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

<b>let </b><font color="Maroon">R</font> be   <a href="relat_1.html#NM1">Relation</a>;<br/><b>let </b><font color="Maroon">FIELD</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">R</font> <a href="relat_2.html#R1">is_reflexive_in</a> <font color="Maroon">FIELD</font> &amp; <font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a> <font color="Maroon">FIELD</font> &amp; <font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a> <font color="Maroon">FIELD</font> )
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green">E19</font></i>: 
 <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> <a href="hidden.html#R1">=</a> <font color="Maroon">FIELD</font>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">FIELD</font> <a href="orders_1.html#R4" target="_self">has_upper_Zorn_property_wrt</a> <font color="Maroon">R</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D7" target="_self">ORDERS_1:def 7</a><br/>

<b>reconsider </b><font color="Maroon">XD</font> = <font color="Maroon">FIELD</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">D</font> being  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:87"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">XD</font>
 ;<br/>
<a NAME="E7:87"/>
( not <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a>, <a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1">(<span class="default"><a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="orders_1.html#M1" target="_self">Choice_Function</a> of <font color="Maroon">M</font>;<br/>
<a NAME="E10:87"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E11:87"/><b>then </b><i><font color="Green">E25</font></i>: 
not  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) or ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> ) );<br/>
<a NAME="E12:87"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:87_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:87_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies  ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:87_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:87"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
;<br/>
<b>consider </b><font color="Maroon">g</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E15:87"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(<a class="txt" href="orders_1.html#E21:87"><i><font color="Green">E56</font></i></a>, );<br/></i>
<a NAME="E16:87"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>proof </b></a><div class="add">

<a NAME="E1:87_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<b>hence </b><a NAME="E2:87_2"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E17:87"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> &amp; <font color="Olive">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:87_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:87_3"/><b>then </b>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">X</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">X</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font>] &amp; not <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E3:87_3"/><b>then </b>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E4:87_3"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E18:87"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> holds  <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">L</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E1:87_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:87_4"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:87_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L1</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:87_5"/><i><font color="Green">E35</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 <b>and </b><br/><a NAME="E2:87_5"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 ;<br/><b>let </b><font color="Maroon">C</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E3:87_5"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 ;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> implies <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E4:87_5"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">A1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">A2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A2</font> <a href="hidden.html#R2">in</a> <font color="Olive">A1</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A2</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A1</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:87_5_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">A2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Olive">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> holds <br/><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A2</font>
 <b>and </b><br/><a NAME="E2:87_5_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:87_5_1"/>
( <font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:87_5_1"/><b>then </b><i><font color="Green">E43</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_5_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_5_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A3</font> = <font color="Maroon">x</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="E3:87_5_1_1"/>
<font color="Maroon">A3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i>, </i>;<br/><a NAME="E4:87_5_1_1"/><b>then </b>
( <font color="Maroon">A3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><a NAME="E5:87_5_1_1"/><b>then </b>
( <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E6:87_5_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:87_5_1"/>
<font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:87_5"/>
for <font color="Olive">A1</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A1</font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i><b>hence </b><a NAME="E6:87_5"/>
<font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font>
 ;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L1</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">L1</font> <a href="hidden.html#R1">=</a> <font color="Olive">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font> holds <br/><font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) );<br/>
<a NAME="E20:87"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>,<font color="Olive">x</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>,<font color="Olive">y</font>] holds <br/><font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_6"><b>proof </b></a><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>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>given </b><font color="Maroon">L1</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E1:87_6"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:87_6"/><i><font color="Green">E47</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 ;<br/>

<b>given </b><font color="Maroon">L2</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E3:87_6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:87_6"/><i><font color="Green">E50</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> implies <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E5:87_6"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">A1</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">A2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A2</font> <a href="hidden.html#R2">in</a> <font color="Olive">A1</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A2</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A1</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A1</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:87_6_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">A2</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Olive">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A2</font>
 <b>and </b><br/><a NAME="E2:87_6_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:87_6_1"/><i><font color="Green">E54</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> )
 
