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<b>let </b><font color="Maroon">A</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<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>  = 1 <a href="ordinal2.html#K10">+^</a> <font color="Maroon">a<sub>1</sub></font>;<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="E2:24"/><i><font color="Green">E44</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">phi</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K5">omega</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>  <a href="ordinal1.html#K5">omega</a>  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">E49</font></i>: 1 <a href="ordinal2.html#K10">+^</a> <a href="ordinal1.html#K5">omega</a>  = 
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">phi</font>
<b>by </b><i>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a>, <a class="ref" href="ordinal2.html#T46">ORDINAL2:46</a></i>
<br/>.= 
 <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="ordinal2.html#T35">ORDINAL2:35</a></i>
;<br/>
<a NAME="E4:24"/><i><font color="Green">E50</font></i>: 
1 <a href="ordinal2.html#K10">+^</a> <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a>  <a href="ordinal1.html#K5">omega</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_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="E1:24_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> 1 <a href="ordinal2.html#K10">+^</a> <a href="ordinal1.html#K5">omega</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">C</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:24_2"/><i><font color="Green">E51</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">B</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="E5:24_2"/><i><font color="Green">E52</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/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="card_1.html#NM1">Cardinal</a> <b>by </b><i>, , <a class="ref" href="card_1.html#T65">CARD_1:65</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as  <a href="finset_1.html#V1">finite</a> <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i>, , </i>;<br/>
<a NAME="E9:24_2"/>
(  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K1">Card</a> <font color="Maroon">x</font> &amp;  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">x</font> &amp;  <a href="card_1.html#K1">Card</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/>
<a NAME="E10:24_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="ordinal2.html#K10">+^</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">x</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">x</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="card_2.html#T49">CARD_2:49</a></i>;<br/>
<a NAME="E11:24_2"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 1 <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x</font> &amp; 1 <a href="ordinal2.html#K10">+^</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="ordinal2.html#K10">+^</a> 1 &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  )
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a>, <a class="ref" href="ordinal2.html#T19">ORDINAL2:19</a></i>;<br/>
<a NAME="E12:24_2"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>;<br/>
<b>hence </b><a NAME="E13:24_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


</div><b>end;</b></div>
<b>assume </b><a NAME="E5:24"/>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E6:24"/><b>then </b><i><font color="Green">E53</font></i>: 
 ex <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <a href="ordinal1.html#K5">omega</a>  <a href="ordinal2.html#K10">+^</a> <font color="Olive">B</font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T30">ORDINAL3:30</a></i>;<br/>
<a NAME="E7:24"/>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> 1 <a href="ordinal2.html#K10">+^</a> <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i><a class="ref" href="ordinal3.html#T27">ORDINAL3:27</a></i>;<br/>
<a NAME="E8:24"/><b>then </b>
 <a href="ordinal1.html#K5">omega</a>  <a href="hidden.html#R1">=</a> 1 <a href="ordinal2.html#K10">+^</a> <a href="ordinal1.html#K5">omega</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E9:24"/>
1 <a href="ordinal2.html#K10">+^</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T33">ORDINAL3:33</a></i>;<br/>


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