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

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  );<br/>
<a NAME="E1:120_1"/>
<font color="Maroon">F<sub>3</sub></font>() is   <a href="ordinal1.html#NM2">Ordinal</a>
 
<b>by </b><i><a class="ref" href="arytm_3.html#T34" target="_self">Th34</a>, <a class="ref" href="ordinal1.html#T23">ORDINAL1:23</a></i>;<br/>
<a NAME="E2:120_1"/><b>then </b><i><font color="Green">E18</font></i>: 
 ex <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">A</font>]
 
<b>by </b><i><a class="ref" href="arytm_3.html#T34" target="_self">Th34</a>, <a class="ref" href="arytm_3.html#T35" target="_self">Th35</a></i>;<br/>
<b>consider </b><font color="Maroon">A</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:120_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">A</font>]
 <b>and </b><br/><a NAME="E5:120_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">B</font>] holds <br/><font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">B</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="ref" href="arytm_3.html#T36" target="_self">Th36</a>);<br/></i>
<a NAME="E6:120_1"/><i><font color="Green">E38</font></i>: 
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="arytm_3.html#T37" target="_self">Th37</a>, <a class="ref" href="ordinal3.html#T10">ORDINAL3:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:120_1_1"/>
<font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><a NAME="E2:120_1_1"/><b>then </b>
 <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T39" target="_self">Th39</a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/><a NAME="E3:120_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="arytm_3.html#T37" target="_self">Th37</a></i>;<br/><b>hence </b><a NAME="E4:120_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">B</font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:120_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>
<a NAME="E10:120_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="arytm_3.html#D8" target="_self">Def8</a>, <a class="ref" href="ordinal1.html#T13">ORDINAL1:13</a></i>;<br/>
<a NAME="E11:120_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; not <font color="Maroon">A</font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="arytm_3.html#T37" target="_self">Th37</a>, <a class="ref" href="ordinal1.html#T7">ORDINAL1:7</a>, <a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">B</font> as  <a href="ordinal1.html#V3">ordinal</a>  <a href="subset_1.html#M1">Element</a> of  <a href="arytm_3.html#K7" target="_self">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_3.html#E9:120_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">s</font>
;<br/>

<b>thus </b><a NAME="E13:120_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">s</font>] )
 <b>by </b><i><a class="ref" href="arytm_3.html#T38" target="_self">Th38</a>, <a class="ref" href="arytm_3.html#D9" target="_self">Def9</a></i>;<br/>

<a NAME="E14:120_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="ordinal2.html#K10">+^</a> 1
 
<b>by </b><i><a class="ref" href="arytm_3.html#D8" target="_self">Def8</a>, <a class="ref" href="ordinal2.html#T48">ORDINAL2:48</a></i>;<br/>
<b>hence </b><a NAME="E15:120_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">s</font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i>, <a class="ref" href="arytm_3.html#T37" target="_self">Th37</a>, <a class="ref" href="arytm_3.html#T21" target="_self">Th21</a></i>;<br/>


</div>
