<?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> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:58"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 ;<br/>

<a NAME="E2:58"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font>, <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</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="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Olive">n</font> &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Olive">r</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><font color="Olive">p</font> <a href="nat_3.html#K11">|-count</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Olive">n</font></span>)</span> <a href="newton.html#K9">choose</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><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> , <font color="Maroon">r</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">p</font> be   <a href="newton.html#NM1">Prime</a>;<br/>



<b>assume </b><a NAME="E1:58_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 &amp; <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> &amp; 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> )
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:58_1"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:58_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 <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E5:58_1"/><i><font color="Green">E48</font></i>: 
<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">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></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E6:58_1"/>
<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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:58"/>
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> implies <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">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></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = 0;<br/>
<a NAME="E2:58_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E3:58_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <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="E4:58_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E5:58_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="newton.html#T9">NEWTON:9</a></i>;<br/>
<a NAME="E6:58_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">0 <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E7:58_2"/>
<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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E8:58_2"/>
<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">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></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:58"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> implies <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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_3"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:58_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = 1;<br/>
<a NAME="E3:58_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:58_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E2:58_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E5:58_3"/>
2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="txt" href="nat_4.html#E3:58_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E6:58_3"/><b>then </b><i><font color="Green">E63</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E7:58_3"/>
<font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E8:58_3"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="txt" href="nat_4.html#E6:58_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E9:58_3"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E10:58_3"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>
<b>hence </b><a NAME="E11:58_3"/>
<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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="txt" href="nat_4.html#E1:58_3"><i><font color="Green">E58</font></i></a>, <a class="txt" href="nat_4.html#E3:58_3"><i><font color="Green">E61</font></i></a>, <a class="txt" href="nat_4.html#E4:58_3"><i><font color="Green">E62</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:58"/>
( <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies <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">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></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_4"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:58_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E4:58_4"/><i><font color="Green">E68</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E5:58_4"/><i><font color="Green">E69</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 <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Olive">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
 <b>and </b><br/><a NAME="E6:58_4"/><i><font color="Green">E70</font></i>: 
<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">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></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/>
<a NAME="E7:58_4"/><i><font color="Green">E71</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 <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4_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="E1:58_4_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:58_4_1"/><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> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:58_4_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:58_4_1"/><i><font color="Green">E74</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <span class="p1">(<span class="default"><font color="Olive">m</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:58_4_1_1"/>
<font color="Maroon">p</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/>
<a NAME="E2:58_4_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
;<br/>
<b>hence </b><a NAME="E3:58_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:58_4_1"/>
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <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="E6:58_4_1"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</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="binop_2.html#K12">/</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/>
<a NAME="E7:58_4_1"/><b>then </b><i><font color="Green">E75</font></i>: 
2 <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="binop_2.html#K12">/</a> <font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:58_4_1"/><i><font color="Green">E76</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> 3</span>)</span> <a href="binop_2.html#K11">*</a> 3 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> 3
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E9:58_4_1"/><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="binop_2.html#K12">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default">3 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/>
<a NAME="E10:58_4_1"/><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="binop_2.html#K12">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 3
 
