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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E37</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><b>then reconsider </b><font color="Maroon">X'</font> = <font color="Maroon">X</font> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="card_4.html#T1">CARD_4:1</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:17_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">X'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, <a class="ref" href="card_2.html#T79">CARD_2:79</a></i>;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">x</font>;<br/><b>take </b><font color="Maroon">y</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E5:17_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) ) )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:17"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; ( for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) ) )
 ;<br/>

<a NAME="E3:17"/>
for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:17"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></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>hence </b><a NAME="E5:17"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, <a class="ref" href="card_2.html#T76">CARD_2:76</a></i>;<br/>


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