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

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ] <b>means </b>for <font color="Olive">n</font> being non <a href="xboole_0.html#V1">empty</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="nat_3.html#NK15" target="_self">ppf</a> <font color="Olive">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">n</font>;<br/>
<a NAME="E1:104"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:104_1"/><i><font color="Green">E43</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> 0
 ;<br/>

<a NAME="E2:104_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="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E3:104_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="nat_3.html#T4" target="_self">Th4</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="E4:104_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font> <a href="pboole.html#R6">=</a>  <a href="polynom1.html#K16">EmptyBag</a> <a href="newton.html#K13">SetPrimes</a> 
 
<b>by </b><i><a class="ref" href="bagorder.html#T20">BAGORDER:20</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">n</font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>assume </b><a NAME="E2:104_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E3:104_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="ref" href="uproots.html#T1">UPROOTS:1</a></i>;<br/><a NAME="E4:104_1_1"/><b>then </b>
<font color="Maroon">k</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">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:104_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">p</font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E7:104_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/><a NAME="E8:104_1_1"/>
not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E9:104_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="nat_3.html#E4:104_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:104_1"/>
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:104"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E2:104_2"/><i><font color="Green">E69</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:104_2"/><i><font color="Green">E74</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_2_1"/>
(  not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> or  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_1"><b>suppose </b></a><a NAME="E1:104_2_1_1"/><i><font color="Green">E103</font></i>: 
not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><div><i><font color="Green">E104</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:104_2_1_1_1"/><i><font color="Green">E105</font></i>: 
not <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:104_2_1_1_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_2_1_1_1_1"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E2:104_2_1_1_1_1"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">x</font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i>, , </i>;<br/>
<a NAME="E4:104_2_1_1_1_1"/><i><font color="Green">E108</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:104_2_1_1_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:104_2_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:104_2_1_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E8:104_2_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:104_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">p</font> = <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1;<br/><b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span>;<br/><a NAME="E3:104_2_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, </i>;<br/><a NAME="E4:104_2_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E5:104_2_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">k</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="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E6:104_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:104_2_1_1"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>set </b><font color="Maroon">t1</font> = <font color="Maroon">t</font>;<br/><b>reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">t</font> = <font color="Maroon">t</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i/>;<br/><a NAME="E10:104_2_1_1"/><i><font color="Green">E113</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:104_2_1_1"/><i><font color="Green">E114</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2_1_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_2_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>

<a NAME="E2:104_2_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:104_2_1_1_2"/><b>then </b><i><font color="Green">E117</font></i>: 
<font color="Maroon">x</font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i>, </i>;<br/>
<a NAME="E5:104_2_1_1_2"/><i><font color="Green">E118</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="nat_3.html#E2:104_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:104_2_1_1_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="nat_3.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E7:104_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:104_2_1_1_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:104_2_1_1_2_1"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 ;<br/><a NAME="E2:104_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:104_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="nat_3.html#T12" target="_self">Th12</a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:104_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:104_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">g</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:104_2_1_1_2_1"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E8:104_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">u</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:104_2_1_1_2_1"/><i><font color="Green">E122</font></i>: 
<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><i><font color="Green">E123</font></i>: <font color="Maroon">n</font> = 
<font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i>, <a class="ref" href="nat_3.html#T13" target="_self">Th13</a>, <a class="txt" href="nat_3.html#E3:104_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="newton.html#K2">|^</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">u</font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><a NAME="E12:104_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">k</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">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T14" target="_self">Th14</a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>hence </b><a NAME="E13:104_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E10:104_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E11:104_2_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E12:104_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:104_2_1_1"/><i><font color="Green">E124</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="nat_3.html#K11" target="_self">|-count</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, , <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E13:104_2_1_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E14:104_2_1_1"/><b>then </b><i><font color="Green">E125</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:104_2_1_1_3"/>
<span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:104_2_1_1_3"/><i><font color="Green">E126</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E4:104_2_1_1_3"/><i><font color="Green">E127</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:104_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:104_2_1_1_3"/><i><font color="Green">E128</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_3.html#E2:104_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E6:104_2_1_1_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:104_2_1_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E8:104_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><a NAME="E16:104_2_1_1"/><b>then </b><i><font color="Green">E129</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font>, <font color="Maroon">t1</font> = <font color="Maroon">t</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E18:104_2_1_1"/>
<font color="Maroon">s1</font>,<font color="Maroon">t1</font> <a href="int_2.html#R2">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_2_1_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_2_1_1_4"/><i><font color="Green">E130</font></i>: 
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="int_2.html#D6">INT_2:def 6</a><br/>

<b>set </b><font color="Maroon">u</font> = <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>;<br/>
<a NAME="E2:104_2_1_1_4"/><i><font color="Green">E131</font></i>: 
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E3:104_2_1_1_4"/>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="int_2.html#T5">INT_2:5</a></i>;<br/>
<a NAME="E4:104_2_1_1_4"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>
 
