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<div class="add">

<b>let </b><font color="Maroon">a</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">b</font> be  non <a href="xboole_0.html#V1">zero</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">an</font> = <font color="Maroon">a</font>, <font color="Maroon">bn</font> = <font color="Maroon">b</font> as  non <a href="xboole_0.html#V1">zero</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:71_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="newton.html#K13">SetPrimes</a> 
 ;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">i</font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i>, <a class="ref" href="newton.html#D6">NEWTON:def 6</a></i>;<br/><a NAME="E3:71_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">an</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">bn</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#T28">NAT_3:28</a></i>;<br/><a NAME="E4:71_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E5:71_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">a</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_1"><b>suppose </b></a><a NAME="E1:71_1_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:71_1_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T23">NAT_1:23</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
0
<b>by </b><i><a class="txt" href="moebius1.html#E4:71_1"><i><font color="Green">E51</font></i></a>, </i>
<br/>.= 
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2"><b>suppose </b></a><a NAME="E1:71_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>thus </b><a NAME="E2:71_1_1_2"/>
<span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:71_1_1_2_1_1"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span><b>by </b><i>, <a class="txt" href="moebius1.html#E4:71_1"><i><font color="Green">E51</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:71_1_1_2_1_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E76</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>then consider </b><font color="Maroon">kb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_2_1_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kb</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">kb</font> = <font color="Maroon">kb</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/><a NAME="E5:71_1_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><a NAME="E6:71_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">lb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:71_1_1_2_1_1_1_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">kb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E9:71_1_1_2_1_1_1_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E10:71_1_1_2_1_1_1_1"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">ka</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:71_1_1_2_1_1_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ka</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">ka</font> = <font color="Maroon">ka</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/><a NAME="E14:71_1_1_2_1_1_1_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><a NAME="E15:71_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">la</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:71_1_1_2_1_1_1_1"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ka</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E18:71_1_1_2_1_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E19:71_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E20:71_1_1_2_1_1_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E3:71_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="moebius1.html#E5:71_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E21:71_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="moebius1.html#E4:71_1"><i><font color="Green">E51</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> 0
<b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:71_1_1_2_1_1_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><div><i><font color="Green">E96</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:71_1_1_2_1_1_2_1"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>consider </b><font color="Maroon">kb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:71_1_1_2_1_1_2_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">kb</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">kb</font> = <font color="Maroon">kb</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/><a NAME="E5:71_1_1_2_1_1_2_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">b</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><a NAME="E6:71_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">lb</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:71_1_1_2_1_1_2_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">kb</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E9:71_1_1_2_1_1_2_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">kb</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">lb</font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:71_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E10:71_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>consider </b><font color="Maroon">ka</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:71_1_1_2_1_1_2_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ka</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T22">NAT_1:22</a></i>;<br/><b>reconsider </b><font color="Maroon">ka</font> = <font color="Maroon">ka</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/><a NAME="E14:71_1_1_2_1_1_2_1"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><a NAME="E15:71_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><b>then consider </b><font color="Maroon">la</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:71_1_1_2_1_1_2_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ka</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E18:71_1_1_2_1_1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">ka</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">la</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E19:71_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><a NAME="E20:71_1_1_2_1_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E3:71_1"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E21:71_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/></div><b>end;</b></div><b>thus </b><span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">p</font>
<b>by </b><i><a class="txt" href="moebius1.html#E4:71_1"><i><font color="Green">E51</font></i></a>, </i>
<br/>.= 
0 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p2">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="polynom1.html#D5">POLYNOM1:def 5</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:71"/>
 <a href="moebius1.html#K5" target="_self">PFactors</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">b</font></span>)</span> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">a</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
