<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">n</font>;<br/>
<a NAME="E1:22"/>
( <a href="xboole_0.html#K1">{}</a>  <a href="ordinal2.html#K10">+^</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> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</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#T44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/>
<a NAME="E2:22"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="ref" href="card_1.html#T51">CARD_1:51</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green">E49</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="22_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>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:22_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">n</font>
 ;<br/>

<b>take </b>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
;<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"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p3">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span></span>)</span> <a href="ordinal2.html#K10">+^</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
<br/>.= 
 <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="card_1.html#T52">CARD_1:52</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>
;<br/>
<b>hence </b><a NAME="E3:22_1"/>
<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"><a href="ordinal1.html#K1">succ</a> 1</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#K10">+^</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E4:22"/><i><font color="Green">E51</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">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="22_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="E1:22_2"/><i><font color="Green">E52</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="E2:22_2"/><i><font color="Green">E53</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="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>]
 ;<br/>

<b>take </b>
0
;<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> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span>;<br/>
<b>consider </b><font color="Maroon">phi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:22_2"/><i><font color="Green">E54</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E5:22_2"/><i><font color="Green">E69</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="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">B</font>)
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<i><font color="Green">E70</font></i>: <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">phi</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="ordinal2.html#T54">ORDINAL2:54</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T35">ORDINAL2:35</a></i>
;<br/>
<b>thus </b><a NAME="E7:22_2"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> 0
  <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="22_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="ordinal1.html#NM2">Ordinal</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_refle.html#D1">ZF_REFLE:def 1</a><br/>

<b>assume </b><a NAME="E1:22_2_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22_2_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</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/>
<b>consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:22_2_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">phi</font> &amp; <font color="Maroon">X</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:22_2_2"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">phi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</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">x</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/>
<a NAME="E10:22_2_2"/><i><font color="Green">E75</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">x</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">n</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E11:22_2_2"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:22_2_2"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , , , </i>;<br/>
<a NAME="E13:22_2_2"/><b>then </b>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/>
<b>hence </b><a NAME="E14:22_2_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> 0
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


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

<a NAME="E8:22_2"/>
1 <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> 1
 
<b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E9:22_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> 1 &amp; 1 <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> 1 )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D2">ORDINAL1:def 2</a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T56">ORDINAL2:56</a></i>;<br/>
<b>hence </b><a NAME="E10:22_2"/>
<font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> 0 <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ordinal2.html#T59">ORDINAL2:59</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:22"/>
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="E6:22"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">A</font> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="ordinal2.html#K10">+^</a> <font color="Olive">n</font>
 ;<br/>


</div>
