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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>3</sub></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<sub>3</sub></font>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:43"/><i><font color="Green">E30</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3">ORDINAL2:sch 3</a>();<br/></i>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4">sup</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:43"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal3.html#E2:43"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal3.html#E4:43"><i><font color="Green">E30</font></i></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">c<sub>4</sub></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">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:43"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:43"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal3.html#E7:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E10:43"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E7:43"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal2.html#T29">ORDINAL2:29</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:43"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E10:43"><i><font color="Green">E32</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal3.html#E12:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<b>take </b><font color="Maroon">c<sub>10</sub></font> =  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font>;<br/>

<b>thus </b><a NAME="E14:43"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:43"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal3.html#E4:43"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ordinal3.html#E12:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>

<i><font color="Green">E34</font></i>: <font color="Maroon">c<sub>10</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="ordinal3.html#E4:43"><i><font color="Green">E30</font></i></a>, <a class="txt" href="ordinal3.html#E12:43"><i><font color="Green">E33</font></i></a></i>
;<br/>
<a NAME="E16:43"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E2:43"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal3.html#T10" target="_self">Th10</a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a></i>;<br/>
<a NAME="E17:43"/><b>then </b>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E7:43"><i><font color="Green">E31</font></i></a>, <a class="txt" href="ordinal3.html#E10:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ordinal3.html#E15:43"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal2.html#T49">ORDINAL2:49</a></i>;<br/>
<b>hence </b><a NAME="E18:43"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>10</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>


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