<?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>thus </b>1 <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">A</font> = 
<span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
<a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">A</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i/>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:74"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</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="74_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><a NAME="E1:74_2"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/><font color="Olive">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">B</font>
 ;<br/>

<div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:74_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:74_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><b>thus </b><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, </i>
<br/>.= 
<font color="Maroon">B</font> <a href="ordinal2.html#K10" target="_self">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal2.html#T11" target="_self">Th11</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K10" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, </i>
;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:74_2_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font> ) )
 ;<br/><a NAME="E2:74_2_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>  <a href="ordinal1.html#NM2">Ordinal</a> = <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>;<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="E4:74_2_2"/><i><font color="Green">E30</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" target="_self">ORDINAL2:sch 3</a>();<br/></i><i><font color="Green">E52</font></i>: <font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" target="_self">sup</a> <font color="Maroon">fi</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal2.html#T11" target="_self">Th11</a>, <a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, <a class="txt" href="ordinal2.html#E2:74_2_2"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font></span>)</span></span>)</span>

;<br/><a NAME="E6:74_2_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:74_2_2_2"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="74_2_2_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:74_2_2_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">x</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<font color="Maroon">x</font> = 
<font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T5" target="_self">Th5</a>, </i>
<br/>.= 
<font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, </i>
;<br/>
<b>hence </b><a NAME="E4:74_2_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:74_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:74_2_2_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> = 
<font color="Maroon">y</font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, </i>
<br/>.= 
<font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, </i>
;<br/>
<b>hence </b><a NAME="E7:74_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, </i>;<br/>


</div><b>end;</b></div><b>hence </b><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> = 
 <a href="tarski.html#K3">union</a> <font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="ordinal2.html#D7" target="_self">Def7</a>, </i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T11" target="_self">Th11</a>, <a class="ref" href="ordinal1.html#D6">ORDINAL1:def 6</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:74_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:74"/>
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="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="ref" href="ordinal2.html#T2" target="_self">Th2</a>);<br/></i>
<b>hence </b><a NAME="E4:74"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 ;<br/>


</div>
