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<b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T14" target="_self">Th14</a>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:61_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E6:61_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:61_1"/>
for <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i/>;<br/>
<a NAME="E8:61_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:61_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">B</font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Olive">C</font> ) )
 <b>by </b><i>, <a class="txt" href="ordinal1.html#E1:61"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E11:61_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E12:61_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">C</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:61_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:61_1_1"/><b>then </b>
not <font color="Maroon">B</font> <a href="ordinal1.html#R1" target="_self">c=</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/><a NAME="E3:61_1_1"/><b>then </b>
not <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E4:61_1_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="txt" href="ordinal1.html#E18"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="ordinal1.html#D4" target="_self">Def4</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:61_1"/><i><font color="Green">E79</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T14" target="_self">Th14</a>, , <a class="txt" href="ordinal1.html#E18"><i><font color="Green">Lemma39</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i/>;<br/><a NAME="E3:61_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a'</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:61_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#D4" target="_self">Def4</a>, , <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E5:61_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E16:61_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#D4" target="_self">Def4</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<a NAME="E17:61_1"/>
( <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span>) &amp; <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" target="_self">|</a> <font color="Maroon">B</font></span>)</span>) )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T14" target="_self">Th14</a>, , <a class="ref" href="ordinal1.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E18:61_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D2" target="_self">Def2</a>, </i>;<br/>


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