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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>,  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>] <b>means </b>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; (  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 implies $2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 holds <br/>$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default">$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>) ) &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> $1 &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  holds <br/>$2 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default">$2 <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>) ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> ex <font color="Olive" title="b1">L</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> st <font color="Maroon">S<sub>1</sub></font>[$1,<font color="Olive" title="b1">L</font>];<br/>
<a NAME="E1:38_1"/><i><font color="Green" title="E14">A1</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:38_1_1"/><i><font color="Green" title="E15">A2</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/> ex <font color="Olive" title="b2">L</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>,<font color="Olive" title="b2">L</font>]
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; $2 is   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">L</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">L</font>] ) );<br/>
<a NAME="E2:38_1_1"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>] holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:38_1_1_1"/><i><font color="Green" title="E17">A4</font></i>: 
( <font color="Maroon" title="c2">a</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; <font color="Maroon" title="c3">b</font> is   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">L</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">L</font>] ) &amp; <font color="Maroon" title="c2">a</font> is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> &amp; <font color="Maroon" title="c4">c</font> is   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> &amp; ( for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">L</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">L</font>] ) )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c2">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c6">b</font> = <font color="Maroon" title="c3">b</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">c</font> = <font color="Maroon" title="c4">c</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E5:38_1_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E6:38_1_1_1"/><i><font color="Green" title="E19">A6</font></i>: 
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c6">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E7:38_1_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> holds <br/><font color="Maroon" title="c6">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>)
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E8:38_1_1_1"/><i><font color="Green" title="E21">A8</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  holds <br/><font color="Maroon" title="c6">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>)
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E9:38_1_1_1"/><i><font color="Green" title="E22">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E10:38_1_1_1"/><i><font color="Green" title="E23">A10</font></i>: 
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> implies <font color="Maroon" title="c7">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E11:38_1_1_1"/><i><font color="Green" title="E24">A11</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> holds <br/><font color="Maroon" title="c7">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>)
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E12:38_1_1_1"/><i><font color="Green" title="E25">A12</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  holds <br/><font color="Maroon" title="c7">c</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font></span>)</span>)
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_1"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<a NAME="E13:38_1_1_1"/>
<font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font>
 
<b>from </b><i><a class="ref" href="ordinal2.html#S4" target="_self" title="ORDINAL2:sch.4">ORDINAL2:sch 4</a>(<a class="txt" href="ordinal2.html#E5:38_1_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="ordinal2.html#E6:38_1_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="ordinal2.html#E7:38_1_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="ordinal2.html#E8:38_1_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="ordinal2.html#E9:38_1_1_1"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="ordinal2.html#E10:38_1_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="txt" href="ordinal2.html#E11:38_1_1_1"><i><font color="Green" title="E24">A11</font></i></a>, <a class="txt" href="ordinal2.html#E12:38_1_1_1"><i><font color="Green" title="E25">A12</font></i></a>);<br/></i>
<b>hence </b><a NAME="E14:38_1_1_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c2">G</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E4:38_1_1"/><i><font color="Green" title="E17">A13</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font> iff ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S1" title="FUNCT_1:sch.1">FUNCT_1:sch 1</a>(<a class="txt" href="ordinal2.html#E2:38_1_1"><i><font color="Green" title="E16">A3</font></i></a>);<br/></i>
<a NAME="E5:38_1_1"/><i><font color="Green" title="E18">A14</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_1_1_2"/>
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">G</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_2_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">G</font>
 ;<br/>

<a NAME="E2:38_1_1_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_2_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:38_1_1_2"/><i><font color="Green" title="E19">A15</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">a'</font> = <font color="Maroon" title="c3">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">L</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E5:38_1_1_2"/><i><font color="Green" title="E20">A16</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">L</font>]
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1"><i><font color="Green" title="E15">A2</font></i></a>, <a class="txt" href="ordinal2.html#E2:38_1_1_2"><i><font color="Green" title="E19">A15</font></i></a></i>;<br/>
<a NAME="E6:38_1_1_2"/>
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">K</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a'</font> &amp; <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">K</font>]
 
