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<a NAME="E3:76_2"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">g1</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span>  holds <br/>(  not <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> or <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  or <font color="Olive">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">g1</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span>;<br/>

<b>assume </b><a NAME="E1:76_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_2_1_1"/>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  or  not <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_1_1_1"><b>suppose </b></a><a NAME="E1:76_2_1_1_1"/>
<font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_2_1_1_1"/>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  or <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_1_1_2"><b>suppose </b></a><a NAME="E1:76_2_1_1_2"/>
not <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 ;<br/><div class="add"><a NAME="E2:76_2_1_1_2"/><b>then </b>
not <font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="card_fil.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:76_2_1_1_2"/><b>then </b>
<font color="Maroon">g1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E4:76_2_1_1_2"/>
( <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span>  or <font color="Maroon">g1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="card_fil.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:76_2"/>
<font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:76_2_2"/>
<font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_2_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 ;<br/>

<a NAME="E2:76_2_2_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="card_fil.html#D7" target="_self">Def7</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">g1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:76_2_2_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E5:76_2_2_1"/><i><font color="Green">E95</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font>
 <b>and </b><br/><a NAME="E6:76_2_2_1"/><i><font color="Green">E96</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">g1</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="card_fil.html#D4" target="_self">Def4</a>, <a class="ref" href="card_2.html#T22">CARD_2:22</a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g2</font> = <font color="Maroon">g1</font> as   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> <b>by </b><i><a class="txt" href="card_fil.html#E14:76"><i><font color="Green">E65</font></i></a>, <a class="ref" href="card_fil.html#T14" target="_self">Th14</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E8:76_2_2_1"/>
<font color="Maroon">g2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="card_fil.html#D6" target="_self">Def6</a>, , <a class="ref" href="card_fil.html#T13" target="_self">Th13</a></i>;<br/>
<b>hence </b><a NAME="E9:76_2_2_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="card_fil.html#T13" target="_self">Th13</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:76_2_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1 ) </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E4:76_2_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E5:76_2_2"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E6:76_2_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="card_fil.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E7:76_2_2"/>
not <font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_2_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:76_2_2_2"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">h1</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E3:76_2_2_2"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">h1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E4:76_2_2_2"/>
<font color="Maroon">h1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E5:76_2_2_2"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">h1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>
<b>thus </b><a NAME="E6:76_2_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E14:76"><i><font color="Green">E65</font></i></a>, <a class="ref" href="card_fil.html#T14" target="_self">Th14</a>, <a class="txt" href="card_fil.html#E15:76"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:76_2_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="card_fil.html#T13" target="_self">Th13</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E136</font></i>: <font color="Maroon">Inv1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Maroon">Inv0</font>
<b>by </b><i><a class="txt" href="card_fil.html#E11:76"><i><font color="Green">E59</font></i></a>, <a class="ref" href="funct_1.html#T138">FUNCT_1:138</a></i>
<br/>.= 
<font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Inv0</font>
<b>by </b><i><a class="txt" href="card_fil.html#E10:76"><i><font color="Green">E58</font></i></a>, <a class="ref" href="relat_1.html#T169">RELAT_1:169</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_4"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_2_4"/>
( <font color="Maroon">Inv0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> or <font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Inv0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 </span><b>by </b><i><a class="ref" href="card_fil.html#D2" target="_self">Def2</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_4_1"><b>suppose </b></a><a NAME="E1:76_2_4_1"/><i><font color="Green">E137</font></i>: 
<font color="Maroon">Inv0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">Z</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">Z</font>
;<br/><b>thus </b><a NAME="E3:76_2_4_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">Z</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_2_4_2"><b>suppose </b></a><a NAME="E1:76_2_4_2"/><i><font color="Green">E138</font></i>: 
<font color="Maroon">M</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Inv0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">O</font> = 1 as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>take </b>
<font color="Maroon">O</font>
;<br/><b>thus </b><a NAME="E3:76_2_4_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">O</font>]
 <b>by </b><i><a class="ref" href="card_fil.html#D7" target="_self">Def7</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E25:76"/><i><font color="Green">E139</font></i>: 
not <font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T68">CARD_1:68</a></i>;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E27:76"/><i><font color="Green">E140</font></i>: 
for <font color="Olive">A</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">A</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="ref" href="card_fil.html#D5" target="_self">Def5</a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">f</font> <a href="relat_1.html#K10">"</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> ) </span>}</span> ;<br/>
<a NAME="E28:76"/>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">N1</font>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2">TREES_2:sch 2</a>();<br/></i>
<a NAME="E29:76"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  <a href="card_1.html#NR1" target="_self">&lt;=`</a> <font color="Maroon">N1</font>
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E30:76"/><b>then </b><i><font color="Green">E141</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  <a href="card_1.html#NR2" target="_self">&lt;`</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T11" target="_self">Th11</a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>
<a NAME="E31:76"/><i><font color="Green">E142</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_3"/><i><font color="Green">E143</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">A</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font><b> such that </b><br/><a NAME="E3:76_3"/><i><font color="Green">E144</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">A</font>) &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> )
 <b>by </b><i><a class="txt" href="card_fil.html#E14:76"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:76_3"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="card_fil.html#D7" target="_self">Def7</a>, <a class="ref" href="card_fil.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">B</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font>;<br/>
<a NAME="E33:76"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">B</font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> 
 
