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

<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E30">A2</font></i>: 
( <font color="Maroon" title="c1">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="c1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1 <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font>;<br/><b>consider </b><font color="Maroon" title="c3">fi</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green" title="E31">A3</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">fi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( 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">A</font> holds <br/><font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">C</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i><i><font color="Green" title="E32">A4</font></i>: <font color="Maroon" title="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">fi</font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:42_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="ordinal2.html#T54" title="ORDINAL2:th.54">ORDINAL2:54</a></i>
<br/>.= 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">fi</font></span>)</span></span>)</span>

;<br/><a NAME="E5:42_1"/>
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">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font> holds <br/> <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="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:42_1_2"/><i><font color="Green" title="E33">A5</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_2"/><i><font color="Green" title="E34">A6</font></i>: 
( <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font> &amp; <font color="Maroon" title="c5">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">fi</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:42_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal3.html#E3:42_1_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E6:42_1_2"/><i><font color="Green" title="E35">A7</font></i>: 
( <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">fi</font> &amp; <font color="Maroon" title="c6">X</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E3:42_1_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="ref" href="ordinal2.html#T29" title="ORDINAL2:th.29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E8:42_1_2"/><i><font color="Green" title="E36">A8</font></i>: 
( <font color="Maroon" title="c8">x</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="c3">fi</font> &amp; <font color="Maroon" title="c7">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E6:42_1_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c8">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal3.html#E8:42_1_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E10:42_1_2"/>
<font color="Maroon" title="c2">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> 
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:42_1_2"><i><font color="Green" title="E33">A5</font></i></a>, <a class="ref" href="ordinal2.html#T55" title="ORDINAL2:th.55">ORDINAL2:55</a></i>;<br/>
<a NAME="E11:42_1_2"/><b>then </b>
 <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="c2">B</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10" target="_self" title="ORDINAL3:th.10">Th10</a></i>;<br/>
<a NAME="E12:42_1_2"/><b>then </b>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</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" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></i>;<br/>
<a NAME="E13:42_1_2"/><b>then </b><i><font color="Green" title="E37">A9</font></i>: 
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="ordinal3.html#E8:42_1_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="ordinal2.html#T50" title="ORDINAL2:th.50">ORDINAL2:50</a></i>;<br/>
<a NAME="E14:42_1_2"/><i><font color="Green" title="E38">A10</font></i>: 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">x</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="c3">fi</font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:42_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="ordinal3.html#E8:42_1_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>
<b>then </b><font color="Maroon" title="c3">fi</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="c9">x</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green" title="E31">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">B</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T53" title="ORDINAL2:th.53">ORDINAL2:53</a></i>
;<br/>
<a NAME="E16:42_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">fi</font> &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D</font> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E3:42_1_2"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="ordinal3.html#E6:42_1_2"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="ordinal3.html#E14:42_1_2"><i><font color="Green" title="E38">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E17:42_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c9">x</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">fi</font></span>)</span> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c7">D</font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/>
<a NAME="E18:42_1_2"/><b>then </b>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">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="c7">D</font> &amp;  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c7">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">fi</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E8:42_1_2"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="ordinal3.html#E13:42_1_2"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a>, <a class="ref" href="ordinal1.html#T34" title="ORDINAL1:th.34">ORDINAL1:34</a></i>;<br/>
<b>hence </b><a NAME="E19:42_1_2"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:42_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:42_1"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:42"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <font color="Maroon" title="c2">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:42"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="ordinal2.html#T52" title="ORDINAL2:th.52">ORDINAL2:52</a></i>;<br/>


</div>