<b>by </b><i><a class="txt" href="ordinal2.html#E5:38_1_1_2"><i><font color="Green" title="E20">A16</font></i></a></i>;<br/>
<a NAME="E7:38_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">L</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E2:38_1_1_2"><i><font color="Green" title="E19">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:38_1_1_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">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" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b2">L</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; ( ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b3">B</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b3">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">B</font></span>)</span>) ) or ( <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">L</font>) ) ) );<br/>
<a NAME="E6:38_1_1"/><i><font color="Green" title="E19">A17</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>] holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:38_1_1_3"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ;<br/>

<b>given </b><font color="Maroon" title="c6">Ab</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c7">Lb</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><a NAME="E2:38_1_1_3"/><i><font color="Green" title="E20">A18</font></i>: 
( <font color="Maroon" title="c6">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c7">Lb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> )
 <b>and </b><br/><a NAME="E3:38_1_1_3"/><i><font color="Green" title="E21">A19</font></i>: 
( ( <font color="Maroon" title="c6">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c6">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">Lb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) or ( <font color="Maroon" title="c6">Ab</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">Ab</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">Ab</font>,<font color="Maroon" title="c7">Lb</font>) ) )
 ;<br/>

<b>given </b><font color="Maroon" title="c8">Ac</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c9">Lc</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><a NAME="E4:38_1_1_3"/><i><font color="Green" title="E22">A20</font></i>: 
( <font color="Maroon" title="c8">Ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c9">Lc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> )
 <b>and </b><br/><a NAME="E5:38_1_1_3"/><i><font color="Green" title="E23">A21</font></i>: 
( ( <font color="Maroon" title="c8">Ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c8">Ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">Lc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) or ( <font color="Maroon" title="c8">Ac</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c8">Ac</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c8">Ac</font>,<font color="Maroon" title="c9">Lc</font>) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_3_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c10">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E1:38_1_1_3_1"/><i><font color="Green" title="E24">A22</font></i>: 
<font color="Maroon" title="c6">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c10">C</font>
 ;<br/><b>consider </b><font color="Maroon" title="c11">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E3:38_1_1_3_1"/><i><font color="Green" title="E25">A23</font></i>: 
( <font color="Maroon" title="c6">Ab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c11">A</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c11">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">Lb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">A</font></span>)</span>) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:38_1_1_3"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_3_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E5:38_1_1_3_1"/><i><font color="Green" title="E26">A24</font></i>: 
( <font color="Maroon" title="c8">Ac</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c12">D</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c12">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">Lc</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">D</font></span>)</span>) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:38_1_1_3"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_3"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="ordinal2.html#E5:38_1_1_3"><i><font color="Green" title="E23">A21</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_3_1"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/><a NAME="E6:38_1_1_3_1"/>
<font color="Maroon" title="c11">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">D</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:38_1_1_3"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_3"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_3_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="ordinal2.html#E5:38_1_1_3_1"><i><font color="Green" title="E26">A24</font></i></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_3_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:38_1_1_3"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_3"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_3_1"><i><font color="Green" title="E25">A23</font></i></a>, <a class="txt" href="ordinal2.html#E5:38_1_1_3_1"><i><font color="Green" title="E26">A24</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:38_1_1_3"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:38_1_1_3"><i><font color="Green" title="E20">A18</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_3"><i><font color="Green" title="E21">A19</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_3"><i><font color="Green" title="E22">A20</font></i></a>, <a class="txt" href="ordinal2.html#E5:38_1_1_3"><i><font color="Green" title="E23">A21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:38_1_1"/><i><font color="Green" title="E20">A25</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/> ex <font color="Olive" title="b2">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:38_1_1_4"/><i><font color="Green" title="E21">A26</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:38_1_1_4"/><i><font color="Green" title="E22">A27</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:38_1_1"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="relat_1.html#D4" title="RELAT_1:def.4">RELAT_1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">L</font> = <font color="Maroon" title="c4">c</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_4"><i><font color="Green" title="E22">A27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">A</font> = <font color="Maroon" title="c3">a</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_4"><i><font color="Green" title="E21">A26</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<div><i><font color="Green" title="E23">A28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_4_1"/><i><font color="Green" title="E24">A29</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>thus </b><a NAME="E2:38_1_1_4_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon">F<sub>2</sub></font>()]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c6">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">L</font>
;<br/>

