<?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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b> <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span>;<br/>
<a NAME="E1:38"/>
(  <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp;  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T60">ORDINAL2:60</a></i>;<br/>
<a NAME="E2:38"/><b>then </b><i><font color="Green">E13</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#T52">ORDINAL2:52</a></i>;<br/>
<a NAME="E3:38"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">B</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_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><a NAME="E1:38_1"/>
 <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>

<b>hence </b> <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T61">ORDINAL2:61</a></i>
<br/>.= 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal4.html#E4:38"><i><font color="Green">E26</font></i></a></i>
<br/>.= 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E4:38"/><i><font color="Green">E26</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#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &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="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="38_2"><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="E1:38_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:38_2"/><i><font color="Green">E28</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> holds <br/> <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<font color="Olive">D</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Olive">D</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span>
 ;<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>  =  <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">a<sub>1</sub></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="E4:38_2"/><i><font color="Green">E29</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">ORDINAL2:sch 3</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>2</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> <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">f1</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:38_2"/><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</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">f1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f2</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E8:38_2"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</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> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> holds <br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">D</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<a NAME="E9:38_2"/><i><font color="Green">E42</font></i>: 
(  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a>, <a class="ref" href="ordinal2.html#T60">ORDINAL2:60</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:38_2_1"/><i><font color="Green">E45</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:38_2_1_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="relat_1.html#T44">RELAT_1:44</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</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="E2:38_2_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">E</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="E4:38_2_1_1"/>
( <font color="Maroon">E</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">E</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="txt" href="ordinal4.html#E6:38_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a>, <a class="ref" href="ordinal2.html#T57">ORDINAL2:57</a></i>;<br/>
<b>hence </b><a NAME="E5:38_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_2_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/><b>then reconsider </b><font color="Maroon">D</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="E3:38_2_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">D</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="txt" href="ordinal4.html#E6:38_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a>, <a class="ref" href="ordinal2.html#T57">ORDINAL2:57</a></i>;<br/><b>thus </b><font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
 <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">D</font>
<b>by </b><i>, <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i>, <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="ordinal4.html#T3" target="_self">Th3</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/></div><b>end;</b></div><a NAME="E4:38_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="txt" href="ordinal4.html#E8:38_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/><a NAME="E5:38_2_1"/>
( <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <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">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, <a class="txt" href="ordinal4.html#E6:38_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="ordinal3.html#T34">ORDINAL3:34</a>, <a class="ref" href="ordinal3.html#T48">ORDINAL3:48</a></i>;<br/><a NAME="E6:38_2_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">f2</font>
 <b>by </b><i>, </i>;<br/><a NAME="E7:38_2_1"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:38_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:38_2_1_3"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</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">y</font> = <font color="Maroon">y</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="E5:38_2_1_3"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> &amp; <font color="Maroon">y</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="txt" href="ordinal4.html#E6:38_2"><i><font color="Green">E36</font></i></a>, , <a class="ref" href="ordinal2.html#T57">ORDINAL2:57</a></i>;<br/>
<b>hence </b><a NAME="E6:38_2_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E8:38_2_1"/><i><font color="Green">E76</font></i>: 
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:38_2_1_4"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>
<b>hence </b><a NAME="E2:38_2_1_4"/>
 <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T26">ORDINAL2:26</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</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="E3:38_2_1_4"/><i><font color="Green">E77</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">f2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">D</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">A1</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E6:38_2_1_4"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">A1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, , <a class="txt" href="ordinal4.html#E9:38_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="ordinal3.html#T49">ORDINAL3:49</a></i>;<br/>
<a NAME="E7:38_2_1_4"/>
<font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:38_2_1_4"/><b>then </b>
<font color="Maroon">A1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f1</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E9:38_2_1_4"/><b>then </b>
<font color="Maroon">A1</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</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">f1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<b>hence </b><a NAME="E10:38_2_1_4"/>
<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">f1</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:38_2_1"/>
<font color="Maroon">f1</font> is <a href="ordinal2.html#V2">increasing</a>
 <b>by </b><i>, <a class="txt" href="ordinal4.html#E6:38_2"><i><font color="Green">E36</font></i></a>, </i>;<br/><a NAME="E10:38_2_1"/><b>then </b>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="ordinal2.html#R1">is_limes_of</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="ref" href="ordinal4.html#T3" target="_self">Th3</a>, , , </i>;<br/><a NAME="E11:38_2_1"/><b>then </b>
 <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K8">lim</a> <font color="Maroon">fi</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#D14">ORDINAL2:def 14</a></i>;<br/><b>hence </b><a NAME="E12:38_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">B</font>]
 <b>by </b><i><a class="ref" href="ordinal4.html#D1" target="_self">Def1</a>, , <a class="ref" href="ordinal2.html#T62">ORDINAL2:62</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:38_2"/>
 <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="ordinal2.html#T63">ORDINAL2:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:38"/>
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="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E6:38"/>
 <a href="ordinal2.html#K12">exp</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<font color="Maroon">A</font></span>)</span>,<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K12">exp</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">A</font></span>)</span>
 ;<br/>


</div>
