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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">f</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</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">f</font> &amp;  <a href="wsierp_1.html#K8">Product</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Olive">f</font></span>)</span>;<br/>
<a NAME="E1:63"/><i><font color="Green">E38</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">f</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">p</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">f</font> &amp;  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</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">E39</font></i>: 
<font color="Maroon">f</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">E40</font></i>: 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f</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>, </i>;<br/>
<a NAME="E4:63_1"/>
<font color="Maroon">p</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">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rvsum_1.html#T124">RVSUM_1:124</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">E46</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:63_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<a NAME="E2:63_2"/>
for <font color="Olive">f</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a>  st <font color="Maroon">n</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">f</font> &amp;  <a href="wsierp_1.html#K8">Product</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Olive">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Olive">f</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">f</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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:63_2_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">n</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">f</font> &amp;  <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/>

<b>consider </b><font color="Maroon">g</font> being    <a href="trees_4.html#M1">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">d</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:63_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T22">FINSEQ_2:22</a></i>;<br/>
<a NAME="E4:63_2_1"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">d</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>, , </i>;<br/>
<a NAME="E5:63_2_1"/><i><font color="Green">E55</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E6:63_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E7:63_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</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">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i>, <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">E58</font></i>: 
<font color="Maroon">n</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">g</font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> = 
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">g</font></span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T126">RVSUM_1:126</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p2">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, , <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#K7">Sum</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">d</font></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">g</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">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">d</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#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">g</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><font color="Maroon">p</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">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E3:63"><i><font color="Green">E60</font></i></a>, </i>
<br/>.= 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p2">(<span class="default"><font color="Maroon">g</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">d</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E10:63_2_1"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><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">n</font> <a href="binop_2.html#K23">+</a> 1]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:63"/><i><font color="Green">E60</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">f</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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E4:63"/><i><font color="Green">E61</font></i>: 
 <a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E5:63"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E6:63"/>
<font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><a href="wsierp_1.html#K8">Product</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div>
