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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:63"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 4 )
 ;<br/>

<a NAME="E2:63"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="fib_num2.html#T14" target="_self">Th14</a></i>;<br/>
<b>assume </b><a NAME="E3:63"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">n</font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:63_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or  ex <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_1"><b>suppose </b></a><a NAME="E1:63_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:63_1_1"/>
 ex <font color="Olive">k</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i><a class="ref" href="fib_num2.html#T14" target="_self">Th14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="63_1_2"><b>suppose </b></a><a NAME="E1:63_1_2"/>
 ex <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> &amp; not <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1 &amp; not <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">k</font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:63_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:63_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E5:63_1_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">k</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/><b>consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:63_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E8:63_1_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 <b>by </b><i>, </i>;<br/><a NAME="E9:63_1_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font>
 <b>by </b><i>, , , <a class="ref" href="xcmplx_1.html#T7">XCMPLX_1:7</a></i>;<br/><a NAME="E10:63_1_2"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E11:63_1_2"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">k</font> is  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i>, , , <a class="ref" href="int_2.html#T3">INT_2:3</a></i>;<br/><a NAME="E12:63_1_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">m</font> is  non <a href="xboole_0.html#V1">empty</a>  <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="E13:63_1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 )
 <b>by </b><i><a class="ref" href="fib_num2.html#T14" target="_self">Th14</a>, </i>;<br/><b>hence </b><a NAME="E14:63_1_2"/>
 ex <font color="Olive">k</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 2 &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font> )
 <b>by </b><i>, , , , <a class="ref" href="fib_num2.html#T15" target="_self">Th15</a>, , <a class="ref" href="fib_num2.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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