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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:21"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>1</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>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>;<br/>
<a NAME="E2:21"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E24</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:21_1"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><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>4</sub></font> ) implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><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>4</sub></font>
 ;<br/>

<a NAME="E2:21_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:21_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/>
<a NAME="E3:21_1_1"/>
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1_1"/>
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:21_1_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="txt" href="nat_4.html#E3:21_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<a NAME="E5:21_1_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:21_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E6:21_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/>
<a NAME="E7:21_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
;<br/>
<a NAME="E8:21_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:21_1_1"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E9:21_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/>
<b>hence </b><a NAME="E10:21_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E2:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:21_1_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="nat_3.html#T52">NAT_3:52</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = 1;<br/>
<b>take </b>
1
;<br/>

<b>thus </b><a NAME="E5:21_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E2:21_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_3.html#T52">NAT_3:52</a>, <a class="txt" href="nat_4.html#E4:21_1_1"><i><font color="Green">E27</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
( ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> &amp; not <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>4</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:21"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="newton.html#NM1">Prime</a> holds <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is  not  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> 
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_1"><b>suppose </b></a><a NAME="E1:21_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:21_2_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E3:21_2_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>hence </b><a NAME="E4:21_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E1:21_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E2:21_2_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="nat_2.html#T6">NAT_2:6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_2_1_2"><b>suppose </b></a><a NAME="E1:21_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:21_2_1_2"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 <b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>;<br/><a NAME="E3:21_2_1_2"/><i><font color="Green">E27</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 0</span>)</span> <a href="binop_2.html#K23">+</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="nat_4.html#E1:21_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="nat_1.html#D1">NAT_1:def 1</a></i>;<br/><a NAME="E4:21_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a>, <a class="txt" href="nat_4.html#E2:21_2_1_2"><i><font color="Green">E26</font></i></a></i>;<br/><b>hence </b><a NAME="E5:21_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E3:21_2_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T54">NAT_1:54</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:21"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K3">div</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:21_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:21_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</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="E3:21_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K3">div</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T68">NAT_1:68</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:21"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="newton.html#NM1">Prime</a><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Olive">b<sub>1</sub></font></span>)</span> holds <br/> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:21_4"/><i><font color="Green">E27</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:21_4"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:21_4"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E4:21_4"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E5:21_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/>
<a NAME="E6:21_4"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E2:21_4"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E7:21_4"/>
<span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="polynom1.html#K8">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:21_4"><i><font color="Green">E29</font></i></a>, <a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/>
<a NAME="E8:21_4"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="nat_3.html#D8">NAT_3:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E10:21_4"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_4_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_4_1"/>
not <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 ;<br/>

<a NAME="E2:21_4_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K8">hcf</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="pepin.html#T2">PEPIN:2</a></i>;<br/>
<a NAME="E3:21_4_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#D5">NAT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:21_4_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>
<a NAME="E6:21_4_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>5</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>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:21_4"><i><font color="Green">E30</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E4:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E2:21_4"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nat_4.html#E5:21_4_1"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<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>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>

<br/>.= 
<font color="Maroon">c<sub>9</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</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>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>
;<br/>
<b>hence </b><a NAME="E8:21_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="nat_4.html#E6:21_4_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#D3">NAT_1:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:21_4"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font> <a href="int_2.html#R2">are_relative_prime</a> 
 
<b>by </b><i><a class="ref" href="euler_2.html#T32">EULER_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/>
<a NAME="E13:21_4"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E11:21_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nat_3.html#T47">NAT_3:47</a></i>;<br/>
<a NAME="E14:21_4"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E6:21_4"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_4.html#E1:21_4"><i><font color="Green">E27</font></i></a>, <a class="txt" href="nat_4.html#E8:21_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_3.html#T42">NAT_3:42</a></i>;<br/>
<a NAME="E15:21_4"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/>
<b>hence </b><a NAME="E16:21_4"/>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:21"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="newton.html#NM1">Prime</a> st ( for <font color="Olive">b<sub>6</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>6</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">b<sub>6</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>6</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span>)</span> holds <br/>for <font color="Olive">b<sub>6</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>6</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">b<sub>6</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>6</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>4</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:21_5"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><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>5</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:21_5"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:21_5"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E4:21_5"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="int_2.html#V1">prime</a>
 ;<br/>

