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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="hidden.html#M1">set</a> ) <b>-&gt; </b>  <a href="ordinal1.html#NM2">Ordinal</a> = <font color="Maroon">A</font>;<br/>
<b>consider </b><font color="Maroon">psi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E2:29"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> 1 &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> 1 holds <br/><font color="Maroon">psi</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>
<b>take </b>
<font color="Maroon">psi</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D5" target="_self">ZFREFLE1:def 5</a><br/>

<b>thus </b><a NAME="E3:29"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E4:29"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</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="E1:29_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> 1 &amp; <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</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">u</font> = <font color="Maroon">u</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="E5:29_1"/>
<font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:29_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>


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

<b>thus </b><a NAME="E5:29"/>
<font color="Maroon">psi</font> is <a href="ordinal2.html#V2">increasing</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_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="ordinal2.html#D16">ORDINAL2:def 16</a><br/><b>let </b><font color="Maroon">C</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:29_2"/>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> )
 ;<br/>

<b>hence </b><a NAME="E2:29_2"/>
<font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="ordinal3.html#T17">ORDINAL3:17</a></i>;<br/>


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

<a NAME="E6:29"/>
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> 1 )
 
<b>by </b><i>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal3.html#T18">ORDINAL3:18</a>, </i>;<br/>
<a NAME="E7:29"/><b>then </b>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a> </span>)</span></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">psi</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font> <a href="hidden.html#R1">=</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">psi</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T14">FUNCT_1:14</a></i>;<br/>
<b>hence </b><a NAME="E8:29"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">psi</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T31">ORDINAL2:31</a></i>;<br/>


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