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

<b>let </b><font color="Maroon">c<sub>1</sub></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> ] means for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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/> <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E1:78"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></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="E1:78_1"/><i><font color="Green">E50</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">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> 0
 ;<br/>

<a NAME="E2:78_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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:78_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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:78_1"><i><font color="Green">E50</font></i></a>, <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="E4:78_1"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>2</sub></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/>
<a NAME="E5:78_1"/>
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E4:78_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="nat_3.html#T20">NAT_3:20</a></i>;<br/>
<b>hence </b><a NAME="E6:78_1"/>
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T53">NAT_1:53</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:78"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:78_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></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="E2:78_2"/><i><font color="Green">E52</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">c<sub>3</sub></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">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:78_2"/><i><font color="Green">E53</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:78_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">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_1"><b>suppose </b></a><a NAME="E1:78_2_1_1"/><i><font color="Green">E54</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">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_2_1_1_1"/><i><font color="Green">E56</font></i>: 
not <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font></span>)</span>
 ;<br/><a NAME="E2:78_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">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></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="E1:78_2_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:78_2_1_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:78_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="moebius1.html#E3:78_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="moebius1.html#E1:78_2_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E4:78_2_1_1_1_1"/><i><font color="Green">E59</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:78_2_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:78_2_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:78_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="moebius1.html#E1:78_2_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:78_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:78_2_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="moebius1.html#E1:78_2_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:78_2_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E8:78_2_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E4:78_2_1_1_1_1"><i><font color="Green">E59</font></i></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="E3:78_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:78_2_1_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E3:78_2_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:78_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="moebius1.html#E2:78_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/><a NAME="E4:78_2_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/><a NAME="E5:78_2_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E3:78_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E6:78_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></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">c<sub>7</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:78_2_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>7</sub></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">c<sub>8</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></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="txt" href="moebius1.html#E8:78_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E10:78_2_1_1"/><i><font color="Green">E60</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">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>10</sub></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="E1:78_2_1_1_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>9</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:78_2_1_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>10</sub></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#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E3:78_2_1_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="moebius1.html#E2:78_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E5:78_2_1_1_2"/><i><font color="Green">E64</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:78_2_1_1_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:78_2_1_1_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></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#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E8:78_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_3.html#T45">NAT_3:45</a></i>;<br/>
<a NAME="E7:78_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">c<sub>9</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:78_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E8:78_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:78_2_1_1_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E9:78_2_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:78_2"><i><font color="Green">E52</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_2_1_1_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E2:78_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:78_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/><a NAME="E3:78_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="moebius1.html#E2:78_2_1_1_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="moebius1.html#E1:78_2_1_1_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:78_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:78_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></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">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:78_2_1_1_2_1"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E8:78_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:78_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:78_2_1_1_2_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E69</font></i>: <font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>8</sub></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"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E8:78_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="moebius1.html#E7:78_2_1_1_2_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="moebius1.html#E10:78_2_1_1_2_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><a NAME="E12:78_2_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E13:78_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E11:78_2_1_1_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>hence </b><a NAME="E14:78_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E5:78_2_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:78_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E9:78_2_1_1_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:78_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E13:78_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:78_2_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:78_2_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="moebius1.html#E3:78_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="moebius1.html#E2:78_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="moebius1.html#E4:78_2_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E12:78_2_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></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#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/><a NAME="E13:78_2_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:78_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="moebius1.html#E11:78_2_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_3.html#T42">NAT_3:42</a></i>;<br/><a NAME="E14:78_2_1_1"/><b>then </b><i><font color="Green">E64</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">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="moebius1.html#E12:78_2_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>;<br/><div><i><font color="Green">E65</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:78_2_1_1_3"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="xboole_0.html#R1">meets</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:78_2_1_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>10</sub></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#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:78_2_1_1_3"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>10</sub></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#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:78_2_1_1_3"/>
<font color="Maroon">c<sub>10</sub></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#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E4:78_2_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/><a NAME="E6:78_2_1_1_3"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>10</sub></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">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E7:78_2_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E13:78_2_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="moebius1.html#E3:78_2_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E8:78_2_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E10:78_2_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="moebius1.html#E6:78_2_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E9:78_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">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><a NAME="E17:78_2_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_2_1_1_4"/>
not <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:78_2_1_1_4"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></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">c<sub>12</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font>;<br/>
<a NAME="E3:78_2_1_1_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E4:78_2_1_1_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></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="E5:78_2_1_1_4"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font>
 
