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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>,  <a href="ordinal1.html#NM3">T-Sequence</a>] <b>means </b>(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; (  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font></span>)</span>) ) &amp; ( for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</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">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>2</sub></font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> ex <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">L</font>];<br/>
<a NAME="E1:40_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">C</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> ex <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>,<font color="Olive">L</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</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> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Maroon">a<sub>2</sub></font> is   <a href="ordinal1.html#NM3">T-Sequence</a> &amp; ( for <font color="Olive">A</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">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Olive">L</font>] ) );<br/>
<a NAME="E2:40_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>] holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_1"><b>proof </b></a><div class="add">

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





<b>assume </b><a NAME="E1:40_1_1_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">a</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Maroon">b</font> is   <a href="ordinal1.html#NM3">T-Sequence</a> &amp; ( for <font color="Olive">A</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">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Olive">L</font>] ) &amp; <font color="Maroon">a</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Maroon">c</font> is   <a href="ordinal1.html#NM3">T-Sequence</a> &amp; ( for <font color="Olive">A</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">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Olive">L</font>] ) )
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="ordinal1.html#NM2">Ordinal</a> ;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i/>;<br/>
<a NAME="E5:40_1_1_1"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:40_1_1_1"/><i><font color="Green">E28</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> implies <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i/>;<br/>
<a NAME="E7:40_1_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font></span>)</span>)
 
<b>by </b><i/>;<br/>
<a NAME="E8:40_1_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font></span>)</span>)
 
<b>by </b><i/>;<br/>
<a NAME="E9:40_1_1_1"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E10:40_1_1_1"/><i><font color="Green">E53</font></i>: 
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> implies <font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i/>;<br/>
<a NAME="E11:40_1_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font></span>)</span>)
 
<b>by </b><i/>;<br/>
<a NAME="E12:40_1_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  holds <br/><font color="Maroon">c</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font></span>)</span>)
 
<b>by </b><i/>;<br/>
<a NAME="E13:40_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 
<b>from </b><i><a class="ref" href="ordinal2.html#S4" target="_self">ORDINAL2:sch 4</a>(, , , , , , , <a class="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a>);<br/></i>
<b>hence </b><a NAME="E14:40_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">G</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:40_1_1"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S1">FUNCT_1:sch 1</a>();<br/></i>
<a NAME="E5:40_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:40_1_1_2"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E2:40_1_1_2_1"/><b>then </b>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:40_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>


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

<b>let </b><font color="Maroon">a</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="E2:40_1_1_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<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="E5:40_1_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a'</font>,<font color="Maroon">L</font>]
 <b>by </b><i>, </i>;<br/>
<a NAME="E6:40_1_1_2"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">K</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font> &amp; <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>,<font color="Olive">K</font>]
 
<b>by </b><i/>;<br/>
<a NAME="E7:40_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a'</font>,<font color="Maroon">L</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:40_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>4</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> ex <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( ( <font color="Olive">A</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">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Olive">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">A</font>,<font color="Olive">L</font>) ) ) );<br/>
<a NAME="E6:40_1_1"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">a</font>,<font color="Olive">c</font>] holds <br/><font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3"><b>proof </b></a><div class="add">

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





<b>assume </b><a NAME="E1:40_1_1_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>given </b><font color="Maroon">Ab</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">Lb</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E2:40_1_1_3"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">Ab</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">Lb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E3:40_1_1_3"/><i><font color="Green">E62</font></i>: 
( ( <font color="Maroon">Ab</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">Ab</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Maroon">Lb</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Maroon">Ab</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Ab</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">Ab</font>,<font color="Maroon">Lb</font>) ) )
 ;<br/>

<b>given </b><font color="Maroon">Ac</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">Lc</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><a NAME="E4:40_1_1_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">Ac</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">Lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>and </b><br/><a NAME="E5:40_1_1_3"/><i><font color="Green">E65</font></i>: 
( ( <font color="Maroon">Ac</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">Ac</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Maroon">Lc</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Maroon">Ac</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Ac</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">Ac</font>,<font color="Maroon">Lc</font>) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_3_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:40_1_1_3_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">Ab</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>
 ;<br/><b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:40_1_1_3_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">Ab</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">Lb</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>) )
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><b>consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E5:40_1_1_3_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">Ac</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">D</font>,<span class="p1">(<span class="default"><font color="Maroon">Lc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>) )
 <b>by </b><i>, , , , <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><a NAME="E6:40_1_1_3_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, , <a class="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="ordinal1.html#T12">ORDINAL1:12</a></i>;<br/><b>hence </b><a NAME="E7:40_1_1_3_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , <a class="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:40_1_1_3"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">c</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:40_1_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/> ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">a</font>,<font color="Olive">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_1_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:40_1_1_4"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> = <font color="Maroon">c</font> as   <a href="ordinal1.html#NM3">T-Sequence</a> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">a</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/>
<div><i><font color="Green">E79</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_4_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>thus </b><a NAME="E2:40_1_1_4_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">F<sub>2</sub></font>()]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4_1_1"><b>proof </b></a><div class="add">

