<?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="E1:66_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>set </b><font color="Maroon">f</font> =  <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font>;<br/>
<a NAME="E2:66_1"/>
 <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="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:66_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K14">Sgm</a> <font color="Maroon">X</font> is    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E4:66_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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> holds <br/><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="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_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="E1:66_1_1"/><i><font color="Green">E40</font></i>: 
<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>
 ;<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="66_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="E1:66_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:66_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="E3:66_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="E4:66_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="E3:66_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="E4:66_1_1"/><b>then </b>
 <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="E5:66_1_1"/><b>then </b>
<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/>
<a NAME="E6:66_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> <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="Olive">y'</font>] )
 
;<br/>
<b>hence </b><a NAME="E7:66_1_1"/>
<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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:66_1"/>
 <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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, </i>;<br/>


</div>