;<br/>
<a NAME="E5:104_2_1_1_4"/><b>then </b>
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:104_2_1_1_4"/><b>then </b>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_3.html#T9" target="_self">Th9</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E7:104_2_1_1_4"/><b>then </b>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:104_2_1_1_4"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">r</font> is <a href="int_2.html#V1">prime</a>
 <b>and </b><br/><a NAME="E10:104_2_1_1_4"/><i><font color="Green">E134</font></i>: 
<font color="Maroon">r</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font>
 <b>by </b><i><a class="ref" href="int_2.html#T48">INT_2:48</a></i>;<br/>
<a NAME="E11:104_2_1_1_4"/>
<font color="Maroon">r</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">s1</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="nat_3.html#T13" target="_self">Th13</a>, <a class="ref" href="nat_1.html#T51">NAT_1:51</a></i>;<br/>
<a NAME="E12:104_2_1_1_4"/><b>then </b>
<font color="Maroon">r</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_3.html#T12" target="_self">Th12</a>, </i>;<br/>
<a NAME="E13:104_2_1_1_4"/><b>then </b><i><font color="Green">E135</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 )
 
<b>by </b><i>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E14:104_2_1_1_4"/>
<font color="Maroon">s1</font> <a href="nat_1.html#K10">hcf</a> <font color="Maroon">t1</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<a NAME="E15:104_2_1_1_4"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="nat_1.html#R1">divides</a> <font color="Maroon">t1</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#T12" target="_self">Th12</a>, <a class="ref" href="nat_3.html#T13" target="_self">Th13</a>, <a class="txt" href="nat_3.html#E3:104_2"><i><font color="Green">E74</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="E16:104_2_1_1_4"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">t</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E17:104_2_1_1_4"/><b>then </b>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D1" target="_self">Def1</a>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>hence </b><a NAME="E18:104_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E19:104_2_1_1"/><b>then </b><i><font color="Green">E137</font></i>: 
 <a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K9">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="nat_3.html#E2:104"><i><font color="Green">E52</font></i></a></i>;<br/><b>consider </b><font color="Maroon">f</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="E21:104_2_1_1"/><i><font color="Green">E138</font></i>: 
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K17">Product</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E22:104_2_1_1"/><i><font color="Green">E139</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="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="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E23:104_2_1_1"/><i><font color="Green">E140</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="newton.html#K13">SetPrimes</a> 
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="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">k</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="ref" href="nat_3.html#T12" target="_self">Th12</a>, <a class="ref" href="uproots.html#T6">UPROOTS:6</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
<b>by </b><i><a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_3.html#T13" target="_self">Th13</a>, <a class="ref" href="circcmb3.html#T36">CIRCCMB3:36</a></i>
;<br/><b>then </b><i><font color="Green">E141</font></i>:  <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">s</font></span>)</span> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="nat_3.html#T11" target="_self">Th11</a>, <a class="ref" href="rvsum_1.html#T125">RVSUM_1:125</a></i>
<br/>.= 
<font color="Maroon">s</font>
<b>by </b><i>, , <a class="txt" href="nat_3.html#E1:104_2"><i><font color="Green">E53</font></i></a></i>
;<br/><a NAME="E26:104_2_1_1"/>
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E27:104_2_1_1"/>
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="txt" href="nat_3.html#E2:104_2"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_3.html#T9" target="_self">Th9</a>, <a class="txt" href="nat_3.html#E3:104_2"><i><font color="Green">E74</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_2_1_2"><b>suppose </b></a><a NAME="E1:104_2_1_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_2_1_2"/>
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:104"/><i><font color="Green">E142</font></i>: 
for <font color="Olive">k</font> being <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(<a class="ref" href="nat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="nat_3.html#T4" target="_self">Th4</a>);<br/></i>
<a NAME="E4:104"/><i><font color="Green">E143</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" target="_self">pfexp</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:104_3"/>
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> or  not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_3_1"><b>suppose </b></a><a NAME="E1:104_3_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:104_3_1"/>
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_3.html#T3" target="_self">Th3</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="104_3_2"><b>suppose </b></a><a NAME="E1:104_3_2"/>
not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">S</font> =  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E3:104_3_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_3_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:104_3_2_1"/><i><font color="Green">E145</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E2:104_3_2_1"/><b>then </b><i><font color="Green">E146</font></i>: 
<font color="Maroon">x</font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">m</font> = <font color="Maroon">x</font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i>, , </i>;<br/>
<a NAME="E4:104_3_2_1"/><i><font color="Green">E147</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="nat_3.html#T5" target="_self">Th5</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:104_3_2_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:104_3_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="pre_circ.html#K1">max</a> <font color="Maroon">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E4:104_3_2"/>
 <a href="nat_3.html#K8" target="_self">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" target="_self">ppf</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