<b>by </b><i>, , , <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:87_6_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A3</font> = <font color="Maroon">x</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="E3:87_6_1_1"/>
( <font color="Maroon">A3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A3</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><a NAME="E4:87_6_1_1"/><b>then </b>
( <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font> &amp; <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp; <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A3</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  )
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E5:87_6_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:87_6_1"/>
<font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:87_6"/>
for <font color="Olive">A1</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A1</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<a NAME="E7:87_6"/><b>then </b>
( <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> &amp; <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/>
<b>hence </b><a NAME="E8:87_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:87"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><br/> for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>,<font color="Olive">d</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>,<font color="Olive">d</font>] holds <br/><font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>given </b><font color="Maroon">L1</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E1:87_7"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E2:87_7"/><i><font color="Green">E58</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 ;<br/>

<b>given </b><font color="Maroon">L2</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E3:87_7"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:87_7"/><i><font color="Green">E60</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 ;<br/>

<div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>assume </b><a NAME="E1:87_7_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:87_7_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E6:87_7_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><b>then </b><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> = 
<font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font>
<b>by </b><i><a class="txt" href="orders_1.html#E5:87_7"><i><font color="Green">E61</font></i></a>, <a class="txt" href="orders_1.html#E5:87_7_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="orders_1.html#E6:87_7_1"><i><font color="Green">E67</font></i></a></i>
<br/>.= 
<font color="Maroon">L1</font>
<b>by </b><i><a class="txt" href="orders_1.html#E5:87_7_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/><a NAME="E4:87_7_1"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font>
 <b>by </b><i><a class="txt" href="orders_1.html#E4:87_7_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="orders_1.html#E6:87_7_1"><i><font color="Green">E67</font></i></a>, , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:87_7_1"/><i><font color="Green">E65</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_7_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:87_7_1_2"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:87_7_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:87_7_1"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ]
 <b>by </b><i><a class="txt" href="orders_1.html#E1:87_7"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E7:87_7_1"/><i><font color="Green">E68</font></i>: 
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_7_1_3"/>
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:87_7_1_3"/><b>then </b>
( not <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ] )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:87_7_1_3"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="orders_1.html#E1:87_7"><i><font color="Green">E57</font></i></a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:87_7_1_3"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><a NAME="E5:87_7_1_3"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 ;<br/><b>hence </b><a NAME="E6:87_7_1_3"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:87_7_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>assume </b><a NAME="E1:87_7_2"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:87_7_2"/><b>then </b><i><font color="Green">E70</font></i>: 
( <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp; <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><b>then </b><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font> = 
<font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font>
<b>by </b><i><a class="txt" href="orders_1.html#E5:87_7"><i><font color="Green">E61</font></i></a>, , </i>
<br/>.= 
<font color="Maroon">L2</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/><a NAME="E4:87_7_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font>
 <b>by </b><i>, , <a class="txt" href="orders_1.html#E1:87_7_2"><i><font color="Green">E69</font></i></a>, <a class="txt" href="orders_1.html#E2:87_7_2"><i><font color="Green">E70</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:87_7_2"/><i><font color="Green">E72</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_7_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:87_7_2_2"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:87_7_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div><a NAME="E6:87_7_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ]
 <b>by </b><i><a class="txt" href="orders_1.html#E1:87_7"><i><font color="Green">E57</font></i></a></i>;<br/><a NAME="E7:87_7_2"/><i><font color="Green">E142</font></i>: 
not <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">d</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_7_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_7_2_3"/>
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:87_7_2_3"/><b>then </b>
( not <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ] )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:87_7_2_3"/><b>then </b>
( <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="orders_1.html#E1:87_7"><i><font color="Green">E57</font></i></a>, <a class="txt" href="orders_1.html#E5:87_7_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:87_7_2_3"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, , </i>;<br/><a NAME="E5:87_7_2_3"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 ;<br/><b>hence </b><a NAME="E6:87_7_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="orders_1.html#E4:87_7_2"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:87_7_2"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:87_7"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">D</font>];<br/>
<a NAME="E22:87"/><i><font color="Green">E144</font></i>: 
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:87_8"/><i><font color="Green">E145</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds  not <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>,<font color="Maroon">D</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">d</font>];<br/>
<a NAME="E2:87_8"/><i><font color="Green">E147</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:87_8_1"/><i><font color="Green">E148</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>,<font color="Olive">d</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">C</font>,<font color="Maroon">a<sub>2</sub></font>] );<br/>
<a NAME="E2:87_8_1"/><i><font color="Green">E149</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:87_8_1"/><i><font color="Green">E150</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:87_8_1_1"/><i><font color="Green">E151</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">C</font> = <font color="Maroon">x</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/>
<b>consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:87_8_1_1"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">C</font>,<font color="Maroon">d</font>]
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">d</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>take </b>
<font color="Maroon">C</font>
;<br/>

