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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> is    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:13_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i>, <a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<a NAME="E2:13_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">A</font> is    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>] )
 <b>and </b><br/><a NAME="E2:13_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> is    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<a NAME="E3:13_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> &amp;  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i>, , <a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a></i>;<br/>
<a NAME="E4:13_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K3">On</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i/>;<br/>
<a NAME="E6:13_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="ordinal4.html#K6">succ</a> <font color="Maroon">a</font>]
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:13_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">A</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:13_1"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  &amp; ( for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:13_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a>  )
 <b>and </b><br/><a NAME="E2:13_1_2"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">B</font> being  <a href="ordinal1.html#NM2">Ordinal</a>  st <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">B</font> is    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>]
 ;<br/>

<b>assume </b><a NAME="E3:13_1_2"/>
<font color="Maroon">A</font> is    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>then reconsider </b><font color="Maroon">a</font> = <font color="Maroon">A</font> as    <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<a NAME="E5:13_1_2"/>
(  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="ordinal4.html#K2">0-element_of</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive">b</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b</font>] ) )
 
<b>by </b><i>, <a class="ref" href="ordinal4.html#T35">ORDINAL4:35</a></i>;<br/>
<b>hence </b><a NAME="E6:13_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:13_1"/>
for <font color="Olive">A</font> being  <a href="ordinal1.html#NM2">Ordinal</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>]
 
<b>from </b><i><a class="ref" href="ordinal2.html#S1">ORDINAL2:sch 1</a>(, , );<br/></i>
<b>hence </b><a NAME="E5:13_1"/>
for <font color="Olive">a</font> being   <a href="ordinal4.html#M1">Ordinal</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>]
 ;<br/>


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