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

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

<b>thus </b> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">A</font>,1 = 
<font color="Maroon" title="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> <font color="Maroon" title="c1">A</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T61" target="_self" title="ORDINAL2:th.61">Th61</a>, <a class="txt" href="ordinal2.html#E68"><i><font color="Green" title="E23">Lm1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font> <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> 1
<b>by </b><i><a class="ref" href="ordinal2.html#T60" target="_self" title="ORDINAL2:th.60">Th60</a></i>
<br/>.= 
<font color="Maroon" title="c1">A</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T56" target="_self" title="ORDINAL2:th.56">Th56</a></i>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1;<br/>
<a NAME="E2:79"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="ref" href="ordinal2.html#T60" target="_self" title="ORDINAL2:th.60">Th60</a></i>;<br/>
<a NAME="E3:79"/><i><font color="Green" title="E37">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:79_2"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>

<b>hence </b> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,<span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">A</font></span>)</span> = 
1 <a href="ordinal2.html#K11" title="ORDINAL2:func.11">*^</a> 1
<b>by </b><i><a class="ref" href="ordinal2.html#T61" target="_self" title="ORDINAL2:th.61">Th61</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="ordinal2.html#T56" target="_self" title="ORDINAL2:th.56">Th56</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:79"/><i><font color="Green" title="E38">A3</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">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="Olive" title="b1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:79_3"/><i><font color="Green" title="E39">A4</font></i>: 
( <font color="Maroon" title="c2">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="c2">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:79_3"/><i><font color="Green" title="E40">A5</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> holds <br/> <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,<font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<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="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> =  <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,$1;<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="E4:79_3"/><i><font color="Green" title="E41">A6</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="c2">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">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">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">B</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" target="_self" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E5:79_3"/><i><font color="Green" title="E42">A7</font></i>: 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal2.html#K8" title="ORDINAL2:func.8">lim</a> <font color="Maroon" title="c3">fi</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:79_3"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="ordinal2.html#E4:79_3"><i><font color="Green" title="E41">A6</font></i></a>, <a class="ref" href="ordinal2.html#T62" target="_self" title="ORDINAL2:th.62">Th62</a></i>;<br/>
<a NAME="E6:79_3"/><i><font color="Green" title="E43">A8</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">fi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:79_3_1"/>
<font color="Maroon" title="c4">x</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>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:79_3_1"/><i><font color="Green" title="E44">A9</font></i>: 
( <font color="Maroon" title="c5">y</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="c4">x</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="c5">y</font> )
 <b>by </b><i><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="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal2.html#E3:79_3_1"><i><font color="Green" title="E44">A9</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<font color="Maroon" title="c4">x</font> = 
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,<font color="Maroon" title="c6">y</font>
<b>by </b><i><a class="txt" href="ordinal2.html#E4:79_3"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="ordinal2.html#E3:79_3_1"><i><font color="Green" title="E44">A9</font></i></a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="ordinal2.html#E2:79_3"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="ordinal2.html#E4:79_3"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="ordinal2.html#E3:79_3_1"><i><font color="Green" title="E44">A9</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:79_3_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_3_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:79_3_2"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><b>let </b><font color="Maroon" title="c4">B</font>, <font color="Maroon" title="c5">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E2:79_3_2"/><i><font color="Green" title="E44">A10</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 ;<br/><b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">A</font>;<br/><b>reconsider </b><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> ;<br/><b>take </b><font color="Maroon" title="c8">D</font> = <font color="Maroon" title="c7">x</font>;<br/><b>thus </b><a NAME="E5:79_3_2"/>
<font color="Maroon" title="c8">D</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="ordinal2.html#E1:79_3"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="ordinal2.html#E4:79_3"><i><font color="Green" title="E41">A6</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c9">E</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/><b>assume </b><a NAME="E6:79_3_2"/>
( <font color="Maroon" title="c8">D</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c9">E</font> &amp; <font color="Maroon" title="c9">E</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> )
 ;<br/><a NAME="E7:79_3_2"/><b>then </b>
<font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</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>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:79_3_2"/>
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">E</font> &amp; <font color="Maroon" title="c3">fi</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">C</font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E6:79_3"><i><font color="Green" title="E43">A8</font></i></a>, <a class="txt" href="ordinal2.html#E2:79_3_2"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:79_3"/><b>then </b>
1 <a href="ordinal2.html#R1" title="ORDINAL2:pred.1">is_limes_of</a> <font color="Maroon" title="c3">fi</font>
 
<b>by </b><i><a class="ref" href="ordinal2.html#D13" target="_self" title="ORDINAL2:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E9:79_3"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:79_3"><i><font color="Green" title="E42">A7</font></i></a>, <a class="ref" href="ordinal2.html#D14" target="_self" title="ORDINAL2:def.14">Def14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:79"/>
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1" target="_self" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal2.html#E2:79"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="ordinal2.html#E3:79"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="ordinal2.html#E4:79"><i><font color="Green" title="E38">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:79"/>
 <a href="ordinal2.html#K12" title="ORDINAL2:func.12">exp</a> 1,<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/>


</div>