<b>thus </b><a NAME="E1:38_1_1_4_1_1"/>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:38_1_1_4"><i><font color="Green" title="E22">A27</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>

<b>thus </b><a NAME="E2:38_1_1_4_1_1"/>
( ( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) or ( <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">L</font>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_4_1"><i><font color="Green" title="E24">A29</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green" title="E24">A30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c7">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E1:38_1_1_4_2"/><i><font color="Green" title="E25">A31</font></i>: 
<font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">C</font>
 ;<br/><b>thus </b><a NAME="E2:38_1_1_4_2"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">b</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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" title="c7">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font></span>)</span>)
;<br/>

<b>take </b>
<font color="Maroon" title="c6">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">L</font>
;<br/>

<b>thus </b><a NAME="E1:38_1_1_4_2_1"/>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:38_1_1_4"><i><font color="Green" title="E22">A27</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>

<b>thus </b><a NAME="E2:38_1_1_4_2_1"/>
( ( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) or ( <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c7">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">C</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">L</font>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_4_2"><i><font color="Green" title="E25">A31</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="38_1_1_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_4_3"/><i><font color="Green" title="E25">A32</font></i>: 
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</font> ) )
 ;<br/><b>thus </b><a NAME="E2:38_1_1_4_3"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">L</font>)]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_4_3_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c6">A</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">L</font>
;<br/>

<b>thus </b><a NAME="E1:38_1_1_4_3_1"/>
( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:38_1_1_4"><i><font color="Green" title="E22">A27</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>

<b>thus </b><a NAME="E2:38_1_1_4_3_1"/>
( ( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c6">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) or ( <font color="Maroon" title="c6">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">A</font>,<font color="Maroon" title="c5">L</font>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_4_3"><i><font color="Green" title="E25">A32</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:38_1_1_4"/>
 ex <font color="Olive" title="b1">b</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">b</font>]
 <b>by </b><i><a class="txt" href="ordinal2.html#E6:38_1_1_4"><i><font color="Green" title="E23">A28</font></i></a>, <a class="txt" href="ordinal2.html#E7:38_1_1_4"><i><font color="Green" title="E24">A30</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E9:38_1_1"/><i><font color="Green" title="E21">A33</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font> &amp; ( for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>] ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S2" title="FUNCT_1:sch.2">FUNCT_1:sch 2</a>(<a class="txt" href="ordinal2.html#E6:38_1_1"><i><font color="Green" title="E19">A17</font></i></a>, <a class="txt" href="ordinal2.html#E7:38_1_1"><i><font color="Green" title="E20">A25</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">L</font> = <font color="Maroon" title="c3">F</font> as   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> <b>by </b><i><a class="txt" href="ordinal2.html#E9:38_1_1"><i><font color="Green" title="E21">A33</font></i></a>, <a class="ref" href="ordinal1.html#D7" title="ORDINAL1:def.7">ORDINAL1:def 7</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">L</font>
;<br/>

<b>thus </b><a NAME="E11:38_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">B</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E9:38_1_1"><i><font color="Green" title="E21">A33</font></i></a></i>;<br/>

<b>thus </b><a NAME="E12:38_1_1"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> implies <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:38_1_1_5"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ;<br/>

<a NAME="E2:38_1_1_5"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b2">K</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b3">B</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b3">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">B</font></span>)</span>) ) or ( <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">K</font>) ) ) )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E9:38_1_1"><i><font color="Green" title="E21">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_5"/>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>


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

<a NAME="E13:38_1_1"/><i><font color="Green" title="E22">A34</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">L1</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Olive" title="b2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>for <font color="Olive" title="b1">L1</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Olive" title="b1">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 holds <br/><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L1</font>;<br/>
<a NAME="E1:38_1_1_6"/><i><font color="Green" title="E23">A35</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">C</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:38_1_1_6_1"/><i><font color="Green" title="E24">A36</font></i>: 
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> holds <br/>for <font color="Olive" title="b2">L1</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Olive" title="b2">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font> holds <br/><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">L1</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c6">L1</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>;<br/>

