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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>] <b>means </b> ex <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <br/>( <font color="Olive" title="b1">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Olive" title="b2">C</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Olive" title="b2">C</font> );<br/>
<a NAME="E1:80"/><i><font color="Green" title="E36">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st ( for <font color="Olive" title="b2">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  st <font color="Olive" title="b2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:80_1"/><i><font color="Green" title="E37">A2</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="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>]
 ;<br/>

<a NAME="E2:80_1"/><i><font color="Green" title="E38">A3</font></i>: 
( ( for <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">B</font> ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:80_1_1"/><i><font color="Green" title="E39">A4</font></i>: 
for <font color="Olive" title="b1">D</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds <font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">D</font>
 ;<br/>

<b>take </b><font color="Maroon" title="c2">B</font> = <font color="Maroon" title="c1">A</font>;<br/>

<b>take </b><font color="Maroon" title="c3">C</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/>

<b>thus </b><a NAME="E2:80_1_1"/>
<font color="Maroon" title="c2">B</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:80_1_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="ordinal1.html#T42" title="ORDINAL1:th.42">ORDINAL1:42</a></i>;<br/>

<b>thus </b><a NAME="E3:80_1_1"/>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a><br/>

<b>thus </b><a NAME="E4:80_1_1"/>
<font color="Maroon" title="c1">A</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="c3">C</font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T44" target="_self" title="ORDINAL2:th.44">Th44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:80_1"/>
(  ex <font color="Olive" title="b1">B</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> st <font color="Maroon" title="c1">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="Olive" title="b1">B</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="80_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c2">B</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><a NAME="E1:80_1_2"/><i><font color="Green" title="E39">A5</font></i>: 
<font color="Maroon" title="c1">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="c2">B</font>
 ;<br/>

<a NAME="E2:80_1_2"/>
<font color="Maroon" title="c2">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:80_1_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">C</font>, <font color="Maroon" title="c4">D</font> being   <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a><b> such that </b><br/><a NAME="E4:80_1_2"/><i><font color="Green" title="E40">A6</font></i>: 
( <font color="Maroon" title="c3">C</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  &amp; <font color="Maroon" title="c4">D</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> &amp; <font color="Maroon" title="c2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c4">D</font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:80_1"><i><font color="Green" title="E37">A2</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">C</font>
;<br/>

<b>take </b><font color="Maroon" title="c5">E</font> =  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c4">D</font>;<br/>

<b>thus </b><a NAME="E5:80_1_2"/>
<font color="Maroon" title="c3">C</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:80_1_2"><i><font color="Green" title="E40">A6</font></i></a></i>;<br/>

<a NAME="E6:80_1_2"/>
( <font color="Maroon" title="c4">D</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  &amp;  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a>  <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:80_1_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:80_1_2"/>
<font color="Maroon" title="c5">E</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T41" title="ORDINAL1:th.41">ORDINAL1:41</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a><br/>

<b>thus </b><a NAME="E8:80_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">C</font> <a href="ordinal2.html#K10" title="ORDINAL2:func.10">+^</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:80_1_2"><i><font color="Green" title="E39">A5</font></i></a>, <a class="txt" href="ordinal2.html#E4:80_1_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="ordinal2.html#T45" target="_self" title="ORDINAL2:th.45">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:80_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:80_1"><i><font color="Green" title="E38">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:80"/>
for <font color="Olive" title="b1">A</font> being  <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2" title="ORDINAL1:sch.2">ORDINAL1:sch 2</a>(<a class="txt" href="ordinal2.html#E1:80"><i><font color="Green" title="E36">A1</font></i></a>);</i><br/>


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