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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM2">Ordinal</a>] means ( <font color="Maroon">a<sub>1</sub></font> is <a href="ordinal1.html#V7">natural</a> implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E1:5_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="xboole_0.html#K1">{}</a> ]
 
<b>by </b><i><a class="txt" href="arytm_3.html#E1:5"><i><font color="Green">E2</font></i></a></i>;<br/>
<div><i><font color="Green">E5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>now </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="E1:5_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1_1_1"/>
 <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> is <a href="ordinal1.html#V7">natural</a>
 ;<br/><a NAME="E2:5_1_1_1"/><b>then </b>
(  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="ordinal1.html#T10">ORDINAL1:10</a>, <a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E3:5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="ordinal1.html#K5">omega</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#T19">ORDINAL1:19</a></i>;<br/><b>hence </b><a NAME="E4:5_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font>]
 <b>by </b><i><a class="txt" href="arytm_3.html#E2:5"><i><font color="Green">E3</font></i></a>, <a class="txt" href="arytm_3.html#E1:5_1_1"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:5_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#K1">succ</a> <font color="Maroon">c<sub>1</sub></font>]
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2"><b>now </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/><a NAME="E1:5_1_2"/><i><font color="Green">E7</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><b>assume </b><a NAME="E2:5_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:5_1_2"/><b>then </b>
 <a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="arytm_3.html#E1:5_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E4:5_1_2"/><b>then </b><i><font color="Green">E8</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="ordinal1.html#T21">ORDINAL1:21</a></i>;<br/><b>assume </b><a NAME="E5:5_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="ordinal1.html#NR2" target="_self">is_limit_ordinal</a> 
 ;<br/><a NAME="E6:5_1_2"/><b>then </b>
(  <a href="ordinal1.html#K5">omega</a>  <a href="ordinal1.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>1</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="arytm_3.html#E4:5_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="ordinal1.html#D12">ORDINAL1:def 12</a></i>;<br/><a NAME="E7:5_1_2"/><b>then </b>
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="ordinal1.html#K5">omega</a> 
 ;<br/><b>hence </b><a NAME="E8:5_1_2"/>
( ( 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>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:5_1"/>
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="ordinal2.html#S1">ORDINAL2:sch 1</a>(<a class="txt" href="arytm_3.html#E1:5_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="arytm_3.html#E2:5_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="arytm_3.html#E3:5_1"><i><font color="Green">E6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:5_1"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="ordinal1.html#V7">natural</a> <a href="ordinal1.html#NM2">Ordinal</a> holds <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>


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