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<b>let </b><font color="Maroon" title="c1">p</font> be  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <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> ;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be  non <a href="xxreal_0.html#NV4" title="XXREAL_0:NV.4">zero</a>  <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> ;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E12">A1</font></i>: 
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a>  &amp; <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">b</font> )
 ;<br/>

<a NAME="E2:16"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2
 
<b>by </b><i><a class="ref" href="nat_3.html#T3" title="NAT_3:th.3">NAT_3:3</a></i>;<br/>
<a NAME="E3:16"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 </span><b>by </b><i><a class="txt" href="moebius1.html#E3:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1"/><b>then </b><i><font color="Green" title="E14">A3</font></i>: 
not <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="pythtrip.html#D2" title="PYTHTRIP:def.2">PYTHTRIP:def 2</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/><a NAME="E3:16_1_1"/>
<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1"/>
not <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:16_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">b</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#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">k</font> being    <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> such that </b><br/><a NAME="E4:16_1_1_1"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/>
<a NAME="E5:16_1_1_1"/>
( <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a>, <a class="txt" href="moebius1.html#E4:16_1_1_1"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E4:16_1_1_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="moebius1.html#E2:16_1_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:16_1_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="euler_1.html#T15" title="EULER_1:th.15">EULER_1:15</a></i>;<br/><a NAME="E5:16_1_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2,<font color="Maroon" title="c3">b</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</a></i>;<br/><b>hence </b><a NAME="E6:16_1_1"/>
( <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="euler_1.html#T14" title="EULER_1:th.14">EULER_1:14</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font>
 ;<br/><div class="add"><a NAME="E2:16_1_2"/><b>then </b><i><font color="Green" title="E14">A5</font></i>: 
not <font color="Maroon" title="c1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="pythtrip.html#D2" title="PYTHTRIP:def.2">PYTHTRIP:def 2</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/><a NAME="E3:16_1_2"/>
<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_2_1"/>
not <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:16_1_2_1"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">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#D6" title="INT_2:def.6">INT_2:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">k</font> being    <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> such that </b><br/><a NAME="E4:16_1_2_1"/><i><font color="Green" title="E15">A6</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="nat_d.html#K8" title="NAT_D:func.8">hcf</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font> &amp; <font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 ;<br/>
<a NAME="E5:16_1_2_1"/>
( <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c4">k</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a>, <a class="txt" href="moebius1.html#E4:16_1_2_1"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E4:16_1_2_1"><i><font color="Green" title="E15">A6</font></i></a>, <a class="txt" href="moebius1.html#E2:16_1_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="int_2.html#D5" title="INT_2:def.5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:16_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="euler_1.html#T15" title="EULER_1:th.15">EULER_1:15</a></i>;<br/><a NAME="E5:16_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2,<font color="Maroon" title="c2">a</font> <a href="int_2.html#R2" title="INT_2:pred.2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="wsierp_1.html#T2" title="WSIERP_1:th.2">WSIERP_1:2</a></i>;<br/><b>hence </b><a NAME="E6:16_1_2"/>
( <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c1">p</font> <a href="newton.html#K16" title="NEWTON:func.16">|^</a> 2 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">b</font> )
 <b>by </b><i><a class="ref" href="euler_1.html#T14" title="EULER_1:th.14">EULER_1:14</a>, <a class="txt" href="moebius1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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