<b>thus </b><a NAME="E6:87_8_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">C</font>,<font color="Maroon">y</font>] )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">h</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E5:87_8_1"/><i><font color="Green">E153</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2">FUNCT_1:sch 2</a>(, );<br/></i>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">h</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#D7">ORDINAL1:def 7</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/>
<a NAME="E7:87_8_1"/><i><font color="Green">E154</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_1_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">h</font>
;<br/>

<b>thus </b><a NAME="E1:87_8_1_2"/>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/>



<b>assume </b><a NAME="E2:87_8_1_2"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E3:87_8_1_2"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">C</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>] )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">L1</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E5:87_8_1_2"/><i><font color="Green">E156</font></i>: 
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E6:87_8_1_2"/><i><font color="Green">E157</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 ;<br/>
<a NAME="E7:87_8_1_2"/><i><font color="Green">E158</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:87_8_1_2_1"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">x</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="E3:87_8_1_2_1"/><i><font color="Green">E160</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A1</font>,<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_1_2_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">K</font>
;<br/>

<b>thus </b><a NAME="E2:87_8_1_2_1_1"/>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">K</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>

<a NAME="E3:87_8_1_2_1_1"/>
<font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:87_8_1_2_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon">A2</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/>



<b>assume </b><a NAME="E5:87_8_1_2_1_1"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A2</font> )
 ;<br/>

<a NAME="E6:87_8_1_2_1_1"/><b>then </b><i><font color="Green">E163</font></i>: 
( <font color="Maroon">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E7:87_8_1_2_1_1"/><b>then </b>
( <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a></i>;<br/>
<b>hence </b><a NAME="E8:87_8_1_2_1_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="orders_1.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:87_8_1_2_1"/>
<font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E5:87_8_1_2_1"/><b>then </b><i><font color="Green">E164</font></i>: 
 ex <font color="Olive">A2</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A2</font>,<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>] )
 
<b>by </b><i/>;<br/>
<a NAME="E6:87_8_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E7:87_8_1_2_1"/>
<font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="txt" href="orders_1.html#E11"><i><font color="Green">Lemma77</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:87_8_1_2"/>
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E9:87_8_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="ordinal1.html#K2">|</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="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E10:87_8_1_2"/><b>then </b>
<font color="Maroon">h</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font>
 
<b>by </b><i>, <a class="ref" href="orders_1.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>hence </b><a NAME="E11:87_8_1_2"/>
<font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:87_8_1"/><b>then </b><i><font color="Green">E165</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <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="E9:87_8_1"/><i><font color="Green">E166</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_8_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:87_8_1_3"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 
;<br/>
<b>hence </b><a NAME="E3:87_8_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:87_8_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">h</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E12:87_8_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">d</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:87_8"/><i><font color="Green">E167</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>,<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E4:87_8"/><i><font color="Green">E168</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>] holds <br/><font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_2"><b>proof </b></a><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>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>given </b><font color="Maroon">A1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:87_8_2"/><i><font color="Green">E169</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A1</font>,<font color="Maroon">x</font>] )
 ;<br/>

