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<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">n</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">f</font> =  <a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">n</font>
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c1">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1"/>
(  not <font color="Maroon" title="c4">a</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> or <font color="Maroon" title="c4">a</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><b>suppose </b></a><a NAME="E1:70_1_1"/>
not <font color="Maroon" title="c4">a</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ;<br/><div class="add"><a NAME="E2:70_1_1"/><b>then </b>
not <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#T33" target="_self" title="NAT_3:th.33">Th33</a></i>;<br/><b>hence </b><a NAME="E3:70_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2"><b>suppose </b></a><a NAME="E1:70_1_2"/><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c4">a</font> is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>
 ;<br/><div class="add"><a NAME="E2:70_1_2"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/><a NAME="E3:70_1_2"/><b>then </b>
<font color="Maroon" title="c4">a</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="nat_3.html#E1:70"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="nat_3.html#E2:70"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="nat_3.html#T23" target="_self" title="NAT_3:th.23">Th23</a></i>;<br/><b>hence </b><a NAME="E4:70_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="nat_3.html#E1:70_1_2"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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