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





<b>given </b><font color="Maroon">xi1</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><a NAME="E1:25"/><i><font color="Green">E37</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">xi1</font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi1</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D5" target="_self">ZFREFLE1:def 5</a><br/>

<b>given </b><font color="Maroon">xi2</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><a NAME="E2:25"/><i><font color="Green">E38</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">xi2</font> is <a href="ordinal2.html#V2">increasing</a> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi2</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zfrefle1.html#D5" target="_self">ZFREFLE1:def 5</a><br/>

<b>consider </b><font color="Maroon">xi</font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:25"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi1</font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">xi2</font> &amp; <font color="Maroon">xi</font> is <a href="ordinal2.html#V2">increasing</a> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , <a class="ref" href="ordinal4.html#T13">ORDINAL4:13</a></i>;<br/>
<b>take </b>
<font color="Maroon">xi</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="E5:25"/><i><font color="Green">E57</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">xi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , , <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>

<a NAME="E6:25"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi1</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<b>hence </b><a NAME="E7:25"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">xi</font> is <a href="ordinal2.html#V2">increasing</a> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>

<a NAME="E8:25"/><b>then </b><i><font color="Green">E58</font></i>: 
(  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</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">xi</font></span>)</span> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi1</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">xi1</font></span>)</span> &amp;  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi2</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">xi2</font></span>)</span> &amp;  <a href="ordinal2.html#K3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K3">sup</a> <font color="Maroon">A</font> &amp;  <a href="ordinal2.html#K3">sup</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="ordinal2.html#T26">ORDINAL2:26</a>, <a class="ref" href="ordinal2.html#T30">ORDINAL2:30</a></i>;<br/>
<b>thus </b><a NAME="E9:25"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:25_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E3:25_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi1</font> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> )
 <b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</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">c</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E8:25_1"/><i><font color="Green">E64</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">xi2</font> &amp; <font color="Maroon">e</font> <a href="tarski.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">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:25_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi2</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="E12:25_1"/>
( <font color="Maroon">xi1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">e</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xi1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> &amp; <font color="Maroon">xi1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> &amp; <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> is   <a href="ordinal1.html#NM2">Ordinal</a> )
 
<b>by </b><i><a class="txt" href="zfrefle1.html#E1"><i><font color="Green">Lemma17</font></i></a>, , , , , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="ordinal4.html#T9">ORDINAL4:9</a></i>;<br/>
<a NAME="E13:25_1"/><b>then </b>
( <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> &amp; <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> is   <a href="ordinal1.html#NM2">Ordinal</a> &amp; <font color="Maroon">a</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">xi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 
<b>by </b><i>, , <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E14:25_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/>


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

<b>thus </b><a NAME="E10:25"/>
 <a href="ordinal2.html#K4">sup</a> <font color="Maroon">xi</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div>
