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<b>let </b><font color="Maroon">X</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:17"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 ;<br/>

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> is <a href="facirc_1.html#V2">with_pair</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:17_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span>
 ;<br/><a NAME="E2:17_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:17_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <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="Olive">a</font>,<font color="Olive">b</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/><b>hence </b><a NAME="E4:17_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="facirc_1.html#D2">FACIRC_1:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:17"/><b>then </b>
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E5:17"/><b>then </b>
<font color="Maroon">X</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:17"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>;<br/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font> = 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>
<br/>.= 
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T41">XBOOLE_1:41</a></i>
;<br/>
<b>hence </b><a NAME="E8:17"/>
<font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">Z</font>
 ;<br/>


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