<b>assume </b><a NAME="E2:38_1_1_6_1"/><i><font color="Green" title="E25">A37</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon" title="c6">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font> )
 ;<br/>

<a NAME="E3:38_1_1_6_1"/><b>then </b><i><font color="Green" title="E26">A38</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">L1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E5:38_1_1"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/>
<a NAME="E4:38_1_1_6_1"/><b>then </b><i><font color="Green" title="E27">A39</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">L1</font>] &amp; <font color="Maroon" title="c5">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">L</font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E9:38_1_1"><i><font color="Green" title="E21">A33</font></i></a>, <a class="txt" href="ordinal2.html#E2:38_1_1_6_1"><i><font color="Green" title="E25">A37</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E5:38_1_1_6_1"/><b>then </b><i><font color="Green" title="E28">A40</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">L1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp; (  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> implies <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">x</font>) ) &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">L2</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Olive" title="b1">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Olive" title="b2">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Olive" title="b1">B</font> holds <br/><font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">B</font>,<font color="Olive" title="b2">L2</font>) ) )
 
<b>by </b><i><a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_6_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1_1_6_1_1"/><i><font color="Green" title="E29">A41</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c7">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/><a NAME="E3:38_1_1_6_1_1"/><i><font color="Green" title="E30">A42</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:38_1_1_6_1"><i><font color="Green" title="E25">A37</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">A1</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c10">L2</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E5:38_1_1_6_1_1"/><i><font color="Green" title="E31">A43</font></i>: 
( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> &amp; <font color="Maroon" title="c10">L2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> )
 <b>and </b><br/><a NAME="E6:38_1_1_6_1_1"/><i><font color="Green" title="E32">A44</font></i>: 
( ( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) or ( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c9">A1</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c9">A1</font>,<font color="Maroon" title="c10">L2</font>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E9:38_1_1"><i><font color="Green" title="E21">A33</font></i></a></i>;<br/><a NAME="E7:38_1_1_6_1_1"/><i><font color="Green" title="E33">A45</font></i>: 
( <font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1"><i><font color="Green" title="E24">A36</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_6_1_1"><i><font color="Green" title="E30">A42</font></i></a>, <a class="txt" href="ordinal2.html#E5:38_1_1_6_1_1"><i><font color="Green" title="E31">A43</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><a NAME="E8:38_1_1_6_1_1"/>
for <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><br/> for <font color="Olive" title="b2">L3</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>  st <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> &amp; <font color="Olive" title="b2">L3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">D</font>,<font color="Olive" title="b2">L3</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_6_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">D</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>let </b><font color="Maroon" title="c12">L3</font> be   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a>;<br/>



<b>assume </b><a NAME="E1:38_1_1_6_1_1_1"/><i><font color="Green" title="E34">A46</font></i>: 
( <font color="Maroon" title="c11">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x'</font> &amp; <font color="Maroon" title="c12">L3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font> )
 ;<br/>

<a NAME="E2:38_1_1_6_1_1_1"/>
<font color="Maroon" title="c8">x'</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:38_1_1_6_1_1_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">L3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1_6_1"><i><font color="Green" title="E27">A39</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>

<b>thus </b><a NAME="E4:38_1_1_6_1_1_1"/>
(  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> implies <font color="Maroon" title="c12">L3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1_6_1"><i><font color="Green" title="E27">A39</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>

<b>thus </b><a NAME="E5:38_1_1_6_1_1_1"/>
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> holds <br/><font color="Maroon" title="c12">L3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">L3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_6_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:38_1_1_6_1_1_1_1"/><i><font color="Green" title="E35">A47</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 ;<br/>

