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

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>6</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:39"><i><font color="Green">E14</font></i></a>, <a class="txt" href="ordinal2.html#E5:39"><i><font color="Green">E18</font></i></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> ] means <font color="Maroon">F<sub>5</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>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:39_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff ( <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E6:39_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="ordinal2.html#E3:39_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="ordinal2.html#E5:39_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E7:39_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="ordinal2.html#E5:39_1"><i><font color="Green">E23</font></i></a></i>;<br/>
<a NAME="E8:39_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:39_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E6:39_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="ordinal1.html#T32">ORDINAL1:32</a></i>;<br/>
<a NAME="E11:39_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E5:39_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ordinal2.html#E10:39_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E12:39_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:39_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:39_1_1"/><b>then </b>
not <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/><a NAME="E3:39_1_1"/><b>then </b>
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E10:39_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_1_1"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:39_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ordinal2.html#E12:39_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal2.html#E1:39_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:39_1"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:39"><i><font color="Green">E14</font></i></a>, <a class="txt" href="ordinal2.html#E5:39"><i><font color="Green">E18</font></i></a>, <a class="txt" href="ordinal2.html#E12:39_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:39_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E3:39_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E13:39_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal2.html#E1:39_1_2"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E4:39_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E14:39_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal2.html#E1:39_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E5:39_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:39_1_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:39_1_3"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:39_1_3"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>) &amp; <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:39"><i><font color="Green">E16</font></i></a>, <a class="txt" href="ordinal2.html#E7:39"><i><font color="Green">E20</font></i></a>, <a class="txt" href="ordinal2.html#E11:39_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ordinal2.html#E1:39_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E3:39_1_3"/><b>then </b>
( <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E4:39_1_3"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E15:39_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ordinal2.html#E2:39_1_3"><i><font color="Green">E33</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1_4"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><a NAME="E2:39_1_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><a NAME="E3:39_1_4"/><b>then </b>
( <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>) &amp; <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="ordinal1.html#K2">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:39"><i><font color="Green">E17</font></i></a>, <a class="txt" href="ordinal2.html#E8:39"><i><font color="Green">E21</font></i></a>, <a class="txt" href="ordinal2.html#E11:39_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ordinal2.html#E1:39_1_4"><i><font color="Green">E32</font></i></a></i>;<br/><b>hence </b><a NAME="E4:39_1_4"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E14:39_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal2.html#E15:39_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:39_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:39"><i><font color="Green">E15</font></i></a>, <a class="txt" href="ordinal2.html#E6:39"><i><font color="Green">E19</font></i></a>, <a class="txt" href="ordinal2.html#E5:39_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="ordinal2.html#E10:39_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="ordinal2.html#E16:39_1"><i><font color="Green">E31</font></i></a></i>;<br/>


</div>