<b>by </b><i/>;<br/>
<a NAME="E11:58_4_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 3 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_4_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:58_4_1_2"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or ( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> ) )
 </span><b>by </b><i><a class="txt" href="nat_4.html#E2:58_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_4_1_2_1"><b>suppose </b></a><a NAME="E1:58_4_1_2_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:58_4_1_2_1"/>
<span class="p1">(<span class="default">1 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T74">XREAL_1:74</a></i>;<br/><a NAME="E3:58_4_1_2_1"/><b>then </b><i><font color="Green">E79</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E4:58_4_1_2_1"/>
3 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/><a NAME="E5:58_4_1_2_1"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> 4 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:58_4"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:58_4_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> 2 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K24">*</a> 2</span>)</span> <a href="binop_2.html#K12">/</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E7:58_4_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</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">p</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E8:58_4_1_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> 2
 <b>by </b><i/>;<br/><a NAME="E9:58_4_1_2_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 2 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E10:58_4_1_2_1"/><i><font color="Green">E81</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a>, <a class="txt" href="nat_4.html#E1:58_4_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="nat_4.html#E3:58_4_1"><i><font color="Green">E73</font></i></a></i>;<br/><font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="nat_4.html#E7:58_4"><i><font color="Green">E71</font></i></a>, , </i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <font color="Maroon">p</font></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>
<br/>.= 
2 <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a>, <a class="txt" href="nat_4.html#E4:58_4"><i><font color="Green">E68</font></i></a>, <a class="txt" href="nat_4.html#E6:58_4"><i><font color="Green">E70</font></i></a>, <a class="txt" href="nat_4.html#E4:58_4_1"><i><font color="Green">E74</font></i></a></i>
;<br/><b>hence </b><a NAME="E12:58_4_1_2_1"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_4_1_2_2"><b>suppose </b></a><a NAME="E1:58_4_1_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> )
 ;<br/><div class="add"><a NAME="E2:58_4_1_2_2"/><b>then </b>
1 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:58_4_1_2_2"/><b>then </b><i><font color="Green">E82</font></i>: 
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:58_4_1_2_2"/>
3 <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <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="E5:58_4_1_2_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> 3 <a href="xxreal_0.html#R1">&lt;</a> 3 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:58_4"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E6:58_4_1_2_2"/><b>then </b>
<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 3</span>)</span> <a href="binop_2.html#K12">/</a> 3 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> 3</span>)</span> <a href="binop_2.html#K12">/</a> 3
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E7:58_4_1_2_2"/><b>then </b>
2 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E8:58_4_1_2_2"/><b>then </b>
3 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:58_4_1_2_2"/><b>then </b>
3 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E10:58_4_1_2_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E5:58_4"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E11:58_4_1_2_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/><a NAME="E12:58_4_1_2_2"/><b>then </b><i><font color="Green">E83</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/><a NAME="E13:58_4_1_2_2"/>
( <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; ( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 2 or 2 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">k</font> ) )
 <b>by </b><i><a class="txt" href="nat_4.html#E7:58_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="xreal_1.html#T1">XREAL_1:1</a>, <a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/><a NAME="E14:58_4_1_2_2"/><b>then </b>
<font color="Maroon">p</font> <a href="power.html#K4">to_power</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="power.html#T44">POWER:44</a></i>;<br/><a NAME="E15:58_4_1_2_2"/><b>then </b>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E16:58_4_1_2_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/><a NAME="E17:58_4_1_2_2"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E1:58_4_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E18:58_4_1_2_2"/><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="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E19:58_4_1_2_2"/><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="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E20:58_4_1_2_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E21:58_4_1_2_2"/>
1 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E22:58_4_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 2
 <b>by </b><i><a class="txt" href="nat_4.html#E1:58_4_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E23:58_4_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="nat_4.html#E3:58_4_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E24:58_4_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T76">XREAL_1:76</a></i>;<br/><a NAME="E25:58_4_1_2_2"/><b>then </b>
<font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 1
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>;<br/><a NAME="E26:58_4_1_2_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binop_2.html#K10">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E27:58_4_1_2_2"/><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a>, <a class="txt" href="nat_4.html#E4:58_4_1"><i><font color="Green">E74</font></i></a></i>;<br/><font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="binop_2.html#K12">/</a> <span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <span class="p2"><a href="int_1.html#K1">[\</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K12">/</a> <span class="p4">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="int_1.html#K1">/]</a></span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
0 <a href="binop_2.html#K10">-</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="int_1.html#D4">INT_1:def 4</a>, <a class="txt" href="nat_4.html#E7:58_4_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="nat_4.html#E8:58_4_1"><i><font color="Green">E76</font></i></a></i>
;<br/><b>hence </b><a NAME="E29:58_4_1_2_2"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0;<br/>
<a NAME="E8:58_4"/><i><font color="Green">E88</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</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="finseq_2.html#T68">FINSEQ_2:68</a></i>;<br/>
<a NAME="E9:58_4"/><i><font color="Green">E90</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</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="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E10:58_4"/>
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> <font color="Maroon">f</font> holds <br/><font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></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="58_4_2"><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="E1:58_4_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:58_4_2"/><b>then </b>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="nat_4.html#E1:58_4"><i><font color="Green">E64</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:58_4_2"/>
<font color="Maroon">f</font> <a href="wsierp_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></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="txt" href="nat_4.html#E2:58_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:58_4"/><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="finseqop.html#K4">|-&gt;</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a>, , <a class="txt" href="nat_4.html#E2:58_4"><i><font color="Green">E66</font></i></a></i>;<br/>
<a NAME="E12:58_4"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f</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="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="E13:58_4"/>
<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">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></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:58"/>
(  <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#R1">&lt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</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="binop_2.html#K12">/</a> 3 implies <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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_5"/><i><font color="Green">E92</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#R1">&lt;</a> <font color="Maroon">p</font>
 ;<br/>

<b>assume </b><a NAME="E2:58_5"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">p</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="binop_2.html#K12">/</a> 3
 ;<br/>