<a NAME="E2:38_1_1_6_1_1_1_1"/>
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c13">C</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<a NAME="E3:38_1_1_6_1_1_1_1"/><b>then </b><i><font color="Green" title="E36">A48</font></i>: 
<font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1_1"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E4:38_1_1_6_1_1_1_1"/><b>then </b><i><font color="Green" title="E37">A49</font></i>: 
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x'</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x'</font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1_1"><i><font color="Green" title="E35">A47</font></i></a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E5:38_1_1_6_1_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c13">C</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">C</font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1_1"><i><font color="Green" title="E35">A47</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_6_1_1_1_1"><i><font color="Green" title="E36">A48</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E6:38_1_1_6_1_1_1_1"/>
<font color="Maroon" title="c12">L3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c13">C</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c13">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">L3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">C</font></span>)</span>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_6_1"><i><font color="Green" title="E26">A38</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_6_1_1_1_1"><i><font color="Green" title="E37">A49</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c13">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E6:38_1_1_6_1_1_1"/><i><font color="Green" title="E35">A50</font></i>: 
( <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c13">C</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E7:38_1_1_6_1_1_1"/><b>then </b>
<font color="Maroon" title="c13">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c8">x'</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="ref" href="ordinal1.html#D2" title="ORDINAL1:def.2">ORDINAL1:def 2</a></i>;<br/>
<a NAME="E8:38_1_1_6_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">L3</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">C</font> &amp; <font color="Maroon" title="c13">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="txt" href="ordinal2.html#E6:38_1_1_6_1_1_1"><i><font color="Green" title="E35">A50</font></i></a>, <a class="ref" href="funct_1.html#T82" title="FUNCT_1:th.82">FUNCT_1:82</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<a NAME="E9:38_1_1_6_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c13">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">L3</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">C</font></span>)</span>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">C</font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_6_1"><i><font color="Green" title="E26">A38</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a>, <a class="txt" href="ordinal2.html#E6:38_1_1_6_1_1_1"><i><font color="Green" title="E35">A50</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>
<b>hence </b><a NAME="E10:38_1_1_6_1_1_1"/>
<font color="Maroon" title="c12">L3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c13">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">L3</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c13">C</font></span>)</span>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_1"><i><font color="Green" title="E34">A46</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:38_1_1_6_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">x'</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_6_1_1"><i><font color="Green" title="E30">A42</font></i></a></i>;<br/><a NAME="E10:38_1_1_6_1_1"/><b>then </b><i><font color="Green" title="E34">A51</font></i>: 
<font color="Maroon" title="c6">L1</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">L2</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:38_1_1_6_1_1"><i><font color="Green" title="E31">A43</font></i></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_6_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c11">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E1:38_1_1_6_1_1_2"/><i><font color="Green" title="E35">A52</font></i>: 
<font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c11">D</font>
 ;<br/><a NAME="E2:38_1_1_6_1_1_2"/><i><font color="Green" title="E36">A53</font></i>: 
<font color="Maroon" title="c11">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x'</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_2"><i><font color="Green" title="E35">A52</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:38_1_1_6_1_1_2"/><i><font color="Green" title="E37">A54</font></i>: 
( <font color="Maroon" title="c9">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c12">C</font> &amp; <font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c12">C</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">L2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">C</font></span>)</span>) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:38_1_1_6_1_1"><i><font color="Green" title="E31">A43</font></i></a>, <a class="txt" href="ordinal2.html#E6:38_1_1_6_1_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_2"><i><font color="Green" title="E35">A52</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/><a NAME="E5:38_1_1_6_1_1_2"/>
( <font color="Maroon" title="c12">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">D</font> &amp; <font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c11">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">D</font></span>)</span>) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_6_1"><i><font color="Green" title="E26">A38</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="txt" href="ordinal2.html#E5:38_1_1_6_1_1"><i><font color="Green" title="E31">A43</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1_2"><i><font color="Green" title="E35">A52</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_6_1_1_2"><i><font color="Green" title="E37">A54</font></i></a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></i>;<br/><b>hence </b><a NAME="E6:38_1_1_6_1_1_2"/>