<b>given </b><font color="Maroon">A2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E2:87_8_2"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A2</font>,<font color="Maroon">x</font>] )
 ;<br/>

<b>consider </b><font color="Maroon">d1</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E4:87_8_2"/><i><font color="Green">E171</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A1</font>,<font color="Maroon">d1</font>]
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">d2</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E6:87_8_2"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A2</font>,<font color="Maroon">d2</font>]
 <b>by </b><i/>;<br/>
<a NAME="E7:87_8_2"/>
( <font color="Maroon">d1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">d2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E8:87_8_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:87_8"/><i><font color="Green">E173</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  ex <font color="Olive">y</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">D</font> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>] ) )
 <b>from </b><i><a class="ref" href="tarski.html#S1">TARSKI:sch 1</a>();<br/></i>
<a NAME="E7:87_8"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_8_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<a NAME="E1:87_8_3"/>
 ex <font color="Olive">d</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A</font>,<font color="Olive">d</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:87_8_3"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:87_8"/>
contradiction
 <b>by </b><i><a class="ref" href="ordinal1.html#T38">ORDINAL1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E24:87"/><i><font color="Green">E174</font></i>: 
( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">A</font>] &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font> ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>,<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="E26:87"/><i><font color="Green">E175</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">D</font> &amp; <font color="Maroon">S<sub>4</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="E27:87"/>
( <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<a NAME="E29:87"/><i><font color="Green">E176</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">S<sub>2</sub></font>[ <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">d</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_9"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:87_9"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = 0;<br/>
<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:87_9"/><i><font color="Green">E178</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L1</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">B</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">L1</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal1.html#S4">ORDINAL1:sch 4</a>();<br/></i>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<a NAME="E4:87_9"/><i><font color="Green">E179</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:87_9"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">d</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> ;<br/>
<a NAME="E3:87_9_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Maroon">d</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T65">RELAT_1:65</a></i>;<br/>
<b>hence </b><a NAME="E4:87_9_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:87_9"/>
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>

<b>thus </b><a NAME="E7:87_9"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E8:87_9"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L1</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E30:87"/>
for <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>   holds not <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font>
 
;<br/>
<a NAME="E31:87"/><b>then </b>
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font>
 
;<br/>
<a NAME="E32:87"/><b>then </b>
(  <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E33:87"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E34:87"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">X</font> = <font color="Maroon">X</font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E36:87"/><i><font color="Green">E180</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>thus </b><a NAME="E1:87_10"/>
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E37:87"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> <a href="orders_1.html#NR3" target="_self">is_linear-order</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_11"><b>proof </b></a><div class="add">

<a NAME="E1:87_11"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V1">reflexive</a>
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D9">RELAT_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E2:87_11"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> is <a href="relat_2.html#V1">reflexive</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T22">WELLORD1:22</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D5" target="_self">ORDERS_1:def 5</a><br/>

<a NAME="E3:87_11"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V8">transitive</a>
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D16">RELAT_2:def 16</a></i>;<br/>
<b>hence </b><a NAME="E4:87_11"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> is <a href="relat_2.html#V8">transitive</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T24">WELLORD1:24</a></i>;<br/>

<a NAME="E5:87_11"/>
<font color="Maroon">R</font> is <a href="relat_2.html#V4">antisymmetric</a>
 
<b>by </b><i>, , <a class="ref" href="relat_2.html#D12">RELAT_2:def 12</a></i>;<br/>
<b>hence </b><a NAME="E6:87_11"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> is <a href="relat_2.html#V4">antisymmetric</a>
 <b>by </b><i><a class="ref" href="wellord1.html#T25">WELLORD1:25</a></i>;<br/>

<b>thus </b><a NAME="E7:87_11"/>
<font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> is <a href="relat_2.html#V6">connected</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="relat_2.html#D6">RELAT_2:def 6</a>,<a class="ref" href="relat_2.html#D14">RELAT_2:def 14</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:87_11_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font></span>)</span> )
 ;<br/>

