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

<b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/><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>let </b><font color="Maroon">m</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:67_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p'</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p'</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] </span>}</span> 
 ;<br/>

<b>assume </b><a NAME="E2:67_1"/><i><font color="Green">E39</font></i>: 
not <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:67_1"/>
for <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a><br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">m</font> being non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p'</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p'</font> <a href="nat_3.html#K11">|-count</a> <font color="Olive">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Olive">m</font>,<font color="Olive">p'</font>] </span>}</span>  holds <br/> <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Olive">X</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>from </b><i><a class="ref" href="nat_4.html#S1" target="_self">NAT_4:sch 1</a>();<br/></i>
<a NAME="E4:67_1"/><b>then </b>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:67_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<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> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></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> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">f</font> = <font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0;<br/>
<a NAME="E6:67_1"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E7:67_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> holds <br/><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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:67_1_1"/><i><font color="Green">E48</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<b>assume </b><a NAME="E2:67_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:67_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:67_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:67_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
( <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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <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="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">XX</font> =  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:67_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:67_1_1_1"/><b>then </b>
 ex <font color="Olive">y'</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">y'</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">y'</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">y'</font>] )
 <b>by </b><i/>;<br/><a NAME="E3:67_1_1_1"/><b>then </b>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:67_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:67_1_1"/><b>then </b>
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:67_1_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p'</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p'</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a></i>;<br/>
<a NAME="E9:67_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:67_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">p'</font> <a href="wsierp_1.html#K2">|^</a> <span class="p3">(<span class="default"><font color="Olive">p'</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">p'</font> is   <a href="newton.html#NM1">Prime</a> : <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Olive">p'</font>] </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being   <a href="newton.html#NM1">Prime</a><b> such that </b><br/><a NAME="E12:67_1_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">p1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">m</font>,<font color="Maroon">p1</font>] )
 <b>by </b><i/>;<br/>
<a NAME="E13:67_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 &amp; <font color="Maroon">p1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E14:67_1_1"/>
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E15:67_1_1"/><b>then </b>
not <font color="Maroon">p</font> <a href="nat_1.html#R1">divides</a> <font color="Maroon">p1</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_3.html#T6">NAT_3:6</a></i>;<br/>
<a NAME="E16:67_1_1"/><b>then </b>
<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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , <a class="ref" href="nat_3.html#T27">NAT_3:27</a></i>;<br/>
<b>hence </b><a NAME="E17:67_1_1"/>
<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"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:67_1"/>
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="nat_4.html#K1" target="_self">|-count</a> <span class="p4">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</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">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K24">*</a> 0
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>;<br/>
<b>hence </b><a NAME="E9:67_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> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, </i>;<br/>


</div>
