<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">n</font> be  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/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>for <font color="Olive">n</font> being 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>   st  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/>for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> holds  <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1;<br/>
<a NAME="E1:84"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  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/>

<b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E42</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> 0
 ;<br/>

<a NAME="E2:84_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:84_1"/><b>then </b>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E4:84_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i><a class="ref" href="bagorder.html#T20">BAGORDER:20</a></i>;<br/>
<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<a NAME="E5:84_1"/>
not <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1_1"/>
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a>  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E2:84_1_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> 1 &amp; 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T20">NAT_3:20</a>, <a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>
<a NAME="E3:84_1_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T52">NAT_1:52</a></i>;<br/>
<b>hence </b><a NAME="E4:84_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:84_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:84"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:84_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">n</font> be  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/>

<b>assume </b><a NAME="E2:84_2"/><i><font color="Green">E72</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:84_2"/><i><font color="Green">E73</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_1"/>
(  not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> or  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1"><b>suppose </b></a><a NAME="E1:84_2_1_1"/><i><font color="Green">E74</font></i>: 
not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_1_1_1"/><i><font color="Green">E76</font></i>: 
not <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:84_2_1_1_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:84_2_1_1_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E2:84_2_1_1_1_1"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">x</font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i>, <a class="ref" href="moebius1.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E4:84_2_1_1_1_1"/><i><font color="Green">E92</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="moebius1.html#T5" target="_self">Th5</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:84_2_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="moebius1.html#T4" target="_self">Th4</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:84_2_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="moebius1.html#T4" target="_self">Th4</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:84_2_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E8:84_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:84_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 as  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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>set </b><font color="Maroon">e</font> = <font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">s</font> = <font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span>;<br/><a NAME="E4:84_2_1_1"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">r</font> is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i>, <a class="txt" href="moebius1.html#E5"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/><a NAME="E5:84_2_1_1"/><b>then </b><i><font color="Green">E94</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/><a NAME="E6:84_2_1_1"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">r</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="E7:84_2_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:84_2_1_1"/><i><font color="Green">E96</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">t</font> = <font color="Maroon">t</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>  <b>by </b><i><a class="ref" href="moebius1.html#T6" target="_self">Th6</a></i>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><a NAME="E11:84_2_1_1"/><i><font color="Green">E97</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:84_2_1_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>

<a NAME="E2:84_2_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:84_2_1_1_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">x</font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i><a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="ref" href="moebius1.html#T8" target="_self">Th8</a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E5:84_2_1_1_2"/><i><font color="Green">E103</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="moebius1.html#T9" target="_self">Th9</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:84_2_1_1_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="moebius1.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_3.html#T45">NAT_3:45</a></i>;<br/>
<a NAME="E7:84_2_1_1_2"/><b>then </b>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:84_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="moebius1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E9:84_2_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_1_1_2_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:84_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/><a NAME="E3:84_2_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="moebius1.html#T8" target="_self">Th8</a>, <a class="ref" href="moebius1.html#T11" target="_self">Th11</a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:84_2_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:84_2_1_1_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:84_2_1_1_2_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</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="E9:84_2_1_1_2_1"/>
<font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E11:84_2_1_1_2_1"/><i><font color="Green">E107</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E108</font></i>: <font color="Maroon">n</font> = 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="moebius1.html#T6" target="_self">Th6</a>, <a class="txt" href="moebius1.html#E2:84_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="moebius1.html#E3:84_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">r</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><a NAME="E13:84_2_1_1_2_1"/>
<font color="Maroon">r</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:84_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>hence </b><a NAME="E14:84_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="moebius1.html#T5" target="_self">Th5</a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:84_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:84_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E13:84_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:84_2_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="txt" href="moebius1.html#E5"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="moebius1.html#T4" target="_self">Th4</a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E13:84_2_1_1"/><i><font color="Green">E110</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/><a NAME="E14:84_2_1_1"/><b>then </b><i><font color="Green">E111</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="moebius1.html#T7" target="_self">Th7</a>, <a class="ref" href="nat_3.html#T42">NAT_3:42</a></i>;<br/><div><i><font color="Green">E115</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:84_2_1_1_3"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:84_2_1_1_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span>
 <b>and </b><br/><a NAME="E4:84_2_1_1_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:84_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a>, <a class="ref" href="moebius1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E6:84_2_1_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E7:84_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T9" target="_self">Th9</a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:84_2_1_1_3"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="moebius1.html#E2:84_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E9:84_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font>, <font color="Maroon">t1</font> = <font color="Maroon">t</font> as  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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E17:84_2_1_1"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">s1</font>,<font color="Maroon">t1</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_2_1_1_4"/>
not <font color="Maroon">s1</font>,<font color="Maroon">t1</font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:84_2_1_1_4"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D6">INT_2:def 6</a></i>;<br/>
<b>set </b><font color="Maroon">u</font> = <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>;<br/>
<a NAME="E3:84_2_1_1_4"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E4:84_2_1_1_4"/>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="int_2.html#T5">INT_2:5</a></i>;<br/>
<a NAME="E5:84_2_1_1_4"/><b>then </b>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>
 