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

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

<b>thus </b><a NAME="E1:40_1_1_4_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>

<b>thus </b><a NAME="E2:40_1_1_4_1_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">L</font>) ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:40_1_1_4_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>
 ;<br/><b>thus </b><a NAME="E2:40_1_1_4_2"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>)
;<br/>

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

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

<b>thus </b><a NAME="E1:40_1_1_4_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>

<b>thus </b><a NAME="E2:40_1_1_4_2_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">L</font>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1_1_4_3"/><i><font color="Green">E83</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> ) )
 ;<br/><b>thus </b><a NAME="E2:40_1_1_4_3"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">L</font>)]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_4_3_1"><b>proof </b></a><div class="add">

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

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

<b>thus </b><a NAME="E1:40_1_1_4_3_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>

<b>thus </b><a NAME="E2:40_1_1_4_3_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">L</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">L</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">L</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A</font>,<font color="Maroon">L</font>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:40_1_1_4"/>
 ex <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">a</font>,<font color="Olive">b</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E9:40_1_1"/><i><font color="Green">E84</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">a</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</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">L</font> = <font color="Maroon">F</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>take </b>
<font color="Maroon">L</font>
;<br/>

<b>thus </b><a NAME="E11:40_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E12:40_1_1"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> implies <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:40_1_1_5"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:40_1_1_5"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">K</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  &amp; ( ( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">A</font>,<font color="Olive">K</font>) ) ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:40_1_1_5"/>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>


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

<a NAME="E13:40_1_1"/><i><font color="Green">E85</font></i>: 
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">B</font> &amp; <font color="Olive">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font> holds <br/><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">L1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_6"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>for <font color="Olive">L1</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">L1</font>;<br/>
<a NAME="E1:40_1_1_6"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">C</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_6_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="E1:40_1_1_6_1"/><i><font color="Green">E87</font></i>: 
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">L1</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">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font> holds <br/><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">L1</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:40_1_1_6_1"/><i><font color="Green">E88</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E3:40_1_1_6_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">A</font>,<font color="Maroon">L1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/>
<a NAME="E4:40_1_1_6_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>,<font color="Maroon">L1</font>] &amp; <font color="Maroon">A</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="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E5:40_1_1_6_1"/><b>then </b><i><font color="Green">E91</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;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; (  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> implies <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<font color="Olive">x</font>) ) &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L2</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">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Olive">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Olive">B</font> holds <br/><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">B</font>,<font color="Olive">L2</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="40_1_1_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="E1:40_1_1_6_1_1"/><i><font color="Green">E92</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">x'</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="E3:40_1_1_6_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a>, , <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>then consider </b><font color="Maroon">A1</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">L2</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E5:40_1_1_6_1_1"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> )
 <b>and </b><br/><a NAME="E6:40_1_1_6_1_1"/><i><font color="Green">E96</font></i>: 
( ( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A1</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">A1</font>,<font color="Maroon">L2</font>) ) )
 <b>by </b><i/>;<br/><a NAME="E7:40_1_1_6_1_1"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , , <a class="ref" href="ordinal2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/><a NAME="E8:40_1_1_6_1_1"/>
for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a><br/> for <font color="Olive">L3</font> being  <a href="ordinal1.html#NM3">T-Sequence</a>  st <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Olive">L3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>,<font color="Olive">L3</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L3</font> be   <a href="ordinal1.html#NM3">T-Sequence</a>;<br/>



<b>assume </b><a NAME="E1:40_1_1_6_1_1_1"/><i><font color="Green">E98</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">L3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font> )
 ;<br/>

<a NAME="E2:40_1_1_6_1_1_1"/>
<font color="Maroon">x'</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/>
<b>hence </b><a NAME="E3:40_1_1_6_1_1_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">L3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>

<b>thus </b><a NAME="E4:40_1_1_6_1_1_1"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> implies <font color="Maroon">L3</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>

