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<b>let </b><font color="Maroon">Z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>thus </b><a NAME="E1:40"/>
(  not <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</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> or <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">Z</font> <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:40_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">Z</font> <a href="tarski.html#R1">c=</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>and </b><br/><a NAME="E2:40_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:40_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E4:40_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:40_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> implies <font color="Maroon">z</font> <a href="hidden.html#R2">in</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><a class="txt" href="zfmisc_1.html#E2"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:40_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E1:40_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmisc_1.html#E4"><i><font color="Green">Lemma28</font></i></a>, </i>;<br/><a NAME="E5:40_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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><a class="txt" href="zfmisc_1.html#E2"><i><font color="Green">Lemma25</font></i></a>, <a class="txt" href="zfmisc_1.html#E2:40_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E6:40_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D1" target="_self">Def1</a>, <a class="txt" href="zfmisc_1.html#E2:40_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmisc_1.html#E3:40_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_2"/><i><font color="Green">E58</font></i>: 
not <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/><b>consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E4:40_1_1_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E1:40_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="zfmisc_1.html#E3"><i><font color="Green">Lemma27</font></i></a>, </i>;<br/><a NAME="E5:40_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</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><a class="txt" href="zfmisc_1.html#E2"><i><font color="Green">Lemma25</font></i></a>, <a class="txt" href="zfmisc_1.html#E3:40_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E6:40_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E2:40_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="zfmisc_1.html#E3:40_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="zfmisc_1.html#E4:40_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><b>assume </b><a NAME="E4:40_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</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>
 ;<br/><b>hence </b><a NAME="E5:40_1_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#D1" target="_self">Def1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:40_1"/>
<font color="Maroon">Z</font> <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><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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

<b>thus </b><a NAME="E2:40"/>
( ( <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x1</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x2</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">Z</font> <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> ) implies <font color="Maroon">Z</font> <a href="tarski.html#R1">c=</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>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


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