<b>set </b><font color="Maroon">r</font> = 1;<br/>
<a NAME="E3:58_5"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E4:58_5"/><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default">1 <a href="binop_2.html#K12">/</a> 3</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> <a href="xxreal_0.html#R1">&lt;=</a> 1 <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>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E5:58_5"/>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1 <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="binop_2.html#K12">/</a> 3
 
<b>by </b><i>, <a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E6:58_5"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E7:58_5"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">n</font> <a href="binop_2.html#K24">*</a> 2 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2 <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="E8:58_5"/>
 <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#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, <a class="ref" href="square_1.html#T93">SQUARE_1:93</a></i>;<br/>
<a NAME="E9:58_5"/><b>then </b><i><font color="Green">E99</font></i>: 
<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> <a href="binop_2.html#K11">*</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> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K11">*</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#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E10:58_5"/>
<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> <a href="binop_2.html#K11">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E11:58_5"/><b>then </b>
<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> <a href="binop_2.html#K11">*</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> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E12:58_5"/><b>then </b>
<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> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/>
<a NAME="E13:58_5"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E14:58_5"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> 1</span>)</span> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="newton.html#T10">NEWTON:10</a></i>;<br/>
<a NAME="E15:58_5"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default">1 <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="newton.html#T11">NEWTON:11</a></i>;<br/>
<b>hence </b><a NAME="E16:58_5"/>
<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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E7:58"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">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>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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>;<br/>
<a NAME="E8:58"/><i><font color="Green">E101</font></i>: 
<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="int_2.html#D5">INT_2:def 5</a></i>;<br/>
<a NAME="E9:58"/><b>then </b><i><font color="Green">E102</font></i>: 
<font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E10:58"/><i><font color="Green">E103</font></i>: 
( <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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="power.html#K6">log</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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#K10">-</a> 1 <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="power.html#K6">log</a> <font color="Maroon">p</font>,<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><a class="ref" href="int_1.html#D4">INT_1:def 4</a></i>;<br/>
<a NAME="E11:58"/><i><font color="Green">E104</font></i>: 
<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> <a href="binop_2.html#K11">*</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E12:58"/>
<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> <a href="binop_2.html#K11">*</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="binop_2.html#K11">*</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#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E13:58"/><b>then </b>
<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> <a href="binop_2.html#K11">*</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E14:58"/><b>then </b>
<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> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="square_1.html#D3">SQUARE_1:def 3</a></i>;<br/>
<a NAME="E15:58"/><b>then </b><i><font color="Green">E105</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> <font color="Maroon">p</font> <a href="binop_2.html#K24">*</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E16:58"/><b>then </b>
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E17:58"/><b>then </b>
 <a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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#NR4" target="_self">&gt;</a>  <a href="power.html#K6">log</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="power.html#T65">POWER:65</a>, </i>;<br/>
<a NAME="E18:58"/><b>then </b>
 <a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i><a class="ref" href="power.html#T60">POWER:60</a>, </i>;<br/>
<a NAME="E19:58"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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#K10">-</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="binop_2.html#K10">-</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <span class="p1"><a href="int_1.html#K1">[\</a><span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i>, , <a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E21:58"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="power.html#K6">log</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default">2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font></span>)</span> or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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="E22:58"/><b>then </b>
<font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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/>
<a NAME="E23:58"/><b>then </b>
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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#T48">POWER:48</a></i>;<br/>
<a NAME="E24:58"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="power.html#D3">POWER:def 3</a>, , </i>;<br/>
<a NAME="E25:58"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<span class="p3">(<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#K10">-</a> 1</span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E26:58"/><b>then </b>
<font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<a NAME="E27:58"/><b>then </b>
<font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><a href="power.html#K6">log</a> <font color="Maroon">p</font>,<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="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/>
<a NAME="E28:58"/><b>then </b><i><font color="Green">E108</font></i>: 
2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="power.html#D3">POWER:def 3</a>, </i>;<br/>
<b>set </b><font color="Maroon">k0</font> = <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">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></span>)</span>;<br/>
<a NAME="E29:58"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E30:58"/><b>then </b>
<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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E31:58"/><b>then </b>
( <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">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></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <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">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></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T1">XREAL_1:1</a></i>;<br/>
<a NAME="E32:58"/><b>then </b>
<font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <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"><span class="p3">(<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></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="power.html#T44">POWER:44</a></i>;<br/>
<a NAME="E33:58"/><b>then </b>
<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> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="power.html#K4">to_power</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/>
<a NAME="E34:58"/><b>then </b>
<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> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="wsierp_1.html#K2">|^</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="power.html#T48">POWER:48</a></i>;<br/>
<b>hence </b><a NAME="E35:58"/>
<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> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="binop_2.html#K24">*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>


</div>