<b>thus </b><a NAME="E5:40_1_1_6_1_1_1"/>
for <font color="Olive">C</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> holds <br/><font color="Maroon">L3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L3</font> <a href="funct_1.html#K1">.</a> <font color="Olive">C</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_6_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:40_1_1_6_1_1_1_1"/><i><font color="Green">E99</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:40_1_1_6_1_1_1_1"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E3:40_1_1_6_1_1_1_1"/><b>then </b><i><font color="Green">E100</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E4:40_1_1_6_1_1_1_1"/><b>then </b><i><font color="Green">E101</font></i>: 
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<a NAME="E5:40_1_1_6_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal2.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E6:40_1_1_6_1_1_1_1"/>
<font color="Maroon">L3</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T17" target="_self">Th17</a></i>;<br/>


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

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

<b>assume </b><a NAME="E6:40_1_1_6_1_1_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E7:40_1_1_6_1_1_1"/><b>then </b>
<font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:40_1_1_6_1_1_1"/><b>then </b>
( <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">L3</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T10" target="_self">Th10</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/>
<a NAME="E9:40_1_1_6_1_1_1"/><b>then </b>
( <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L3</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font></span>)</span>) &amp; <span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, , <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E10:40_1_1_6_1_1_1"/>
<font color="Maroon">L3</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L3</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">C</font></span>)</span>)
 <b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:40_1_1_6_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x'</font>,<span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:40_1_1_6_1_1"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">L1</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L2</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T1" target="_self">Th1</a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_6_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:40_1_1_6_1_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:40_1_1_6_1_1_2"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:40_1_1_6_1_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">C</font> &amp; <font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">L2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font></span>)</span>) )
 <b>by </b><i><a class="ref" href="ordinal2.html#T1" target="_self">Th1</a>, <a class="ref" href="ordinal2.html#T2" target="_self">Th2</a>, <a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><a NAME="E5:40_1_1_6_1_1_2"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</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> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">D</font>,<span class="p1">(<span class="default"><font color="Maroon">L1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>) )
 <b>by </b><i>, , , <a class="ref" href="ordinal2.html#T1" target="_self">Th1</a>, <a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, <a class="ref" href="ordinal1.html#T12">ORDINAL1:12</a></i>;<br/><b>hence </b><a NAME="E6:40_1_1_6_1_1_2"/>
<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">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T11" target="_self">Th11</a>, <a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:40_1_1_6_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> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="ordinal2.html#T1" target="_self">Th1</a>, <a class="ref" href="ordinal2.html#T2" target="_self">Th2</a>, <a class="ref" href="ordinal2.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal2.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:40_1_1_6_1"/>
<font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L1</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:40_1_1_6"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">A</font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>();</i><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E14:40_1_1"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st  <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">A</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Olive">A</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1_7"><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="E1:40_1_1_7"/><i><font color="Green">E107</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a>, <font color="Maroon">K</font> being   <a href="ordinal1.html#NM3">T-Sequence</a><b> such that </b><br/><a NAME="E3:40_1_1_7"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> &amp; <font color="Maroon">K</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"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:40_1_1_7"/><i><font color="Green">E109</font></i>: 
( ( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">C</font>,<font color="Maroon">K</font>) ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:40_1_1_7"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">D</font>,<span class="p1">(<span class="default"><font color="Maroon">K</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>) )
 <b>by </b><i>, <a class="txt" href="ordinal2.html#E2:40_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<a NAME="E7:40_1_1_7"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , , <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#T12">ORDINAL1:12</a></i>;<br/>
<b>hence </b><a NAME="E8:40_1_1_7"/>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">A</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A</font></span>)</span>)
 <b>by </b><i>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>


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

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

<b>assume </b><a NAME="E15:40_1_1"/><i><font color="Green">E111</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <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="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E16:40_1_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> ex <font color="Olive">K</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> &amp; ( ( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">B</font>,<span class="p1">(<span class="default"><font color="Olive">K</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font></span>)</span>) ) or ( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">A</font>,<font color="Olive">K</font>) ) ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:40_1_1"/>
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">D</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="ordinal1.html#K2">|</a> <font color="Maroon">D</font></span>)</span>)
 <b>by </b><i>, , <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:40_1"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>2</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>hence </b><a NAME="E3:40_1"/>
 ex <font color="Olive">L</font> being  <a href="ordinal1.html#NM3">T-Sequence</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>(),<font color="Olive">L</font>]
 ;<br/>


</div>
