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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:24"/><i><font color="Green">E18</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> or  not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/><i><font color="Green">E19</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>4</sub></font> = 0;<br/><a NAME="E2:24_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1"/>
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> ) ) ) )
 <b>by </b><i><a class="txt" href="moebius1.html#E1:24_1_1"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/><i><font color="Green">E19</font></i>: 
not  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_1"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</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#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:24_1_2_1"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:24_1_2_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:24_1_2_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/>
<a NAME="E3:24_1_2_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i><a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="moebius1.html#E2:24_1_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E5:24_1_2_1_1"/><i><font color="Green">E25</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a>, <a class="txt" href="moebius1.html#E3:24_1_2_1_1"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E6:24_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:24"><i><font color="Green">E18</font></i></a>, <a class="txt" href="moebius1.html#E1:24_1_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:24_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:24_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="moebius1.html#E1:24_1_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E8:24_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E9:24_1_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:24_1_2_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:24_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:24_1_2"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 as  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><b>set </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/><a NAME="E4:24_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a>, <a class="txt" href="moebius1.html#E2:24_1_2"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E5:24_1_2"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is   <a href="newton.html#NM1">Prime</a>
 <b>by </b><i><a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/><a NAME="E6:24_1_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>by </b><i><a class="txt" href="moebius1.html#E5:24_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E7:24_1_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:24_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">zero</a> <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="moebius1.html#E9:24_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/><a NAME="E11:24_1_2"/><i><font color="Green">E24</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/><a NAME="E12:24_1_2"/><i><font color="Green">E25</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:24_1_2_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:24_1_2_2"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/>
<a NAME="E3:24_1_2_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is   <a href="newton.html#NM1">Prime</a>
 
<b>by </b><i><a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font> as  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="moebius1.html#E2:24_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E5:24_1_2_2"/><i><font color="Green">E29</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E3:24_1_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E6:24_1_2_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E9:24_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="nat_3.html#T45">NAT_3:45</a></i>;<br/>
<a NAME="E7:24_1_2_2"/><b>then </b>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E11:24_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/>
<a NAME="E8:24_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:24_1_2_2"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:24_1_2_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="moebius1.html#E1:24"><i><font color="Green">E18</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_2_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:24_1_2_2_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:24_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/><a NAME="E3:24_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="txt" href="moebius1.html#E2:24_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="moebius1.html#E1:24_1_2_2_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><a NAME="E4:24_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E5:24_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:24_1_2_2_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E8:24_1_2_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E6:24_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:24_1_2_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><font color="Maroon">c<sub>2</sub></font> = 
<font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="moebius1.html#E9:24_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="moebius1.html#E7:24_1_2_2_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="moebius1.html#E10:24_1_2_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="newton.html#T11">NEWTON:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>13</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/><a NAME="E12:24_1_2_2_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/><b>hence </b><a NAME="E13:24_1_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E6:24_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:24_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="moebius1.html#E9:24_1_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:24_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>hence </b><a NAME="E13:24_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:24_1_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:24_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_2_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:24_1_2_3"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a>, <a class="txt" href="moebius1.html#E9:24_1_2"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:24_1_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="moebius1.html#E1:24_1_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="moebius1.html#E12:24_1_2"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:24_1_2"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#D9">NAT_3:def 9</a></i>;<br/><a NAME="E15:24_1_2"/><b>then </b><i><font color="Green">E27</font></i>: 
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="moebius1.html#E5:24_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="moebius1.html#E13:24_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_3.html#T42">NAT_3:42</a></i>;<br/><div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:24_1_2_4"/>
 <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="xboole_0.html#R1">meets</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:24_1_2_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:24_1_2_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E5:24_1_2_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E15:24_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="moebius1.html#E3:24_1_2_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:24_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E12:24_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="moebius1.html#E4:24_1_2_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E7:24_1_2_4"/>
contradiction
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/></div><b>end;</b></div><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="E17:24_1_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:24_1_2_5_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E2:24_1_2_5_1"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="moebius1.html#E1:24_1_2_5_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a>, <a class="txt" href="moebius1.html#E16:24_1_2"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E3:24_1_2_5_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_3.html#T28">NAT_3:28</a>, <a class="txt" href="moebius1.html#E9:24_1_2"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E4:24_1_2_5_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="moebius1.html#E2:24_1_2_5_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="moebius1.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E5:24_1_2_5_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="moebius1.html#E3:24_1_2_5_1"><i><font color="Green">E31</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:24_1_2_5"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/>

<b>hence </b><a NAME="E3:24_1_2_5"/>
<font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="moebius1.html#T15" target="_self">Th15</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E18:24_1_2"/>
(  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="newton.html#NM1">Prime</a> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> ) &amp; ( not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom2.html#K1">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK14" target="_self">ppf</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> implies <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>2</sub></font> ) ) ) )
 <b>by </b><i><a class="txt" href="moebius1.html#E12:24_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="moebius1.html#E9:24_1_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
