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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">e1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">e2</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:14"/><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="tarski.html#K2">}</a></span> <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">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="E3:14"/><i><font color="Green">E22</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; <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">e1</font>,<font color="Maroon">e2</font></span><a href="tarski.html#K2">}</a></span> <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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:14_1"/>
( ( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) or ( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> ) or ( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 </span><b>by </b><i>, <a class="ref" href="zfmisc_1.html#T10">ZFMISC_1:10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_1"><b>suppose </b></a><a NAME="E1:14_1_1"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><div class="add"><a NAME="E2:14_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:14_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:14_1_1"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_2"><b>suppose </b></a><a NAME="E1:14_1_2"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_2"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_3"><b>suppose </b></a><a NAME="E1:14_1_3"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:14_1_3"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="14_1_4"><b>suppose </b></a><a NAME="E1:14_1_4"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 ;<br/><div class="add"><a NAME="E2:14_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="sgraph1.html#K2" target="_self">TWOELEMENTSETS</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="enumset1.html#T69">ENUMSET1:69</a></i>;<br/><b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:14_1_4"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:14_1_4"/>
( <font color="Maroon">e1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">e2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e2</font> )
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T9">ZFMISC_1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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