<?xml version="1.0"?>
<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> , <font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Maroon">n</font>
 ;<br/>

<b>assume </b><a NAME="E2:71"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 ;<br/>

<a NAME="E3:71"/><b>then </b><i><font color="Green">E40</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="binop_2.html#K24">*</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E4:71"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default">4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:71"/><b>then </b>
<font color="Maroon">m</font> <a href="binop_2.html#K24">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">4 <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E6:71"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> = <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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">f1</font> =  <a href="finseq_1.html#K14">Sgm</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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">f2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E7:71"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </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"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></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="71_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="E1:71_1"/>
<font color="Maroon">x</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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 ;<br/><a NAME="E2:71_1"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">p</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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E3:71_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>, </i>;<br/><b>hence </b><a NAME="E4:71_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="E9:71"/><b>then </b><i><font color="Green">E48</font></i>: 
<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  <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/>
<div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="71_2"><b>now </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="E1:71_2"/><i><font color="Green">E54</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> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span>
 ;<br/><a NAME="E2:71_2"/><b>then </b>
<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"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </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="E3:71_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<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"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/><a NAME="E4:71_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D13">FINSEQ_1:def 13</a>, </i>;<br/><a NAME="E5:71_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:71_2"/><b>then </b>
 ex <font color="Olive">p</font> being  <a href="newton.html#NM1">Prime</a> st <br/>( <font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">p</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> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><b>hence </b><a NAME="E7:71_2"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</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"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="wsierp_1.html#K3">.</a> <font color="Maroon">k</font> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </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 )
 <b>by </b><i>, , , <a class="txt" href="nat_4.html#E11:71"><i><font color="Green">E56</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:71"/><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> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </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="E12:71"/><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K5">rng</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"><a href="finseq_1.html#K14">Sgm</a> <span class="p4">{<span class="default"> <span class="p5">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p0">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></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/>
<b>reconsider </b><font color="Maroon">f1</font> =  <a href="finseq_1.html#K14">Sgm</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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  as    <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/>
<b>reconsider </b><font color="Maroon">f2</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p3">{<span class="default"> <span class="p4">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p5">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> as    <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="E15:71"/>
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f2</font>
 
<b>by </b><i>, , <a class="txt" href="nat_4.html#E20:71"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E16:71"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="rvsum_1.html#K16">Product</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="nat_4.html#E21:71"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E17:71"/>
<font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3 <a href="binop_2.html#K24">*</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E18:71"/><b>then </b><i><font color="Green">E61</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<b>reconsider </b><font color="Maroon">X</font> = <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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span>  as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, </i>;<br/>
<a NAME="E20:71"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T44">FINSEQ_3:44</a>, </i>;<br/>
<a NAME="E21:71"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<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> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">r</font> =  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>;<br/>
<a NAME="E22:71"/><i><font color="Green">E64</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E23:71"/>
for <font color="Olive">n</font>, <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a><br/> for <font color="Olive">X</font> being <a href="finset_1.html#V1">finite</a>  <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"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>,<font color="Olive">m</font>) where <font color="Olive">k</font> is   <a href="newton.html#NM1">Prime</a> : ( <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">m</font>] ) </span>}</span>  &amp; <font color="Olive">r</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 holds <br/> <a href="card_1.html#K4">card</a> <font color="Olive">X</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><font color="Olive">r</font></span><a href="int_1.html#K1">/]</a></span>
 
<b>from </b><i><a class="ref" href="nat_4.html#S4" target="_self">NAT_4:sch 4</a>();<br/></i>
<a NAME="E24:71"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E25:71"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, </i>;<br/>
<a NAME="E26:71"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<a NAME="E27:71"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K3">to_power</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<b>hence </b><a NAME="E28:71"/>
 <a href="wsierp_1.html#K8">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K14">Sgm</a> <span class="p2">{<span class="default"> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p4">(<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="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) </span>}</span> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


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