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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>( $1 is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a> implies <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:86_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 
<b>by </b><i><a class="txt" href="ordinal2.html#E1:86"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<div><i><font color="Green" title="E39">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><b>now </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="E1:86_1_1"/><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:86_1_1_1"/>
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 ;<br/><a NAME="E2:86_1_1_1"/><b>then </b>
(  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</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" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T10" title="ORDINAL1:th.10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:86_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">A</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T19" title="ORDINAL1:th.19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E4:86_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>]
 <b>by </b><i><a class="txt" href="ordinal2.html#E2:86"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="ordinal2.html#E1:86_1_1"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:86_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c1">A</font>]
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E40">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_2"><b>now </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/><a NAME="E1:86_1_2"/><i><font color="Green" title="E41">A7</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></i>;<br/><b>assume </b><a NAME="E2:86_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E3:86_1_2"/><b>then </b>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="ordinal2.html#E1:86_1_2"><i><font color="Green" title="E41">A7</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E4:86_1_2"/><b>then </b><i><font color="Green" title="E42">A8</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21" title="ORDINAL1:th.21">ORDINAL1:21</a></i>;<br/><b>assume </b><a NAME="E5:86_1_2"/>
<font color="Maroon" title="c1">A</font> <a href="ordinal1.html#NR2" title="ORDINAL1:NR.2">is_limit_ordinal</a> 
 ;<br/><a NAME="E6:86_1_2"/><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="c1">A</font> &amp; not <font color="Maroon" title="c1">A</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#E4:86_1_2"><i><font color="Green" title="E42">A8</font></i></a>, <a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></i>;<br/><a NAME="E7:86_1_2"/><b>then </b>
not <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ordinal1.html#K5" title="ORDINAL1:func.5">omega</a> 
 ;<br/><b>hence </b><a NAME="E8:86_1_2"/>
( ( 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>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">A</font>] )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:86_1"/>
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="ordinal2.html#S1" target="_self" title="ORDINAL2:sch.1">ORDINAL2:sch 1</a>(<a class="txt" href="ordinal2.html#E1:86_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="ordinal2.html#E2:86_1"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="ordinal2.html#E3:86_1"><i><font color="Green" title="E40">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:86_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>1</sub></font>()]
 ;<br/>


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