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<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b>( <font color="Maroon">P<sub>1</sub></font>[$1] &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  );<br/>
<a NAME="E1:109_1"/>
<font color="Maroon">F<sub>3</sub></font>() is   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:109"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="ordinal1.html#T23" title="ORDINAL1:th.23">ORDINAL1:23</a></i>;<br/>
<a NAME="E2:109_1"/><b>then </b><i><font color="Green" title="E86">A6</font></i>: 
 ex <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">A</font>]
 
<b>by </b><i><a class="txt" href="arytm_3.html#E3:109"><i><font color="Green" title="E83">A3</font></i></a>, <a class="txt" href="arytm_3.html#E5:109"><i><font color="Green" title="E85">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c3">A</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:109_1"/><i><font color="Green" title="E87">A7</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c3">A</font>]
 <b>and </b><br/><a NAME="E5:109_1"/><i><font color="Green" title="E88">A8</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">B</font>] holds <br/><font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Olive" title="b1">B</font>
 <b>from </b><i><a class="ref" href="ordinal1.html#S1" title="ORDINAL1:sch.1">ORDINAL1:sch 1</a>(<a class="txt" href="arytm_3.html#E2:109_1"><i><font color="Green" title="E86">A6</font></i></a>);<br/></i>
<a NAME="E6:109_1"/><i><font color="Green" title="E89">A9</font></i>: 
 <a href="arytm_3.html#K11" title="ARYTM_3:func.11">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E2:109"><i><font color="Green" title="E82">A2</font></i></a>, <a class="txt" href="arytm_3.html#E4:109"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="arytm_3.html#E4:109_1"><i><font color="Green" title="E87">A7</font></i></a>, <a class="ref" href="ordinal3.html#T10" title="ORDINAL3:th.10">ORDINAL3:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="109_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:109_1_1"/>
<font color="Maroon" title="c3">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:109_1_1"/><b>then </b>
 <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E6:109_1"><i><font color="Green" title="E89">A9</font></i></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/><a NAME="E3:109_1_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E4:109_1"><i><font color="Green" title="E87">A7</font></i></a></i>;<br/><b>hence </b><a NAME="E4:109_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c4">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E9:109_1"/><i><font color="Green" title="E90">A10</font></i>: 
<font color="Maroon" title="c3">A</font> <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">B</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>
<a NAME="E10:109_1"/>
<font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 
<b>by </b><i><a class="txt" href="arytm_3.html#E9:109_1"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="ordinal1.html#T13" title="ORDINAL1:th.13">ORDINAL1:13</a></i>;<br/>
<a NAME="E11:109_1"/><b>then </b><i><font color="Green" title="E91">A11</font></i>: 
( <font color="Maroon" title="c4">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; not <font color="Maroon" title="c3">A</font> <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c4">B</font> )
 
<b>by </b><i><a class="txt" href="arytm_3.html#E4:109_1"><i><font color="Green" title="E87">A7</font></i></a>, <a class="ref" href="ordinal1.html#T7" title="ORDINAL1:th.7">ORDINAL1:7</a>, <a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">s</font> = <font color="Maroon" title="c4">B</font> as  <a href="ordinal1.html#V3" title="ORDINAL1:attr.3">ordinal</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="arytm_3.html#K5" title="ARYTM_3:func.5">RAT+</a>  <b>by </b><i><a class="txt" href="arytm_3.html#E48"><i><font color="Green" title="E34">Lm5</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">s</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">s</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon">F<sub>2</sub></font>()] )</span><br/>

<b>thus </b><a NAME="E13:109_1"/>
( <font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">s</font>] )
 <b>by </b><i><a class="txt" href="arytm_3.html#E5:109_1"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="arytm_3.html#E11:109_1"><i><font color="Green" title="E91">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon">F<sub>2</sub></font>()]</span><br/>

<a NAME="E14:109_1"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> 1
 
<b>by </b><i><a class="txt" href="arytm_3.html#E9:109_1"><i><font color="Green" title="E90">A10</font></i></a>, <a class="ref" href="ordinal2.html#T48" title="ORDINAL2:th.48">ORDINAL2:48</a></i>;<br/>
<b>hence </b><a NAME="E15:109_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">s</font> <a href="arytm_3.html#K9" title="ARYTM_3:func.9">+</a> <font color="Maroon">F<sub>2</sub></font>()]
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:109"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="arytm_3.html#E4:109_1"><i><font color="Green" title="E87">A7</font></i></a>, <a class="ref" href="arytm_3.html#T64" target="_self" title="ARYTM_3:th.64">Th64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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