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

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





<b>assume </b><a NAME="E1:75"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:75"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:75"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E4:75"/><b>then </b><i><font color="Green">E7</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></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</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> );<br/>
<a NAME="E5:75"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st ( 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">B</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">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:75_1"/><i><font color="Green">E26</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">B</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> holds <br/><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Olive">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:75_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:75_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:75_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><div><i><font color="Green">E52</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:75_1_1_1"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">D</font> iff ( <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">D</font> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>assume </b><a NAME="E2:75_1_1_1"/>
not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E3:75_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font></span>)</span> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, , <a class="ref" href="ordinal2.html#T28" target="_self">Th28</a>, , , <a class="txt" href="ordinal2.html#E3:75_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/><b>hence </b><a NAME="E4:75_1_1_1"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E2:75_1_1_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">D</font></span>)</span> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, <a class="ref" href="ordinal2.html#T11" target="_self">Th11</a>, <a class="ref" href="ordinal2.html#T26" target="_self">Th26</a>, , , <a class="txt" href="ordinal2.html#E3:75_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:75_1_1_2"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:75_1_1"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:75_1_2"/>
for <font color="Olive">D</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  holds <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">D</font>
 ;<br/><a NAME="E2:75_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">B</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> <font color="Maroon">C</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="E4:75_1_2"/><i><font color="Green">E55</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</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">B</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">E56</font></i>: <font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> = 
 <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#T17" target="_self">Th17</a>, , <a class="ref" href="ordinal2.html#T26" target="_self">Th26</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="E6:75_1_2"/><i><font color="Green">E57</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T17" target="_self">Th17</a>, , <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/><a NAME="E7:75_1_2"/><b>then </b>
<font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <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" target="_self">*^</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E8:75_1_2"/><b>then </b>
<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" target="_self">*^</a> <font color="Maroon">C</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#T26" target="_self">Th26</a>, <a class="ref" href="ordinal2.html#T28" target="_self">Th28</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:75_1_2"/><b>then </b>
<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" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:75_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<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" target="_self">*^</a> <font color="Maroon">C</font> <a href="ordinal1.html#R1">c=</a>  <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>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E11:75_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">C</font> &amp; <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" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T7" target="_self">Th7</a>, , , <a class="txt" href="ordinal2.html#E3:75_1"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E12:75_1_2"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T27" target="_self">Th27</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:75_1"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#D7" target="_self">Def7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:75"/>
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 
<b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="ref" href="ordinal2.html#T10" target="_self">Th10</a>);<br/></i>
<b>hence </b><a NAME="E7:75"/>
(  not <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">C</font> )
 ;<br/>


</div>
