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<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <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="txt" href="arytm_3.html#E3:120"><i><font color="Green">E89</font></i></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">E92</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:120"><i><font color="Green">E89</font></i></a>, <a class="txt" href="arytm_3.html#E5:120"><i><font color="Green">E91</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:120_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 <b>and </b><br/><a NAME="E5:120_1"/><i><font color="Green">E94</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1">ORDINAL1:sch 1</a>(<a class="txt" href="arytm_3.html#E2:120_1"><i><font color="Green">E92</font></i></a>);<br/></i>
<a NAME="E6:120_1"/><i><font color="Green">E95</font></i>: 
 <a href="arytm_3.html#K13" target="_self">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:120"><i><font color="Green">E88</font></i></a>, <a class="txt" href="arytm_3.html#E4:120"><i><font color="Green">E90</font></i></a>, <a class="txt" href="arytm_3.html#E4:120_1"><i><font color="Green">E93</font></i></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">c<sub>3</sub></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">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E6:120_1"><i><font color="Green">E95</font></i></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">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:120_1"><i><font color="Green">E93</font></i></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">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E9:120_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></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">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E9:120_1"><i><font color="Green">E96</font></i></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">E97</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E4:120_1"><i><font color="Green">E93</font></i></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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="ref" href="arytm_3.html#T34" target="_self">Th34</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E13:120_1"/>
( <font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font>] )
 <b>by </b><i><a class="txt" href="arytm_3.html#E5:120_1"><i><font color="Green">E94</font></i></a>, <a class="txt" href="arytm_3.html#E11:120_1"><i><font color="Green">E97</font></i></a></i>;<br/>

<a NAME="E14:120_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="ordinal2.html#K10">+^</a> 1
 
<b>by </b><i><a class="txt" href="arytm_3.html#E9:120_1"><i><font color="Green">E96</font></i></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">c<sub>5</sub></font> <a href="arytm_3.html#K11" target="_self">+</a> <font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:120"><i><font color="Green">E87</font></i></a>, <a class="txt" href="arytm_3.html#E4:120_1"><i><font color="Green">E93</font></i></a>, <a class="ref" href="arytm_3.html#T64" target="_self">Th64</a></i>;<br/>


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