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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>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:67"/><i><font color="Green">E25</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="67_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><b>that </b><br/><a NAME="E1:67_1"/><i><font color="Green">E26</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> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:67_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><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="E1:67_1_1"/><i><font color="Green">E29</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="E2:67_1_1"/><i><font color="Green">E30</font></i>: 
<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#E1:67_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>now </b></a><div class="add"><a NAME="E1:67_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>5</sub></font> iff ( <font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> ) )
 <b>by </b><i><a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><b>assume </b><a NAME="E2:67_1_1_1"/>
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E3:67_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:67_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal2.html#E1:67_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal2.html#E1:67_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="ordinal2.html#T45" target="_self">Th45</a>, <a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a>, <a class="ref" href="ordinal1.html#T34">ORDINAL1:34</a></i>;<br/><b>hence </b><a NAME="E4:67_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:67_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:67_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:67_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="ordinal2.html#E2:67_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="ordinal2.html#T45" target="_self">Th45</a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/><b>hence </b><a NAME="E3:67_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:67_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:67_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:67_1_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:67_1_2"/>
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="E2:67_1_2"/><b>then </b><i><font color="Green">E29</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">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">a<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="E4:67_1_2"/><i><font color="Green">E30</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">E31</font></i>: <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="ordinal2.html#K4" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="ordinal2.html#E2:67_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal2.html#E2:67_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="ordinal2.html#E4:67_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="ordinal2.html#T46" target="_self">Th46</a></i>
<br/>.= 
 <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>

;<br/><a NAME="E6:67_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:67_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal2.html#E4:67_1_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E7:67_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</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>
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:67_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="ordinal2.html#E4:67_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E8:67_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E5:67_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="ordinal2.html#T27" target="_self">Th27</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:67_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E3:67_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:67"/>
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#E1:67"><i><font color="Green">E25</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:67"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


</div>
