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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" 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"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">n</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> $1 holds <br/> <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="Olive" title="b1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font>;<br/>
<a NAME="E1:101"/><span class="lab"><font color="Green" title="E53">A1</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" target="_self" title="NAT_3:def.9">Def9</a></span>;<br/>
<a NAME="E2:101"/><span class="lab"><font color="Green" title="E54">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> implies  <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> <font color="Maroon" title="c2">n</font> )</span><br/>

<a NAME="E1:101_1"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:101_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 ; <a class="txt" 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"><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> <font color="Maroon" title="c2">n</font></span><br/>

<a NAME="E3:101_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A4</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E57">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c3">k</font> = <font color="Maroon" title="c2">n</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">assume </span><a NAME="E2:101_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:101_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T53" title="NAT_1:th.53">NAT_1:53</a></span>;<br/><a NAME="E4:101_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:101_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">p</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Olive" title="b1">p</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></span>;<br/><a NAME="E6:101_1_1"/><span class="kw">then </span>
 not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T37" target="_self" title="NAT_3:th.37">Th37</a></span>;<br/><span class="kw">hence </span><a NAME="E7:101_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="nat_3.html#D9" target="_self" title="NAT_3:def.9">Def9</a></span>; <a class="txt" 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="E5:101_1"/>
 <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K16" title="PRE_POLY:func.16">EmptyBag</a> <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="bagorder.html#T19" title="BAGORDER:th.19">BAGORDER:19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:101_1"/>
 <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> <font color="Maroon" title="c2">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="ref" href="nat_3.html#T20" target="_self" title="NAT_3:th.20">Th20</a></span>; <a class="txt" 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="E3:101"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_2"/><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> implies  <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="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:101_2"/><span class="lab"><font color="Green" title="E57">A8</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 ; <a class="txt" 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"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font></span><br/>

<a NAME="E3:101_2"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" target="_self" title="NAT_3:def.9">Def9</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_2_1"/>
(  not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font> or  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_2_1_1"/><span class="lab"><font color="Green" title="E59">A10</font></span>: 
 not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" 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"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1;<br/><span class="kw">set </span><font color="Maroon" title="c5">e</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">s</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font></span>)</span>;<br/><div><span class="lab"><font color="Green" title="E60">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:101_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A12</font></span>: 
 not <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:101_2_1_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> or <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_2_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A13</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">m</font> = <font color="Maroon" title="c7">x</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E3:101_2_1_1_1_1"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_2"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="txt" href="#E1:101_2_1_1_1_1"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E4:101_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2_1_1_1"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="txt" href="#E1:101_2_1_1_1_1"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E5:101_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A14</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E6:101_2_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E1:101_2_1_1_1_1"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="nat_3.html#T34" target="_self" title="NAT_3:th.34">Th34</a></span>;<br/>
<a NAME="E7:101_2_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:101_2_1_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_2_1_1_1_1"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>; <a class="txt" 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><span class="kw">hence </span><a NAME="E3:101_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2_1_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a></span>; <a class="txt" 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="E3:101_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A15</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="ref" href="nat_3.html#T34" target="_self" title="NAT_3:th.34">Th34</a></span>;<br/><a NAME="E4:101_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A16</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/><a NAME="E5:101_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E7:101_2_1_1"/><span class="lab"><font color="Green" title="E63">A17</font></span>: 
