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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_1_1_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_1_1_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</font>;<br/><b>thus </b><a NAME="E2:9_1_1_1_1"/>
<font color="Maroon" title="c2">k</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:9_1_1_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="subset_1.html#D2" title="SUBSET_1:def.2">SUBSET_1:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_1_1_1_2"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>let </b><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">n</font>;<br/><a NAME="E2:9_1_1_1_2"/>
( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c1">n</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <b>by </b><i><a class="txt" href="stirl2_1.html#E1:9_1_1_1_2"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="stirl2_1.html#T8" target="_self" title="STIRL2_1:th.8">Th8</a></i>;<br/><a NAME="E3:9_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c2">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 ;<br/><b>hence </b><a NAME="E4:9_1_1_1_2"/>
<font color="Maroon" title="c2">k</font> is <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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