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



<b>assume </b><a NAME="E1:42"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</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>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_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>  = <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green">E29</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</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>1</sub></font> holds <br/><font color="Maroon">c<sub>3</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><i><font color="Green">E30</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</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>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:42_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green">E29</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>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T35">ORDINAL2:35</a></i>
;<br/><a NAME="E5:42_1"/>
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>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font> holds <br/> <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:42_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<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="E3:42_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>4</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>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:42_1"><i><font color="Green">E30</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#E3:42_1_2"><i><font color="Green">E32</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="E6:42_1_2"/><i><font color="Green">E33</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>3</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#E3:42_1_2"><i><font color="Green">E32</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="E8:42_1_2"/><i><font color="Green">E34</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>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</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#E6:42_1_2"><i><font color="Green">E33</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#E8:42_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E10:42_1_2"/>
<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> 
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:42_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal2.html#T55">ORDINAL2:55</a></i>;<br/>
<a NAME="E11:42_1_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="ordinal3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E12:42_1_2"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <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>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> &amp; <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>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, <a class="ref" href="ordinal2.html#T44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T45">ORDINAL2:45</a></i>;<br/>
<a NAME="E13:42_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="ordinal1.html#K1">succ</a> <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>2</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <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>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal3.html#E8:42_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal2.html#T50">ORDINAL2:50</a></i>;<br/>
<a NAME="E14:42_1_2"/><i><font color="Green">E36</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:42_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal3.html#E8:42_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="ordinal3.html#E3:42_1"><i><font color="Green">E29</font></i></a></i>
<br/>.= 
<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>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="ordinal2.html#T53">ORDINAL2:53</a></i>
;<br/>
<a NAME="E16:42_1_2"/><b>then </b>
( <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>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</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#E3:42_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="ordinal3.html#E6:42_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="ordinal3.html#E14:42_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E17:42_1_2"/><b>then </b>
( <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>2</sub></font></span>)</span> <a href="ordinal2.html#K10">+^</a> <font color="Maroon">c<sub>2</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>3</sub></font></span>)</span> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font> )
 
<b>by </b><i><a class="ref" href="ordinal1.html#T33">ORDINAL1:33</a>, <a class="ref" href="ordinal2.html#T27">ORDINAL2:27</a></i>;<br/>
<a NAME="E18:42_1_2"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>7</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>3</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="ordinal3.html#E8:42_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="ordinal3.html#E13:42_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/>
<b>hence </b><a NAME="E19:42_1_2"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:42_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:42_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:42"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal2.html#K11">*^</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:42"><i><font color="Green">E27</font></i></a>, <a class="ref" href="ordinal2.html#T52">ORDINAL2:52</a></i>;<br/>


</div>