<a NAME="E5:21_5"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<a NAME="E6:21_5"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E7:21_5"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/>
<a NAME="E8:21_5"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:21_5"><i><font color="Green">E29</font></i></a>, <a class="txt" href="nat_4.html#E7:21_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nat_4.html#E4:21_5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_3.html#T31">NAT_3:31</a></i>;<br/>
<a NAME="E9:21_5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E6:21_5"><i><font color="Green">E33</font></i></a>, <a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/>
<a NAME="E10:21_5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:21_5"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nat_4.html#E9:21_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nat_4.html#E4:21_5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_3.html#T31">NAT_3:31</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_5_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_5_1_1"><b>suppose </b></a><a NAME="E1:21_5_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:21_5_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:21_5_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="nat_4.html#E5:21_5"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_3.html#T25">NAT_3:25</a></i>;<br/><b>thus </b><a NAME="E3:21_5_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E8:21_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_2.html#T10">NAT_2:10</a>, <a class="txt" href="nat_4.html#E2:21_5_1_1"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_5_1_2"><b>suppose </b></a><a NAME="E1:21_5_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E4:21_5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_3.html#T32">NAT_3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> 0
<b>by </b><i><a class="txt" href="nat_4.html#E5:21_5"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_4.html#E1:21_5_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_3.html#T24">NAT_3:24</a></i>
<br/>.= 
0

;<br/><b>then </b><i><font color="Green">E39</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 0
<b>by </b><i><a class="txt" href="nat_4.html#E8:21_5"><i><font color="Green">E35</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>3</sub></font>

;<br/><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E4:21_5"><i><font color="Green">E31</font></i></a>, <a class="ref" href="nat_3.html#T32">NAT_3:32</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> 0
<b>by </b><i><a class="txt" href="nat_4.html#E5:21_5"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_4.html#E1:21_5_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_3.html#T24">NAT_3:24</a></i>
<br/>.= 
0

;<br/><b>then </b><i><font color="Green">E40</font></i>: <font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> 0
<b>by </b><i><a class="txt" href="nat_4.html#E10:21_5"><i><font color="Green">E37</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font>

;<br/><b>thus </b><a NAME="E6:21_5_1_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:21_5"><i><font color="Green">E28</font></i></a>, <a class="txt" href="nat_4.html#E4:21_5"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_4.html#E3:21_5_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="nat_4.html#E5:21_5_1_2"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E7:21"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:21_6"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<a NAME="E2:21_6"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1 &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/>ex <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:21_6_1"/><i><font color="Green">E30</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1
 ;<br/>

<b>assume </b><a NAME="E2:21_6_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</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>5</sub></font>
 ;<br/>

<a NAME="E3:21_6_1"/>
 <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:21_6_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="card_4.html#T17">CARD_4:17</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21_6_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="newton.html#NM1">Prime</a> <b>by </b><i><a class="txt" href="nat_4.html#E5:21_6_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="nat_3.html#T34">NAT_3:34</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K3">div</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  <b>by </b><i><a class="txt" href="nat_4.html#E3:21"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E8:21_6_1"/><i><font color="Green">E33</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:21_6_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="nat_4.html#E5:21_6_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="nat_4.html#E5:21"><i><font color="Green">E26</font></i></a></i>;<br/>
<a NAME="E9:21_6_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> is <a href="int_2.html#V1">prime</a> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>10</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>11</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:21_6_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="nat_4.html#E6:21"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>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="E11:21_6_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E8:21_6_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="nat_4.html#E9:21_6_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="nat_4.html#E1:21_6"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E12:21_6_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:21_6_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E13:21_6_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E14:21_6_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E15:21_6_1"/><b>then </b><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="nat_1.html#R1">divides</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_3.html#D7">NAT_3:def 7</a></i>;<br/>
<i><font color="Green">E38</font></i>: <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K3">div</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E13:21_6_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="nat_4.html#E4:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E15:21_6_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="nat_4.html#E11:21_6_1"><i><font color="Green">E35</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K3">div</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></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>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T13">NEWTON:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K3">div</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E4:21"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nat_4.html#E15:21_6_1"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>;<br/>
<b>take </b>
<font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</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>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span>
;<br/>

<b>thus </b><a NAME="E17:21_6_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binop_2.html#K24">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_3.html#K10">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E16:21_6_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_6"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:21"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="nat_4.html#E2:21"><i><font color="Green">E23</font></i></a>, <a class="txt" href="nat_4.html#E7:21"><i><font color="Green">E28</font></i></a>);<br/></i>
<a NAME="E9:21"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#NK12" target="_self">pfexp</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E10:21"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:21"><i><font color="Green">E22</font></i></a>, <a class="txt" href="nat_4.html#E8:21"><i><font color="Green">E29</font></i></a></i>;<br/>


</div>