<font color="Maroon" title="c3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c8">s</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font></span>)</span>, <font color="Maroon" title="c9">t</font> = <font color="Maroon" title="c7">t</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_2_1_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a></span>;<br/><a NAME="E9:101_2_1_1"/><span class="lab"><font color="Green" title="E64">A18</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="newton.html#K12" title="NEWTON:func.12">SetPrimes</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/><a NAME="E10:101_2_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></span>;<br/><a NAME="E11:101_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A19</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E2:101_2_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">s1</font> = <font color="Maroon" title="c8">s</font>, <font color="Maroon" title="c11">t1</font> = <font color="Maroon" title="c9">t</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><a NAME="E13:101_2_1_1"/><span class="lab"><font color="Green" title="E66">A20</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" target="_self" title="NAT_3:def.9">Def9</a></span>;<br/><a NAME="E14:101_2_1_1"/><span class="lab"><font color="Green" title="E67">A21</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c12">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c9">t</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span> or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_2_1_1_2"/><span class="lab"><font color="Green" title="E68">A22</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">m</font> = <font color="Maroon" title="c13">x</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E3:101_2_1_1_2"/><span class="lab"><font color="Green" title="E69">A23</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2_1_1_2"><span class="lab"><font color="Green" title="E68">A22</font></span></a>, <a class="ref" href="nat_3.html#D9" target="_self" title="NAT_3:def.9">Def9</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E70">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:101_2_1_1_2_1"/><span class="lab"><font color="Green" title="E71">A25</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:101_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c9">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="nat_3.html#D8" target="_self" title="NAT_3:def.8">Def8</a></span>;<br/><a NAME="E3:101_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1_2"><span class="lab"><font color="Green" title="E69">A23</font></span></a>, <a class="txt" href="#E1:101_2_1_1_2_1"><span class="lab"><font color="Green" title="E71">A25</font></span></a>, <a class="ref" href="pre_poly.html#D7" title="PRE_POLY:def.7">PRE_POLY:def 7</a></span>;<br/><a NAME="E4:101_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c9">t</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">g</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:101_2_1_1_2_1"/><span class="lab"><font color="Green" title="E72">A26</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E7:101_2_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_2_1_1"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">u</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E9:101_2_1_1_2_1"/><span class="lab"><font color="Green" title="E73">A27</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c9">t</font></span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><font color="Maroon" title="c3">n</font> = 
<font color="Maroon" title="c8">s</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</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" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_2_1_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E6:101_2_1_1_2_1"><span class="lab"><font color="Green" title="E72">A26</font></span></a>, <a class="txt" href="#E9:101_2_1_1_2_1"><span class="lab"><font color="Green" title="E73">A27</font></span></a>, <a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">s</font> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">u</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> <font color="Maroon" title="c16">u</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="newton.html#T6" title="NEWTON:th.6">NEWTON:6</a></span>
;<br/><a NAME="E11:101_2_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E12:101_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_2_1_1"><span class="lab"><font color="Green" title="E62">A16</font></span></a>, <a class="ref" href="nat_3.html#D7" target="_self" title="NAT_3:def.7">Def7</a></span>; <a class="txt" 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="E5:101_2_1_1_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E7:101_2_1_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E13:101_2_1_1"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="ref" href="nat_3.html#T45" target="_self" title="NAT_3:th.45">Th45</a></span>;<br/>
<a NAME="E6:101_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2_1_1_2"><span class="lab"><font color="Green" title="E68">A22</font></span></a></span>;<br/>
<a NAME="E7:101_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_2"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<a NAME="E8:101_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_2_1_1_2"><span class="lab"><font color="Green" title="E70">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E9:101_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A28</font></span>: 
<font color="Maroon" title="c14">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E10:101_2_1_1_2"/>
<font color="Maroon" title="c13">x</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1_2"><span class="lab"><font color="Green" title="E69">A23</font></span></a>, <a class="ref" href="nat_3.html#T34" target="_self" title="NAT_3:th.34">Th34</a></span>;<br/>
<a NAME="E11:101_2_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:101_2_1_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:101_2_1_1_2"><span class="lab"><font color="Green" title="E71">A28</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>; <a class="txt" 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="E15:101_2_1_1"/>
<font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c11">t1</font> <a href="int_2.html#R1" title="INT_2:pred.1">are_relative_prime</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c12">u</font> = <font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font>;<br/>
<a NAME="E1:101_2_1_1_3"/><span class="lab"><font color="Green" title="E68">A29</font></span>: 
