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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Olive">b<sub>2</sub></font> is <a href="ordinal1.html#V7">natural</a> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E1:82"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/>

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<a NAME="E2:82_1"/><i><font color="Green">E38</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>take </b><font color="Maroon">c<sub>3</sub></font> =  <a href="xboole_0.html#K1">{}</a> ;<br/>

<b>thus </b><a NAME="E2:82_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:82_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T42">ORDINAL1:42</a></i>;<br/>

<b>thus </b><a NAME="E3:82_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a><br/>

<b>thus </b><a NAME="E4:82_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="ordinal2.html#T44" target="_self">Th44</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:82_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_2"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon">c<sub>2</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><a NAME="E1:82_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:82_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:82_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> being   <a href="ordinal1.html#NM2">Ordinal</a><b> such that </b><br/><a NAME="E4:82_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="ordinal1.html#V7">natural</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:82_1"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>take </b><font color="Maroon">c<sub>5</sub></font> =  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>4</sub></font>;<br/>

<b>thus </b><a NAME="E5:82_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 <b>by </b><i><a class="txt" href="ordinal2.html#E4:82_1_2"><i><font color="Green">E40</font></i></a></i>;<br/>

<a NAME="E6:82_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp;  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 
<b>by </b><i><a class="txt" href="ordinal2.html#E4:82_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E7:82_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T41">ORDINAL1:41</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a><br/>

<b>thus </b><a NAME="E8:82_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="ordinal2.html#K10" target="_self">+^</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:82_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="ordinal2.html#E4:82_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="ordinal2.html#T45" target="_self">Th45</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:82_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:82_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:82"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="ordinal1.html#S2">ORDINAL1:sch 2</a>(<a class="txt" href="ordinal2.html#E1:82"><i><font color="Green">E36</font></i></a>);</i><br/>


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