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<b>take </b><font color="Maroon" title="c1">X</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<a NAME="E1:113_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#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="card_2.html#T13" title="CARD_2:th.13">CARD_2:13</a></i>;<br/>
<a NAME="E2:113_1_1"/><b>then </b>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <font color="Maroon" title="c1">X</font> <a href="aofa_000.html#NR1" title="AOFA_000:NR.1">nin</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T84" title="CARD_1:th.84">CARD_1:84</a></i>;<br/>
<b>hence </b><a NAME="E3:113_1_1"/>
not <font color="Maroon" title="c1">X</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="ref" href="card_4.html#T3" title="CARD_4:th.3">CARD_4:3</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="113_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:113_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:113_1_1_1"/><i><font color="Green" title="E73">A1</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c4">b</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/><b>assume </b><a NAME="E4:113_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><a NAME="E5:113_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">x</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> is <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a> )
 ;<br/><b>hence </b><a NAME="E6:113_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="aofa_000.html#E3:113_1_1_1"><i><font color="Green" title="E73">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:113_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="freealg.html#D1" title="FREEALG:def.1">FREEALG:def 1</a><br/>


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