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

<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> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="newton.html#NM1">Prime</a> st <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="wsierp_1.html#K9">Product</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Olive">b<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Olive">b<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:63"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:63_1"/>
( 0 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<a NAME="E2:63_1"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E3:63_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a>, <a class="ref" href="nat_4.html#T50" target="_self">Th50</a></i>;<br/>
<a NAME="E4:63_1"/>
<font color="Maroon">c<sub>2</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/>
<b>hence </b><a NAME="E5:63_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E2:63_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</a>, <a class="txt" href="nat_4.html#E3:63_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_3.html#T21">NAT_3:21</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:63"/><i><font color="Green">E63</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="63_2"><b>proof </b></a><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>assume </b><a NAME="E1:63_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/>

<a NAME="E2:63_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="newton.html#NM1">Prime</a> st <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="wsierp_1.html#K9">Product</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Olive">b<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Olive">b<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:63_2_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>5</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:63_2_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K4">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:63_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
<a NAME="E4:63_2_1"/><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K24">*</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="ref" href="rvsum_1.html#T126">RVSUM_1:126</a>, <a class="txt" href="nat_4.html#E1:63_2_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="nat_4.html#E3:63_2_1"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E5:63_2_1"/><i><font color="Green">E68</font></i>: 
 <a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E4:63_2_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E6:63_2_1"/><i><font color="Green">E69</font></i>: 
<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#E4:63_2_1"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E7:63_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="nat_4.html#E3:63_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E8:63_2_1"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:63_2_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E3:63_2_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</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>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E5:63_2_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nat_4.html#E6:63_2_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="nat_3.html#T28">NAT_3:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</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>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E1:63_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_4.html#E8:63_2_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E5:63_2_1"><i><font color="Green">E68</font></i></a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><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>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#T52" target="_self">Th52</a>, <a class="txt" href="nat_4.html#E6:63_2_1"><i><font color="Green">E69</font></i></a></i>
<br/>.= 
 <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="wsierp_1.html#K4">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="nat_4.html#T51" target="_self">Th51</a></i>
;<br/>
<b>hence </b><a NAME="E10:63_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:63_2_1"><i><font color="Green">E66</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:63_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:63"/><i><font color="Green">E64</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#E1:63"><i><font color="Green">E62</font></i></a>, <a class="txt" href="nat_4.html#E2:63"><i><font color="Green">E63</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E4:63"/><i><font color="Green">E65</font></i>: 
 <a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E5:63"/>
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="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E6:63"/>
<font color="Maroon">c<sub>2</sub></font> <a href="nat_3.html#K10">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K9">Product</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K8">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:63"><i><font color="Green">E64</font></i></a>, <a class="txt" href="nat_4.html#E4:63"><i><font color="Green">E65</font></i></a></i>;<br/>


</div>