;<br/>
<a NAME="E6:78_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">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E7:78_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E2:78_2_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E8:78_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></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">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:78_2_1_1_4"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E11:78_2_1_1_4"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/>
<a NAME="E12:78_2_1_1_4"/>
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:78_2_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="moebius1.html#E11:78_2_1_1_4"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E13:78_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E10:78_2_1_1_4"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_3.html#T5">NAT_3:5</a></i>;<br/>
<a NAME="E14:78_2_1_1_4"/><b>then </b><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:78_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E15:78_2_1_1_4"/>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E16:78_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E10:78_2_1_1_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="moebius1.html#E11:78_2_1_1_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="moebius1.html#E14:78_2_1_1_4"><i><font color="Green">E71</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="E17:78_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></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="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:78_2_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="nat_3.html#T37">NAT_3:37</a></i>;<br/>
<a NAME="E18:78_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></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">c<sub>9</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E19:78_2_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E10:78_2_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E20:78_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><b>consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> <b> such that </b><br/><a NAME="E19:78_2_1_1"/><i><font color="Green">E67</font></i>: 
 <a href="nat_3.html#K8">Product</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K17">Product</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E20:78_2_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p3">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D5">NAT_3:def 5</a></i>;<br/><a NAME="E21:78_2_1_1"/><i><font color="Green">E69</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="newton.html#K13">SetPrimes</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E22:78_2_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>2</sub></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="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E12:78_2_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="moebius1.html#E13:78_2_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="txt" href="moebius1.html#E14:78_2_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="moebius1.html#E20:78_2_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="uproots.html#T6">UPROOTS:6</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
<b>by </b><i><a class="txt" href="moebius1.html#E2:78_2_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="moebius1.html#E21:78_2_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="circcmb3.html#T36">CIRCCMB3:36</a></i>
;<br/><b>then </b><i><font color="Green">E71</font></i>:  <a href="nat_3.html#K8">Product</a> <span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="moebius1.html#K5" target="_self">PFactors</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E19:78_2_1_1"><i><font color="Green">E67</font></i></a>, <a class="ref" href="rvsum_1.html#T125">RVSUM_1:125</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="txt" href="moebius1.html#E22:78_2_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>
;<br/><a NAME="E25:78_2_1_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></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#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/><a NAME="E26:78_2_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></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">c<sub>8</sub></font></span>)</span> &amp;  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="moebius1.html#E12:78_2_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E27:78_2_1_1"/><i><font color="Green">E73</font></i>: 
(  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="moebius1.html#E24:78_2_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="moebius1.html#E11:78_2_1_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_3.html#T3">NAT_3:3</a>, <a class="txt" href="moebius1.html#E1:78_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="moebius1.html#E10:78_2_1_1"><i><font color="Green">E60</font></i></a></i>;<br/> <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>3</sub></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">c<sub>8</sub></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">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E17:78_2_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="moebius1.html#T50" target="_self">Th50</a>, <a class="txt" href="moebius1.html#E8:78_2_1_1"><i><font color="Green">E59</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E26:78_2_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="moebius1.html#E15:78_2_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_3.html#T19">NAT_3:19</a></i>
;<br/><b>hence </b><a NAME="E29:78_2_1_1"/>
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E8:78_2_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="moebius1.html#E27:78_2_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="nat_3.html#T1">NAT_3:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_2_1_2"><b>suppose </b></a><a NAME="E1:78_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">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:78_2_1_2"/>
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:78_2"><i><font color="Green">E51</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:78"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="txt" href="moebius1.html#E1:78"><i><font color="Green">E49</font></i></a>, <a class="txt" href="moebius1.html#E2:78"><i><font color="Green">E50</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:78"/><i><font color="Green">E52</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="moebius1.html#T14" target="_self">Th14</a></i>;<br/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="moebius1.html#D6" target="_self">Def6</a></i>
;<br/>
<b>hence </b><a NAME="E7:78"/>
 <a href="moebius1.html#K6" target="_self">Radical</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:78"><i><font color="Green">E51</font></i></a>, <a class="txt" href="moebius1.html#E5:78"><i><font color="Green">E52</font></i></a></i>;<br/>


</div>