<a NAME="E2:87_11_1"/><b>then </b><i><font color="Green">E181</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">X</font> )
 
<b>by </b><i><a class="ref" href="wellord1.html#T19">WELLORD1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i/>;<br/>
<a NAME="E4:87_11_1"/><i><font color="Green">E184</font></i>: 
( <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> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <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> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">X</font>,<font color="Maroon">X</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/>
<b>consider </b><font color="Maroon">A1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:87_11_1"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A1</font>,<font color="Maroon">x</font>] )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">A2</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:87_11_1"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A2</font>,<font color="Maroon">y</font>] )
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">L1</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E10:87_11_1"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E11:87_11_1"/><i><font color="Green">E188</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">L2</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E13:87_11_1"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E14:87_11_1"/><i><font color="Green">E190</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 <b>by </b><i/>;<br/>
<div><i><font color="Green">E191</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_11_1_1"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>assume </b><a NAME="E1:87_11_1_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:87_11_1_1"/><b>then </b><i><font color="Green">E193</font></i>: 
( <font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp; <font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A1</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><b>then </b><font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font> = 
<font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A1</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">L1</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/><a NAME="E4:87_11_1_1"/><b>then </b><i><font color="Green">E194</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font>
 <b>by </b><i>, , , <a class="ref" href="orders_1.html#T12" target="_self">Th12</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:87_11_1_1"/><i><font color="Green">E195</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><a NAME="E6:87_11_1_1"/>
not <font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y'</font>
 ;<br/><a NAME="E7:87_11_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <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="orders_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:87_11_1_1"/><b>then </b>
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="orders_1.html#T5" target="_self">Th5</a>, <a class="ref" href="orders_1.html#T94" target="_self">Th94</a></i>;<br/><a NAME="E9:87_11_1_1"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L2</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 ;<br/><a NAME="E10:87_11_1_1"/><b>then </b>
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T93" target="_self">Th93</a></i>;<br/><b>hence </b><a NAME="E11:87_11_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> or <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E196</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_11_1_2"><b>now </b></a><div class="add"><b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A2</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/><b>assume </b><a NAME="E1:87_11_1_2"/><i><font color="Green">E197</font></i>: 
<font color="Maroon">A2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font>
 ;<br/><a NAME="E2:87_11_1_2"/><b>then </b><i><font color="Green">E198</font></i>: 
( <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A2</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp; <font color="Maroon">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A2</font> )
 <b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/><b>then </b><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A2</font> = 
<font color="Maroon">L2</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A2</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">L2</font>
<b>by </b><i>, <a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>
;<br/><a NAME="E4:87_11_1_2"/><b>then </b><i><font color="Green">E199</font></i>: 
<font color="Maroon">y'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font>
 <b>by </b><i>, <a class="ref" href="orders_1.html#T5" target="_self">Th5</a>, <a class="ref" href="orders_1.html#T12" target="_self">Th12</a>, <a class="ref" href="orders_1.html#T93" target="_self">Th93</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:87_11_1_2"/><i><font color="Green">E200</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><a NAME="E6:87_11_1_2"/>
for <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>  holds not <font color="Olive">d1</font> <a href="hidden.html#R2">in</a> <font color="Olive">d1</font>
 ;<br/><a NAME="E7:87_11_1_2"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <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="E8:87_11_1_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="orders_1.html#T95" target="_self">Th95</a></i>;<br/><a NAME="E9:87_11_1_2"/><b>then </b>
 ex <font color="Olive">d1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> st <br/>( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Olive">d1</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) ) )
 ;<br/><a NAME="E10:87_11_1_2"/><b>then </b>
<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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="orders_1.html#T94" target="_self">Th94</a></i>;<br/><b>hence </b><a NAME="E11:87_11_1_2"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> or <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E17:87_11_1"/>
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; not <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> implies <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E18:87_11_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> or <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> or <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> <a href="wellord1.html#K2">|_2</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="txt" href="orders_1.html#E11"><i><font color="Green">Lemma77</font></i></a>, , <a class="ref" href="ordinal1.html#T24">ORDINAL1:24</a></i>;<br/>


