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

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





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b><span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>;<br/>
<a NAME="E1:52"/>
( <a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/>
<a NAME="E2:52"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
;<br/>
<a NAME="E3:52"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:52_1"/><i><font color="Green">E22</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ordinal3.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default">1 <a href="ordinal2.html#K11">*^</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> 1</span>)</span> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T27" target="_self">Th27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:52"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">D</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:52_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:52_2"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">D</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>

<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2_1"/>
(  not 1 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or  not 1 <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/><a NAME="E2:52_2_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="ordinal1.html#R1">c=</a> 1 or <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a> 1 )
 <b>by </b><i><a class="ref" href="ordinal1.html#T26">ORDINAL1:26</a></i>;<br/><a NAME="E3:52_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/><a NAME="E4:52_2_1"/>
( <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; 1 <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a>, <a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/><b>hence </b><a NAME="E5:52_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_2_2"/><i><font color="Green">E40</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; 1 <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 ;<br/><a NAME="E2:52_2_2"/>
1 <a href="hidden.html#R1">=</a> 1 <a href="ordinal2.html#K11">*^</a> 1
 <b>by </b><i><a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/><a NAME="E3:52_2_2"/><b>then </b><i><font color="Green">E41</font></i>: 
( 1 <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="ordinal3.html#E5:52_2_2"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font>;<br/><b>consider </b><font color="Maroon">fi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E5:52_2_2"/><i><font color="Green">E42</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i><a NAME="E6:52_2_2"/>
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; ( for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Olive">D</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:52_2_2_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T10" target="_self">Th10</a></i>;<br/>

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

<b>assume </b><a NAME="E2:52_2_2_1"/>
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E3:52_2_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> &amp; <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:52_2_2"><i><font color="Green">E40</font></i></a>, , <a class="ref" href="ordinal3.html#T10" target="_self">Th10</a></i>;<br/>
<b>hence </b><a NAME="E4:52_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>


</div><b>end;</b></div><a NAME="E7:52_2_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font> &amp; <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <span class="p1">(<span class="default"><font color="Maroon">fi</font> <a href="ordinal3.html#K4" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> &amp; <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>by </b><i>, <a class="txt" href="ordinal3.html#E5:52_2_2"><i><font color="Green">E42</font></i></a>, , , <a class="ref" href="ordinal3.html#T17" target="_self">Th17</a>, <a class="txt" href="ordinal3.html#E4:52"><i><font color="Green">E35</font></i></a></i>;<br/><b>hence </b><a NAME="E8:52_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="ordinal3.html#E5:52_2_2"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="ordinal3.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:52_2"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:52"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E6:52"/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span>
 ;<br/>


</div>
