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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E17">A1</font></i>: 
( <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 ;<br/>

<a NAME="E2:29"/>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:29"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="ordinal3.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<a NAME="E4:29"/><b>then </b><i><font color="Green" title="E18">A2</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> $1;<br/>
<b>consider </b><font color="Maroon" title="c3">L</font> being   <a href="ordinal2.html#NM1" title="ORDINAL2:NM.1">Ordinal-Sequence</a><b> such that </b><br/><a NAME="E6:29"/><i><font color="Green" title="E19">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp; ( for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">C</font>) ) )
 <b>from </b><i><a class="ref" href="ordinal2.html#S3" title="ORDINAL2:sch.3">ORDINAL2:sch 3</a>();<br/></i>
<i><font color="Green" title="E20">A4</font></i>: <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> = 
 <a href="ordinal2.html#K4" title="ORDINAL2:func.4">sup</a> <font color="Maroon" title="c3">L</font>
<b>by </b><i><a class="txt" href="ordinal3.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="ordinal3.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="ordinal2.html#T46" title="ORDINAL2:th.46">ORDINAL2:46</a></i>
<br/>.= 
 <a href="ordinal2.html#K3" title="ORDINAL2:func.3">sup</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">L</font></span>)</span>

;<br/>
<a NAME="E8:29"/>
for <font color="Olive" title="b1">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> holds <br/> <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">C</font> be   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font>
 ;<br/>

<div><i><font color="Green" title="E22">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_2_1"/>
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/><a NAME="E2:29_2_1"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T33" title="ORDINAL1:th.33">ORDINAL1:33</a></i>;<br/><a NAME="E3:29_2_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> &amp; <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font></span>)</span> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E4:29"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="ordinal2.html#T44" title="ORDINAL2:th.44">ORDINAL2:44</a>, <a class="ref" href="ordinal2.html#T49" title="ORDINAL2:th.49">ORDINAL2:49</a>, <a class="ref" href="ordinal2.html#T51" title="ORDINAL2:th.51">ORDINAL2:51</a></i>;<br/><b>hence </b><a NAME="E4:29_2_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T22" title="ORDINAL1:th.22">ORDINAL1:22</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_2_2"/>
not <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 ;<br/><a NAME="E2:29_2_2"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:29_2_2"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="ref" href="ordinal3.html#T30" target="_self" title="ORDINAL3:th.30">Th30</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:29_2_2_1"/>
not <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><a NAME="E2:29_2_2_1"/><b>then </b>
<font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T26" title="ORDINAL1:th.26">ORDINAL1:26</a>, <a class="ref" href="ordinal2.html#T50" title="ORDINAL2:th.50">ORDINAL2:50</a></i>;<br/><a NAME="E3:29_2_2_1"/><b>then </b>
<font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:29_2"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="ordinal3.html#E4:29_2_2"><i><font color="Green" title="E23">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:29_2_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E6:29_2_2"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D</font> )
 <b>by </b><i><a class="txt" href="ordinal3.html#E1:29"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="ordinal3.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/><b>then </b><font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">D</font></span>)</span> = 
<font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <span class="p1">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="txt" href="ordinal3.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a></i>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">D</font></span>)</span>
<b>by </b><i><a class="ref" href="ordinal2.html#T45" title="ORDINAL2:th.45">ORDINAL2:45</a></i>
;<br/><a NAME="E8:29_2_2"/><b>then </b>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">L</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E6:29"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="ordinal3.html#E4:29_2_2"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="ordinal3.html#E6:29_2_2"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:29_2_2"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E7:29"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="ordinal2.html#T27" title="ORDINAL2:th.27">ORDINAL2:27</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:29_2"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="ordinal3.html#E2:29_2"><i><font color="Green" title="E22">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:29"/>
<font color="Maroon" title="c2">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>;<br/>


</div>