<font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E2:101_2_1_1_3"/>
<font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T5" title="INT_2:th.5">INT_2:5</a></span>;<br/>
<a NAME="E3:101_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A30</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:101_2_1_1_3"/>
<font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="int_2.html#D3" title="INT_2:def.3">INT_2:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:101_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1_3"><span class="lab"><font color="Green" title="E69">A30</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E6:101_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">r</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:101_2_1_1_3"/><span class="lab"><font color="Green" title="E70">A31</font></span>: 
<font color="Maroon" title="c13">r</font> is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E9:101_2_1_1_3"/><span class="lab"><font color="Green" title="E71">A32</font></span>: 
<font color="Maroon" title="c13">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#T31" title="INT_2:th.31">INT_2:31</a></span>;<br/>
<a NAME="E10:101_2_1_1_3"/>
<font color="Maroon" title="c10">s1</font> <a href="int_2.html#K3" title="INT_2:func.3">gcd</a> <font color="Maroon" title="c11">t1</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#D5" title="NAT_D:def.5">NAT_D:def 5</a></span>;<br/>
<a NAME="E11:101_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">s1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:101_2_1_1_3"><span class="lab"><font color="Green" title="E71">A32</font></span></a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<a NAME="E12:101_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">r</font> <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_2_1_1_3"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="ref" href="nat_3.html#T5" target="_self" title="NAT_3:th.5">Th5</a></span>;<br/>
<a NAME="E13:101_2_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c13">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<a NAME="E14:101_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c11">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_2_1_1_3"><span class="lab"><font color="Green" title="E70">A31</font></span></a>, <a class="txt" href="#E9:101_2_1_1_3"><span class="lab"><font color="Green" title="E71">A32</font></span></a>, <a class="txt" href="#E1:101_2_1_1_3"><span class="lab"><font color="Green" title="E68">A29</font></span></a>, <a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a>, <a class="ref" href="nat_d.html#T4" title="NAT_D:th.4">NAT_D:4</a></span>;<br/>
<a NAME="E15:101_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="ref" href="nat_3.html#T37" target="_self" title="NAT_3:th.37">Th37</a></span>;<br/>
<a NAME="E16:101_2_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:101_2_1_1"><span class="lab"><font color="Green" title="E66">A20</font></span></a>, <a class="txt" href="#E14:101_2_1_1"><span class="lab"><font color="Green" title="E67">A21</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E17:101_2_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" 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="E16:101_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A33</font></span>: 
 <a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c3">n</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="c8">s</font></span>)</span> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c9">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_2_1_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="ref" href="nat_3.html#T58" target="_self" title="NAT_3:th.58">Th58</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">f</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="E18:101_2_1_1"/><span class="lab"><font color="Green" title="E69">A34</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="c8">s</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="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E19:101_2_1_1"/><span class="lab"><font color="Green" title="E70">A35</font></span>: 
<font color="Maroon" title="c12">f</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="c8">s</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" title="UPROOTS:func.1">canFS</a> <span class="p2">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c8">s</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D5" target="_self" title="NAT_3:def.5">Def5</a></span>;<br/><a NAME="E20:101_2_1_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c8">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" target="_self" title="NAT_3:def.9">Def9</a></span>;<br/><a NAME="E21:101_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A36</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c8">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E11:101_2_1_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="ref" href="nat_3.html#T42" target="_self" title="NAT_3:th.42">Th42</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c12">f</font> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:101_2_1_1"><span class="lab"><font color="Green" title="E70">A35</font></span></a>, <a class="ref" href="uproots.html#T4" title="UPROOTS:th.4">UPROOTS:4</a></span>
<br/>.= 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_2_1_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E9:101_2_1_1"><span class="lab"><font color="Green" title="E64">A18</font></span></a>, <a class="ref" href="finseq_2.html#T34" title="FINSEQ_2:th.34">FINSEQ_2:34</a></span>
;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A37</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="c8">s</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">s</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:101_2_1_1"><span class="lab"><font color="Green" title="E69">A34</font></span></a>, <a class="ref" href="rvsum_1.html#T95" title="RVSUM_1:th.95">RVSUM_1:95</a></span>
<br/>.= 
<font color="Maroon" title="c8">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1"><span class="lab"><font color="Green" title="E61">A15</font></span></a>, <a class="txt" href="#E11:101_2_1_1"><span class="lab"><font color="Green" title="E65">A19</font></span></a>, <a class="ref" href="nat_3.html#T59" target="_self" title="NAT_3:th.59">Th59</a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_2_1_1_6"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:101_2_1_1_6"/>
