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<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">E</font> =  <a href="euler_1.html#K1" title="EULER_1:func.1">Euler</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">b0</font> =  <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">F1</font> =  <a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">b1</font> = <span class="p1">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">f0</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E2:54"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K20" title="RVSUM_1:func.20">Product</a> <font color="Maroon" title="c7">f0</font>
 <span class="kw">and </span><br/><a NAME="E3:54"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon" title="c7">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D5" onmouseover="rs('nat_3/D5')" onmouseout="rh()">NAT_3:def 5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">f1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw"> such that </span><br/><a NAME="E5:54"/><span class="lab"><font color="Green" title="E49">A3</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K20" title="RVSUM_1:func.20">Product</a> <font color="Maroon" title="c8">f1</font>
 <span class="kw">and </span><br/><a NAME="E6:54"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D5" onmouseover="rs('nat_3/D5')" onmouseout="rh()">NAT_3:def 5</a></span>;<br/>
<a NAME="E7:54"/><span class="lab"><font color="Green" title="E51">A5</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<a NAME="E8:54"/><span class="lab"><font color="Green" title="E52">A6</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T12" target="_self" onmouseover="rs('number08/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E9:54"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
 <a href="pre_poly.html#K13" title="PRE_POLY:func.13">support</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#T28" target="_self" onmouseover="rs('number08/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<a NAME="E10:54"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="number08.html#D2" target="_self" onmouseover="rs('number08/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<a NAME="E11:54"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="number08.html#T13" target="_self" onmouseover="rs('number08/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E12:54"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E13:54"/><span class="lab"><font color="Green" title="E57">A11</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="number08.html#T13" target="_self" onmouseover="rs('number08/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E14:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">f0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
<a NAME="E15:54"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">f0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f0</font> = <font color="Maroon" title="c7">f0</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E17:54"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c8">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f1</font> = <font color="Maroon" title="c8">f1</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E19:54"/>
 for <font color="Olive" title="b1">x</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f1</font> holds <br/><font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">f0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f1</font> implies <font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">f0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54_1"/><span class="lab"><font color="Green" title="E59">A13</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">and </span><br/><a NAME="E2:54_1"/><span class="lab"><font color="Green" title="E60">A14</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">f1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">f0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">p</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>;<br/>
<a NAME="E3:54_1"/><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/>
<a NAME="E4:54_1"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T25" onmouseover="rs('relat_1/T25')" onmouseout="rh()">RELAT_1:25</a></span>;<br/>
<a NAME="E5:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A16</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p4">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">c</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:54_1"/><span class="lab"><font color="Green" title="E63">A17</font></span>: 
<font color="Maroon" title="c13">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">and </span><br/><a NAME="E8:54_1"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
<span class="p1">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">c</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="ref" href="number08.html#D2" target="_self" onmouseover="rs('number08/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="lab"><font color="Green" title="E65">A19</font></span>: <font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">c</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E5:54_1"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E8:54_1"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
;<br/>
<span class="lab"><font color="Green" title="E66">A20</font></span>: <font color="Maroon" title="c9">f0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E11:54"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="txt" href="#E13:54"><span class="lab"><font color="Green" title="E57">A11</font></span></a>, <a class="txt" href="#E1:54_1"><span class="lab"><font color="Green" title="E59">A13</font></span></a>, <a class="txt" href="#E2:54_1"><span class="lab"><font color="Green" title="E60">A14</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K20" title="FINSEQ_1:func.20">canFS</a> <span class="p3">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p5">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Maroon" title="c13">c</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54"><span class="lab"><font color="Green" title="E51">A5</font></span></a>, <a class="txt" href="#E8:54"><span class="lab"><font color="Green" title="E52">A6</font></span></a>, <a class="txt" href="#E9:54"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="txt" href="#E10:54"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="txt" href="#E12:54"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="txt" href="#E5:54_1"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="txt" href="#E7:54_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>
;<br/>
<a NAME="E11:54_1"/>
<font color="Maroon" title="c13">c</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">c</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:54_1"/>
<font color="Maroon" title="c10">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c9">f0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:54_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="txt" href="#E10:54_1"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="ref" href="newton.html#T89" onmouseover="rs('newton/T89')" onmouseout="rh()">NEWTON:89</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:54"/><span class="kw">then </span>
 <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c10">f1</font> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a>  <a href="rvsum_1.html#K19" title="RVSUM_1:func.19">Product</a> <font color="Maroon" title="c9">f0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:54"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="ref" href="number08.html#T32" target="_self" onmouseover="rs('number08/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:54"/>
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="number08.html#K2" title="NUMBER08:func.2">Euler_factorization_1</a> <font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="int_1.html#R1" title="INT_1:pred.1">divides</a> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E5:54"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