;<br/>
<a NAME="E6:84_2_1_1_4"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:84_2_1_1_4"/><b>then </b>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="moebius1.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E8:84_2_1_1_4"/><b>then </b>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">w</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:84_2_1_1_4"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">w</font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E11:84_2_1_1_4"/><i><font color="Green">E123</font></i>: 
<font color="Maroon">w</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/>
<a NAME="E12:84_2_1_1_4"/>
<font color="Maroon">w</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:84_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="moebius1.html#E1:84_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E13:84_2_1_1_4"/><b>then </b>
<font color="Maroon">w</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:84_2"><i><font color="Green">E73</font></i></a>, <a class="ref" href="nat_3.html#T5">NAT_3:5</a></i>;<br/>
<a NAME="E14:84_2_1_1_4"/><b>then </b><i><font color="Green">E124</font></i>: 
( <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E15:84_2_1_1_4"/>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E16:84_2_1_1_4"/><b>then </b>
<font color="Maroon">r</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:84_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="moebius1.html#E1:84_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="moebius1.html#E2:84_2_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E17:84_2_1_1_4"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_3.html#T37">NAT_3:37</a></i>;<br/>
<a NAME="E18:84_2_1_1_4"/><b>then </b>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E19:84_2_1_1_4"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E20:84_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E18:84_2_1_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/><a NAME="E19:84_2_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">s</font></span>)</span> &amp;  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span> )
 <b>by </b><i><a class="ref" href="moebius1.html#T8" target="_self">Th8</a>, </i>;<br/><i><font color="Green">E126</font></i>:  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font> = 
 <a href="nat_3.html#K8">Product</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">s</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">t</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="moebius1.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">t</font></span>)</span>
<b>by </b><i><a class="ref" href="moebius1.html#T11" target="_self">Th11</a>, <a class="ref" href="moebius1.html#T10" target="_self">Th10</a>, <a class="ref" href="nat_3.html#T19">NAT_3:19</a></i>
;<br/><a NAME="E21:84_2_1_1"/><i><font color="Green">E127</font></i>: 
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">t</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font> )
 <b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E22:84_2_1_1"/><i><font color="Green">E128</font></i>: 
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</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"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_2_1_1_6"/>
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:84_2_1_1_6"/><b>then </b>
( <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:84_2"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E3:84_2_1_1_6"/><b>then </b>
( <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E4:84_2_1_1_6"/><b>then </b>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">s</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E5:84_2_1_1_6"/>
contradiction
 <b>by </b><i><a class="ref" href="moebius1.html#T11" target="_self">Th11</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="ref" href="moebius1.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:84_2_1_1"/><i><font color="Green">E129</font></i>: 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1_1_7"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:84_2_1_1_7_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:84_2_1_1_7_1_1"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:84_2_1_1_7_1_1"/><b>then </b><i><font color="Green">E135</font></i>: 
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="moebius1.html#T7" target="_self">Th7</a>, </i>;<br/><a NAME="E3:84_2_1_1_7_1_1"/>
<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/><b>hence </b><a NAME="E4:84_2_1_1_7_1_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, <a class="ref" href="moebius1.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_3.html#T22">NAT_3:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:84_2_1_1_7_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:84_2_1_1_7_1_2"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <b>by </b><i><a class="ref" href="int_2.html#T47">INT_2:47</a>, </i>;<br/><a NAME="E3:84_2_1_1_7_1_2"/><b>then </b><i><font color="Green">E136</font></i>: 
not <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="pythtrip.html#D2">PYTHTRIP:def 2</a></i>;<br/><a NAME="E4:84_2_1_1_7_1_2"/><i><font color="Green">E137</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/><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">s</font> = 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T32">NAT_3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K24">*</a> 0
<b>by </b><i>, <a class="ref" href="moebius1.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_3.html#T27">NAT_3:27</a></i>
<br/>.= 
0

;<br/><b>hence </b><a NAME="E6:84_2_1_1_7_1_2"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="moebius1.html#T1" target="_self">Th1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E24:84_2_1_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</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> <span class="p2">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">t</font></span>)</span></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> <span class="p2">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">s</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E2:84_2"><i><font color="Green">E72</font></i></a>, <a class="ref" href="nat_3.html#T28">NAT_3:28</a></i>;<br/><b>hence </b><a NAME="E25:84_2_1_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E2:84_2_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="moebius1.html#E1:84_2_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="moebius1.html#E1:84_2"><i><font color="Green">E56</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="84_2_1_2"><b>suppose </b></a><a NAME="E1:84_2_1_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:84_2_1_2"/>
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> holds  <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E1:84_2"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:84"/><i><font color="Green">E138</font></i>: 
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>consider </b><font color="Maroon">mS</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:84"/><i><font color="Green">E139</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">mS</font>
 <b>by </b><i/>;<br/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> = 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>
<br/>.= 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E7:84"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, </i>;<br/>


</div>
