<?xml version="1.0"?>
<div class="add">

<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="E1:77"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E34</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> 
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E2:77_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1"><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="E1:77_1_1"/><i><font color="Green">E36</font></i>: 
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>4</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:77_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="ordinal2.html#T52" target="_self">Th52</a></i>;<br/><b>hence </b><a NAME="E3:77_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:77_1_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:77_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:77_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><a NAME="E3:77_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:77_1_1"><i><font color="Green">E36</font></i></a></i>;<br/><a NAME="E4:77_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:77"><i><font color="Green">E33</font></i></a>, <a class="txt" href="ordinal2.html#E1:77_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal2.html#T50" target="_self">Th50</a>, <a class="ref" href="ordinal2.html#T51" target="_self">Th51</a>, <a class="ref" href="ordinal2.html#T53" target="_self">Th53</a></i>;<br/><b>hence </b><a NAME="E5:77_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:77_1_1_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/><a NAME="E2:77_1_1_3"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</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="ordinal1.html#NM2">Ordinal</a> = <font color="Maroon">a<sub>1</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="ordinal2.html#NM1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E4:77_1_1_3"/><i><font color="Green">E41</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</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>4</sub></font> holds <br/><font color="Maroon">c<sub>5</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" target="_self">ORDINAL2:sch 3</a>();<br/></i><i><font color="Green">E42</font></i>: <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K4" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal2.html#E1:77_1_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ordinal2.html#E2:77_1_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="ordinal2.html#E4:77_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="ref" href="ordinal2.html#T54" target="_self">Th54</a></i>
<br/>.= 
 <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>

;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_3_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>assume </b><a NAME="E1:77_1_1_3_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:77_1_1_3_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="ordinal1.html#NM2">Ordinal</a> <b>by </b><i><a class="txt" href="ordinal2.html#E3:77_1_1_3_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/><a NAME="E5:77_1_1_3_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:77_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ordinal2.html#E3:77_1_1_3_2"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E6:77_1_1_3_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:77_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="ordinal2.html#E1:77_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="ordinal2.html#E4:77_1_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="ordinal2.html#E3:77_1_1_3_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="ordinal2.html#T57" target="_self">Th57</a></i>;<br/><b>hence </b><a NAME="E7:77_1_1_3_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:77_1_1_3_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="ordinal1.html#T22">ORDINAL1:22</a></i>;<br/></div><b>end;</b></div><a NAME="E7:77_1_1_3"/><b>then </b>
(  <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="tarski.html#K3">union</a> <span class="p1">(<span class="default"><a href="ordinal2.html#K3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="ordinal1.html#R1">c=</a>  <a href="ordinal2.html#K3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="ordinal2.html#T5" target="_self">Th5</a>, <a class="ref" href="ordinal2.html#T28" target="_self">Th28</a></i>;<br/><b>hence </b><a NAME="E8:77_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:77_1_1_3"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:77_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:77_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="ordinal2.html#E3:77_1_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:77_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="txt" href="ordinal2.html#E2:77_1"><i><font color="Green">E35</font></i></a>);<br/></i><b>hence </b><a NAME="E4:77_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:77"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K11" target="_self">*^</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:77"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>


</div>
