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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" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon" title="c1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:61_1"/><i><font color="Green" title="E34">A3</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E1:61"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="ordinal1.html#E2:61"><i><font color="Green" title="E33">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b><font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c2">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:61_1"/><i><font color="Green" title="E35">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E6:61_1"/><i><font color="Green" title="E36">A5</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal1.html#E3:61_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="txt" href="ordinal1.html#E5:61_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/>
<a NAME="E7:61_1"/>
for <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="ordinal1.html#E5:61_1"><i><font color="Green" title="E35">A4</font></i></a></i>;<br/>
<a NAME="E8:61_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E10:61_1"/><i><font color="Green" title="E37">A6</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> holds <br/><font color="Maroon" title="c3">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">C</font> ) )
 <b>by </b><i><a class="txt" href="ordinal1.html#E6:61_1"><i><font color="Green" title="E36">A5</font></i></a>, <a class="ref" href="ordinal1.html#T32" target="_self" title="ORDINAL1:th.32">Th32</a></i>;<br/>
<a NAME="E11:61_1"/><i><font color="Green" title="E38">A7</font></i>: 
( <font color="Maroon" title="c3">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">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" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E5:61_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="ordinal1.html#E10:61_1"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<a NAME="E12:61_1"/><b>then </b><i><font color="Green" title="E39">A8</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2" target="_self" title="ORDINAL1:def.2">Def2</a></i>;<br/>
<div><i><font color="Green" title="E40">A9</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" title="c4">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:61_1_1"/><i><font color="Green" title="E41">A10</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/><a NAME="E2:61_1_1"/><b>then </b>
not <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E10:61_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="ref" href="ordinal1.html#T7" target="_self" title="ORDINAL1:th.7">Th7</a></i>;<br/><b>hence </b><a NAME="E3:61_1_1"/>
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E5:61_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="ordinal1.html#E12:61_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="txt" href="ordinal1.html#E1:61_1_1"><i><font color="Green" title="E41">A10</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:61_1"/><i><font color="Green" title="E41">A11</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E1:61"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="ordinal1.html#E2:61"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="ordinal1.html#E12:61_1"><i><font color="Green" title="E39">A8</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">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" title="c4">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1_2"/><i><font color="Green" title="E42">A12</font></i>: 
<font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c4">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" target="_self" title="ORDINAL1:th.23">Th23</a></i>;<br/><a NAME="E3:61_1_2"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">a'</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E13:61_1"><i><font color="Green" title="E40">A9</font></i></a>, <a class="txt" href="ordinal1.html#E1:61_1_2"><i><font color="Green" title="E42">A12</font></i></a></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" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> )
 <b>by </b><i><a class="txt" href="ordinal1.html#E14:61_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="ordinal1.html#E1:61_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">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" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="ordinal1.html#E3:61_1_2"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E16:61_1"/><b>then </b><i><font color="Green" title="E42">A14</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="ordinal1.html#E14:61_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">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" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span>) &amp; <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c3">B</font></span>)</span>) )
 
<b>by </b><i><a class="txt" href="ordinal1.html#E1:61"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="ordinal1.html#E2:61"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="ordinal1.html#E11:61_1"><i><font color="Green" title="E38">A7</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:61_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal1.html#E5:61_1"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="ordinal1.html#E10:61_1"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="ordinal1.html#E16:61_1"><i><font color="Green" title="E42">A14</font></i></a></i>;<br/>


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