;<br/>
<a NAME="E34:76"/><b>then </b><i><font color="Green">E145</font></i>: 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T9" target="_self">Th9</a>, , <a class="ref" href="card_fil.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E35:76"/><i><font color="Green">E146</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_4"/><i><font color="Green">E147</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">X1</font> = <font color="Maroon">X</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E14:76"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E3:76_4"/>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/>
<b>then consider </b><font color="Maroon">h1</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:76_4"/><i><font color="Green">E148</font></i>: 
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font>
 <b>and </b><br/><a NAME="E6:76_4"/><i><font color="Green">E149</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E7:76_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">h1</font> <a href="tarski.html#R1">c=</a> 2
 <b>by </b><i><a class="ref" href="card_fil.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<a NAME="E8:76_4"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">Z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> holds <br/><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">Z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:76_4_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">Z1</font> = <font color="Maroon">Z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> <b>by </b><i><a class="txt" href="card_fil.html#E21:76"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E3:76_4_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">Z1</font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> 
 
;<br/>
<a NAME="E4:76_4_1"/><b>then </b>
<font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">Z1</font>)
 
<b>by </b><i><a class="txt" href="card_fil.html#E15:76"><i><font color="Green">E67</font></i></a>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>
<a NAME="E5:76_4_1"/><b>then </b>
<font color="Maroon">f1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">X1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">h</font> where <font color="Olive">h</font> is   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span> : ( <font color="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Z1</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">h</font> being   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">N1</font>,<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,1</span><a href="tarski.html#K2">}</a></span><b> such that </b><br/><a NAME="E7:76_4_1"/><i><font color="Green">E153</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font>
 <b>and </b><br/><a NAME="E8:76_4_1"/>
<font color="Maroon">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E9:76_4_1"/><i><font color="Green">E155</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Z1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Z1</font>
 ;<br/>
<b>thus </b><a NAME="E10:76_4_1"/>
<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="txt" href="card_fil.html#E22:76"><i><font color="Green">E72</font></i></a>, <a class="txt" href="card_fil.html#E24:76"><i><font color="Green">E73</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:76_4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E10:76_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="txt" href="card_fil.html#E18:76"><i><font color="Green">E69</font></i></a>, <a class="txt" href="card_fil.html#E20:76"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">MEET</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> ;<br/>
<a NAME="E36:76"/>
 <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E10:76"><i><font color="Green">E58</font></i></a>, <a class="txt" href="card_fil.html#E14:76"><i><font color="Green">E65</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E37:76"/><b>then </b><i><font color="Green">E156</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p3">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="relat_1.html#T90">RELAT_1:90</a></i>;<br/>
<a NAME="E38:76"/><i><font color="Green">E158</font></i>: 
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p3">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T148">RELAT_1:148</a></i>;<br/>
<a NAME="E39:76"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">g</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:76_5"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:76_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:76_5"/><i><font color="Green">E161</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> 
 <b>and </b><br/><a NAME="E5:76_5"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="txt" href="card_fil.html#E20:76"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:76_5"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T14" target="_self">Th14</a>, <a class="txt" href="card_fil.html#E21:76"><i><font color="Green">E71</font></i></a>, <a class="txt" href="card_fil.html#E22:76"><i><font color="Green">E72</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:76_5"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">g</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E40:76"/>
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T84">FUNCT_1:84</a></i>;<br/>
<a NAME="E41:76"/><b>then </b><i><font color="Green">E163</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K1">meet</a> <span class="p2">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">g</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="card_fil.html#E15:76"><i><font color="Green">E67</font></i></a>, , <a class="txt" href="card_fil.html#E18:76"><i><font color="Green">E69</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">MEET</font> =  <a href="setfam_1.html#K1">meet</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">A</font>) where <font color="Olive">A</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">N1</font> : <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">N1</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">M</font> <b>by </b><i><a class="txt" href="card_fil.html#E14:76"><i><font color="Green">E65</font></i></a></i>;<br/>
<a NAME="E43:76"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">MEET</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="card_fil.html#T10" target="_self">Th10</a>, <a class="txt" href="card_fil.html#E14:76"><i><font color="Green">E65</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E44:76"/>
contradiction
 <b>by </b><i><a class="txt" href="card_fil.html#E20:76"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div>
