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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:100_1_1_1"/>
( <font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_1"><b>suppose </b></a><a NAME="E1:100_1_1_1_1"/>
<font color="Maroon">R</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:100_1_1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="100_1_1_1_2"><b>suppose </b></a><a NAME="E1:100_1_1_1_2"/>
<font color="Maroon">R</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">R</font> = <font color="Maroon">R</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="relat_1.html#NM1">Relation</a> ;<br/><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </span>}</span> ;<br/><a NAME="E3:100_1_1_1_2"/><i><font color="Green">E31</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="100_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:100_1_1_1_2_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </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="100_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</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="E1:100_1_1_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 ;<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="E3:100_1_1_1_2_1_1"/><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<a NAME="E4:100_1_1_1_2_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:100_1_1_1_2_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </span>}</span> 
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">e</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="E2:100_1_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E4:100_1_1_1_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> 
 <b>and </b><br/><a NAME="E5:100_1_1_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 ;<br/>
<b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:100_1_1_1_2_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">z</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E8:100_1_1_1_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font>
 
<b>by </b><i>, , <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<b>hence </b><a NAME="E9:100_1_1_1_2_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:100_1_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">R</font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E5:100_1_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a> </span>)</span> where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">R</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i><b>hence </b><a NAME="E6:100_1_1_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">R</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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