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



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">z</font> being  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:12_1"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> 2 )
 ;<br/><b>thus </b><a NAME="E4:12_1"/>
<font color="Maroon">e</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font>
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">e'</font> = <font color="Maroon">e</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><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="E7:12_1"/><i><font color="Green">E22</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">e'</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/><a NAME="E8:12_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">e'</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E9:12_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/><b>thus </b><a NAME="E10:12_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font> &amp; <font color="Maroon">e</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/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:12"/>
( <font color="Maroon">e</font> is  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> &amp;  ex <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">y</font> &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K2">}</a></span> ) )
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:12"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">e</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> )
 ;<br/>
<b>reconsider </b><font color="Maroon">xy</font> = <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> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E6:12"/>
 ex <font color="Olive">z</font> being <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">A</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font> &amp;  <a href="card_1.html#K4">card</a> <font color="Olive">z</font> <a href="hidden.html#R1">=</a> 2 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">xy</font>
;<br/>

<b>thus </b><a NAME="E1:12_2"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xy</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:12_2"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">xy</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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">A</font>
 ;<br/>


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