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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>thus </b> <a href="ordinal2.html#K12" target="_self">exp</a> <font color="Maroon">c<sub>1</sub></font>,1 = 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" target="_self">exp</a> <font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T61" target="_self">Th61</a>, <a class="txt" href="ordinal2.html#E68"><i><font color="Green">Lemma23</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> 1
<b>by </b><i><a class="ref" href="ordinal2.html#T60" target="_self">Th60</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="ordinal2.html#T56" target="_self">Th56</a></i>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means  <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1;<br/>
<a NAME="E2:81"/><i><font color="Green">E36</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="ordinal2.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E3:81"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:81_2"/>
 <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>

<b>hence </b> <a href="ordinal2.html#K12" target="_self">exp</a> 1,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
1 <a href="ordinal2.html#K11" target="_self">*^</a> 1
<b>by </b><i><a class="ref" href="ordinal2.html#T61" target="_self">Th61</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="ordinal2.html#T56" target="_self">Th56</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:81"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:81_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:81_3"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<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> =  <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:81_3"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" target="_self">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E5:81_3"/><i><font color="Green">E42</font></i>: 
 <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8" target="_self">lim</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:81_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ordinal2.html#E4:81_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal2.html#T62" target="_self">Th62</a></i>;<br/>
<a NAME="E6:81_3"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:81_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:81_3_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal2.html#E3:81_3_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<font color="Maroon">c<sub>4</sub></font> = 
 <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="ordinal2.html#E4:81_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ordinal2.html#E3:81_3_1"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="ordinal2.html#E2:81_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="ordinal2.html#E4:81_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ordinal2.html#E3:81_3_1"><i><font color="Green">E44</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:81_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="81_3_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:81_3_2"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E2:81_3_2"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> 1 &amp; 1 <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal2.html#E1:81_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E5:81_3_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:81_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ordinal2.html#E4:81_3"><i><font color="Green">E41</font></i></a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E6:81_3_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E7:81_3_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:81_3_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E6:81_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="ordinal2.html#E2:81_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:81_3"/><b>then </b>
1 <a href="ordinal2.html#R1" target="_self">is_limes_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#D13" target="_self">Def13</a></i>;<br/>
<b>hence </b><a NAME="E9:81_3"/>
 <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:81_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal2.html#D14" target="_self">Def14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:81"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" target="_self">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal2.html#E2:81"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal2.html#E3:81"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal2.html#E4:81"><i><font color="Green">E38</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:81"/>
 <a href="ordinal2.html#K12" target="_self">exp</a> 1,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/>


</div>
