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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 );<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_1"/>
<font color="Maroon">h</font> <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 ;<br/><b>take </b><font color="Maroon">n</font> = 0;<br/><b>thus </b><a NAME="E2:71_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1_1_2"/>
not <font color="Maroon">h</font> <a href="group_1.html#NR2" target="_self">is_of_order_0</a> 
 ;<br/><b>then consider </b><font color="Maroon">n</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i><a class="txt" href="group_1.html#E8:71_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E4:71_1_1_2"/>
<font color="Maroon">n</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E5:71_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:71_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E8:71_1_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><b>take </b><font color="Maroon">k</font> = <font color="Maroon">k</font>;<br/><b>thus </b><a NAME="E9:71_1_1_2"/>
( <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 <b>by </b><i/>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">m</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>assume </b><a NAME="E11:71_1_1_2"/>
( <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E12:71_1_1_2"/><b>then </b>
( <font color="Maroon">h</font> <a href="group_1.html#K6" target="_self">|^</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2" target="_self">1.</a> <font color="Maroon">G</font> &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><a NAME="E13:71_1_1_2"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E14:71_1_1_2"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/></div>
<b>end;</b></div>


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