<font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E7:38_1_1_6_1_1"><i><font color="Green" title="E33">A45</font></i></a>, <a class="txt" href="ordinal2.html#E10:38_1_1_6_1_1"><i><font color="Green" title="E34">A51</font></i></a>, <a class="txt" href="ordinal2.html#E2:38_1_1_6_1_1_2"><i><font color="Green" title="E36">A53</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_6_1_1_2"><i><font color="Green" title="E37">A54</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:38_1_1_6_1_1"/>
<font color="Maroon" title="c6">L1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:38_1_1"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_6_1"><i><font color="Green" title="E26">A38</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_6_1_1"><i><font color="Green" title="E29">A41</font></i></a>, <a class="txt" href="ordinal2.html#E5:38_1_1_6_1_1"><i><font color="Green" title="E31">A43</font></i></a>, <a class="txt" href="ordinal2.html#E6:38_1_1_6_1_1"><i><font color="Green" title="E32">A44</font></i></a>, <a class="txt" href="ordinal2.html#E7:38_1_1_6_1_1"><i><font color="Green" title="E33">A45</font></i></a>, <a class="txt" href="ordinal2.html#E10:38_1_1_6_1_1"><i><font color="Green" title="E34">A51</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:38_1_1_6_1"/>
<font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">L1</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:38_1_1_6_1"><i><font color="Green" title="E28">A40</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:38_1_1_6"/>
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">A</font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="ordinal2.html#E1:38_1_1_6"><i><font color="Green" title="E23">A35</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E14:38_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> holds <br/><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">A</font></span>)</span>)
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:38_1_1_7"/><i><font color="Green" title="E23">A55</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">C</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>, <font color="Maroon" title="c7">K</font> being   <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a><b> such that </b><br/><a NAME="E3:38_1_1_7"/><i><font color="Green" title="E24">A56</font></i>: 
( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c7">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:38_1_1_7"/><i><font color="Green" title="E25">A57</font></i>: 
( ( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">B</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">B</font></span>)</span>) ) or ( <font color="Maroon" title="c6">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">C</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c7">K</font>) ) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E9:38_1_1"><i><font color="Green" title="E21">A33</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:38_1_1_7"/><i><font color="Green" title="E26">A58</font></i>: 
( <font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c8">D</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c8">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">D</font></span>)</span>) )
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:38_1_1_7"><i><font color="Green" title="E24">A56</font></i></a>, <a class="txt" href="ordinal2.html#E4:38_1_1_7"><i><font color="Green" title="E25">A57</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>
<a NAME="E7:38_1_1_7"/>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">D</font> &amp; <font color="Maroon" title="c7">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font></span>)</span> &amp; <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font> )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E13:38_1_1"><i><font color="Green" title="E22">A34</font></i></a>, <a class="txt" href="ordinal2.html#E1:38_1_1_7"><i><font color="Green" title="E23">A55</font></i></a>, <a class="txt" href="ordinal2.html#E3:38_1_1_7"><i><font color="Green" title="E24">A56</font></i></a>, <a class="txt" href="ordinal2.html#E6:38_1_1_7"><i><font color="Green" title="E26">A58</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#T12" title="ORDINAL1:th.12">ORDINAL1:12</a></i>;<br/>
<b>hence </b><a NAME="E8:38_1_1_7"/>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">A</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">A</font></span>)</span>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E6:38_1_1_7"><i><font color="Green" title="E26">A58</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c5">D</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E15:38_1_1"/><i><font color="Green" title="E23">A59</font></i>: 
( <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">B</font> &amp; <font color="Maroon" title="c5">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">D</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E16:38_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ex <font color="Olive" title="b2">K</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D</font> &amp; <font color="Olive" title="b2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> &amp; ( ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>() ) or  ex <font color="Olive" title="b3">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b3">B</font> &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b3">B</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">K</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">B</font></span>)</span>) ) or ( <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">K</font>) ) ) )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E9:38_1_1"><i><font color="Green" title="E21">A33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:38_1_1"/>
<font color="Maroon" title="c4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c5">D</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">L</font> <a href="ordinal1.html#K2" title="ORDINAL1:func.2">|</a> <font color="Maroon" title="c5">D</font></span>)</span>)
 <b>by </b><i><a class="txt" href="ordinal2.html#E13:38_1_1"><i><font color="Green" title="E22">A34</font></i></a>, <a class="txt" href="ordinal2.html#E15:38_1_1"><i><font color="Green" title="E23">A59</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:38_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="ordinal2.html#E1:38_1"><i><font color="Green" title="E14">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:38_1"/>
 ex <font color="Olive" title="b1">L</font> being  <a href="ordinal1.html#NM3" title="ORDINAL1:NM.3">T-Sequence</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>(),<font color="Olive" title="b1">L</font>]
 ;<br/>


</div>