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


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E39:87"/><i><font color="Green">E201</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) )
 <b>by </b><i>, , , </i>;<br/>
<b>take </b>
<font color="Maroon">x</font>
;<br/>

<b>thus </b><a NAME="E40:87"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font>
 <b>by </b><i>, , </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="orders_1.html#D11" target="_self">ORDERS_1:def 11</a><br/>

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

<b>assume </b><a NAME="E41:87"/><i><font color="Green">E202</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> )
 ;<br/>

<b>consider </b><font color="Maroon">L</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E43:87"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E44:87"/><i><font color="Green">E204</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L1</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">B</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  ) )
 <b>by </b><i/>;<br/>
<a NAME="E45:87"/><i><font color="Green">E205</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">z</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_12"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:87_12"/><i><font color="Green">E206</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</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="E5:87_12"/><i><font color="Green">E207</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_12_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">K</font>
;<br/>

<b>thus </b><a NAME="E2:87_12_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">K</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>

<a NAME="E3:87_12_1"/>
<font color="Maroon">y</font> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:87_12_1"/><i><font color="Green">E208</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L1</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/>



<b>assume </b><a NAME="E5:87_12_1"/><i><font color="Green">E209</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> &amp; <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font> )
 ;<br/>

<a NAME="E6:87_12_1"/><b>then </b><i><font color="Green">E210</font></i>: 
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E7:87_12_1"/>
<font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="orders_1.html#E11"><i><font color="Green">Lemma77</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:87_12_1"/><b>then </b>
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , <a class="txt" href="orders_1.html#E11"><i><font color="Green">Lemma77</font></i></a>, <a class="ref" href="funct_1.html#T82">FUNCT_1:82</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>hence </b><a NAME="E9:87_12_1"/>
<font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L1</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/>
<div><i><font color="Green">E211</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_12_2"/>
<span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:87_12_2"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E3:87_12_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="orders_1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E4:87_12_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:87_12"/><i><font color="Green">E212</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:87_12"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E9:87_12"/><b>then </b>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d2</font> where <font color="Olive">d2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">y</font></span>)</span> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E10:87_12"/>
<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="orders_1.html#T5" target="_self">Th5</a>, <a class="txt" href="orders_1.html#E11"><i><font color="Green">Lemma77</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> ;<br/>
<div><i><font color="Green">E213</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_13"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_13"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:87_13"/><b>then </b>
( <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> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="relat_2.html#R4">is_antisymmetric_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:87_13"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D4">RELAT_2:def 4</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i>, , </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_14"><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>assume </b><a NAME="E1:87_14"/><i><font color="Green">E214</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font>
 ;<br/><b>hence </b><a NAME="E2:87_14"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y'</font>
 <b>by </b><i>, </i>;<br/><b>reconsider </b><font color="Maroon">z'</font> = <font color="Maroon">z</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="orders_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E4:87_14"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z'</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> &amp; <font color="Maroon">R</font> <a href="relat_2.html#R8">is_transitive_in</a>  <a href="relat_1.html#K3">field</a> <font color="Maroon">R</font> &amp; <font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 <b>by </b><i>, , , , , <a class="ref" href="orders_1.html#T5" target="_self">Th5</a></i>;<br/><b>hence </b><a NAME="E5:87_14"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font>
 <b>by </b><i>, , , , <a class="ref" href="relat_2.html#D8">RELAT_2:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E49:87"/><b>then </b><i><font color="Green">E216</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  &amp; <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E50:87"/><b>then </b>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">d1</font> where <font color="Olive">d1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> : for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">L</font> holds <br/>( <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">d1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">d1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">R</font> ) </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E51:87"/>
contradiction
 <b>by </b><i><a class="ref" href="orders_1.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a></i>;<br/>


</div>