<span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c9">t</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:101_2_1_1_6"/><span class="lab"><font color="Green" title="E73">A38</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c8">s</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="polynom2.html#K1" title="POLYNOM2:func.1">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c9">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></span>;<br/><a NAME="E4:101_2_1_1_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c8">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1_6"><span class="lab"><font color="Green" title="E73">A38</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E5:101_2_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A39</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:101_2_1_1"><span class="lab"><font color="Green" title="E71">A36</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E6:101_2_1_1_6"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_2_1_1_6"><span class="lab"><font color="Green" title="E73">A38</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E7:101_2_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:101_2_1_1"><span class="lab"><font color="Green" title="E67">A21</font></span></a>, <a class="txt" href="#E5:101_2_1_1_6"><span class="lab"><font color="Green" title="E74">A39</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><span class="kw">hence </span><a NAME="E8:101_2_1_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>; <a class="txt" 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="E25:101_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A40</font></span>: 
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c8">s</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c9">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E26:101_2_1_1"/>
 <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="c9">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="txt" href="#E14:101_2_1_1"><span class="lab"><font color="Green" title="E67">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E27:101_2_1_1"/>
 <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="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_2_1_1"><span class="lab"><font color="Green" title="E63">A17</font></span></a>, <a class="txt" href="#E25:101_2_1_1"><span class="lab"><font color="Green" title="E73">A40</font></span></a>, <a class="txt" href="#E16:101_2_1_1"><span class="lab"><font color="Green" title="E68">A33</font></span></a>, <a class="txt" href="#E23:101_2_1_1"><span class="lab"><font color="Green" title="E72">A37</font></span></a>, <a class="ref" href="nat_3.html#T19" target="_self" title="NAT_3:th.19">Th19</a></span>; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_2_1_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c3">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k</font>
 ; <a class="txt" 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"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:101_2_1_2"/>
 <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="c3">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_2"><span class="lab"><font color="Green" title="E56">A7</font></span></a></span>; <a class="txt" 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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:101"/><span class="lab"><font color="Green" title="E56">A41</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E54">A2</font></span></a>, <a class="txt" href="#E3:101"><span class="lab"><font color="Green" title="E55">A6</font></span></a>);<br/></span>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:101_3"/>
(  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  or  not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_1"><span class="kw">suppose </span></a><a NAME="E1:101_3_1"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" 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"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:101_3_1"/>
 <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="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101"><span class="lab"><font color="Green" title="E54">A2</font></span></a></span>; <a class="txt" 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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_3_2"><span class="kw">suppose </span></a><a NAME="E1:101_3_2"/>
 not  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> 
 ; <a class="txt" 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"><a href="nat_3.html#NK15" title="NAT_3:NK.15">ppf</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">S</font> =  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E3:101_3_2"/>
 <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c2">S</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_3_2_1"/><span class="lab"><font color="Green" title="E57">A42</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom2.html#K1" title="POLYNOM2:func.1">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="c1">n</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c2">S</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">m</font> = <font color="Maroon" title="c3">x</font> as   <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  ;<br/>
<a NAME="E3:101_3_2_1"/>
<font color="Maroon" title="c3">x</font> is   <a href="newton.html#NM1" title="NEWTON:NM.1">Prime</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101"><span class="lab"><font color="Green" title="E53">A1</font></span></a>, <a class="txt" href="#E1:101_3_2_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="nat_3.html#T34" target="_self" title="NAT_3:th.34">Th34</a></span>;<br/>
<a NAME="E4:101_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A43</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_2.html#D4" title="INT_2:def.4">INT_2:def 4</a></span>;<br/>
<a NAME="E5:101_3_2_1"/>
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_3_2_1"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:101_3_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c2">S</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_3_2_1"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>; <a class="txt" 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><span class="kw">hence </span><a NAME="E4:101_3_2"/>
 <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="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101"><span class="lab"><font color="Green" title="E56">A41</font></span></a></span>; <a class="txt" 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></div><span class="kw">end;</span></div>

</div>
