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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>assume </b><a NAME="E1:43"/><i><font color="Green">E19</font></i>: 
( <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">C</font> &amp; <font color="Maroon">B</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E21</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">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a>, <a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a></i>;<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>  = <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</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="E4:43"/><i><font color="Green">E22</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>
<font color="Maroon">B</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">fi</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal3.html#T22" target="_self">Th22</a>, <a class="ref" href="ordinal3.html#T24" target="_self">Th24</a>, <a class="ref" href="ordinal3.html#T25" target="_self">Th25</a>, <a class="ref" href="ordinal2.html#T54">ORDINAL2:54</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T35">ORDINAL2:35</a></i>
;<br/>
<b>then consider </b><font color="Maroon">X</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <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">fi</font></span>)</span> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T22" target="_self">Th22</a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</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="ordinal3.html#T26" target="_self">Th26</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">D</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:43"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">fi</font> &amp; <font color="Maroon">X</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T26" target="_self">Th26</a>, <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="E12:43"/><i><font color="Green">E37</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">fi</font> &amp; <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">fi</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="ordinal3.html#T27" target="_self">Th27</a>, <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="ordinal3.html#T29" target="_self">Th29</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>take </b><font color="Maroon">E</font> =  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">x</font>;<br/>

<b>thus </b><a NAME="E14:43"/>
<font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T22" target="_self">Th22</a>, <a class="ref" href="ordinal3.html#T25" target="_self">Th25</a>, <a class="ref" href="ordinal3.html#T29" target="_self">Th29</a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>

<i><font color="Green">E38</font></i>: <font color="Maroon">E</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
<br/>.= 
<font color="Maroon">D</font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="ordinal3.html#T25" target="_self">Th25</a>, <a class="ref" href="ordinal3.html#T29" target="_self">Th29</a></i>
;<br/>
<a NAME="E16:43"/>
( <font color="Maroon">D</font> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T24" target="_self">Th24</a>, , <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/>
<a NAME="E17:43"/><b>then </b>
( <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="ordinal3.html#T26" target="_self">Th26</a>, <a class="ref" href="ordinal3.html#T27" target="_self">Th27</a>, <a class="ref" href="ordinal3.html#T30" target="_self">Th30</a>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/>
<b>hence </b><a NAME="E18:43"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


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