<?xml version="1.0"?>
<div class="add">

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



<b>assume </b><a NAME="E1:121"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> or <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span> )
 ;<br/>

<b>assume </b><a NAME="E2:121"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/><b>consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:121_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E11:121_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:121_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 <b>and </b><br/><a NAME="E7:121_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> )
 <b>by </b><i><a class="ref" href="tarski.html#T7">TARSKI:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:121_1_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E6:121_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<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="E4:121_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E5:121_1_1"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><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>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E6:121_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<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> <font color="Maroon">Y1</font>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E7:121_1_1"/>
<font color="Maroon">Y1</font> <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E52"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="zfmisc_1.html#E2"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E8:121_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="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E55"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E9:121_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> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E10:121_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E50"><i><font color="Green">Lemma61</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green">Lemma64</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E9:121_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E12:121_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">Y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E11:121_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>and </b><br/><a NAME="E12:121_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E13:121_1"/>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<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="E15:121_1"/><i><font color="Green">E92</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E16:121_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i/>;<br/><a NAME="E17:121_1"/>
<font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><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>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E18:121_1"/><b>then </b>
( <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <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> or <font color="Maroon">Y1</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="txt" href="zfmisc_1.html#E52"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E19:121_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="tarski.html#K3">union</a> <font color="Maroon">X</font></span>)</span> )
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E55"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E20:121_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E50"><i><font color="Green">Lemma61</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">Y</font> &amp;  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) );<br/><b>consider </b><font color="Maroon">Z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_2"/><i><font color="Green">E96</font></i>: 
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> iff ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i><a NAME="E4:121_2"/>
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:121_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">Y2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:121_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z</font>
 <b>and </b><br/><a NAME="E9:121_2"/><i><font color="Green">E99</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z</font> or  not <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font> )
 <b>by </b><i>, <a class="ref" href="tarski.html#T7">TARSKI:7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_2_1"/><i><font color="Green">E100</font></i>: 
for <font color="Olive">Y2</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or  ex <font color="Olive">Y1</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">Y1</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y2</font> &amp;  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y1</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) ) )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:121_2_1_1"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:121_2_1_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>and </b><br/><a NAME="E4:121_2_1_1"/><i><font color="Green">E103</font></i>: 
 ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33" target="_self">Th33</a></i>;<br/><a NAME="E5:121_2_1_1"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E55"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" target="_self">Th38</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/><a NAME="E6:121_2_1_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T43" target="_self">Th43</a></i>;<br/><a NAME="E7:121_2_1_1"/><b>then </b>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E8:121_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E52"><i><font color="Green">Lemma62</font></i></a>, <a class="ref" href="zfmisc_1.html#T38" target="_self">Th38</a></i>;<br/></div><b>end;</b></div><a NAME="E3:121_2_1"/><b>then </b>
<font color="Maroon">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E50"><i><font color="Green">Lemma61</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:121_2_1"/><b>then </b>
 ex <font color="Olive">X2</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">Y2</font> <a href="hidden.html#R1">=</a> <font color="Olive">X2</font> &amp;  ex <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">X2</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">z</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:121_2_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y2</font>
 <b>and </b><br/><a NAME="E7:121_2_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E8:121_2_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Z</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T38" target="_self">Th38</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E9:121_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E52"><i><font color="Green">Lemma62</font></i></a>, <a class="txt" href="zfmisc_1.html#E55"><i><font color="Green">Lemma64</font></i></a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">Y1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:121_2"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E13:121_2"/><i><font color="Green">E107</font></i>: 
for <font color="Olive">Y2</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">Y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> or for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   holds <br/>(  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y2</font> or  not <font color="Olive">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> ) )
 ;<br/><div><i><font color="Green">E108</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="121_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E1:121_2_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:121_2_2"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">Y1</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">x</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/><a NAME="E3:121_2_2"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T55" target="_self">Th55</a></i>;<br/><a NAME="E4:121_2_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">Y1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E5:121_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="zfmisc_1.html#E55"><i><font color="Green">Lemma64</font></i></a>, <a class="ref" href="zfmisc_1.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div><a NAME="E15:121_2"/>
<font color="Maroon">Y1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" target="_self">[:</a><span class="default"><font color="Maroon">Y</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2" target="_self">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T33" target="_self">Th33</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E16:121_2"/><b>then </b>
 ex <font color="Olive">y</font>, <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">Y1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E17:121_2"/>
contradiction
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T38" target="_self">Th38</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:121"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div>
