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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">z1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">z2</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">n</font>;<br/>





<a NAME="E1:64"/><i><font color="Green">E59</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/>
<a NAME="E2:64"/><b>then </b><i><font color="Green">E60</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<i><font color="Green">E102</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T98">RVSUM_1:98</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K10">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T119">RVSUM_1:119</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
;<br/>
<a NAME="E4:64"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:64_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">r2</font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/>
<b>reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">c2</font> = <font color="Maroon">z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">c12</font> = <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">abs1</font> = <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">abs2</font> = <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">abs12</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">abs'12</font> = <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, </i>;<br/>
<a NAME="E7:64_2"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c1</font> <a href="binop_2.html#K3">+</a> <font color="Maroon">c2</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c1</font></span><a href="complex1.html#K17">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c2</font></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T142">COMPLEX1:142</a></i>;<br/>
<a NAME="E8:64_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c12</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c1</font></span><a href="complex1.html#K17">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c2</font></span><a href="complex1.html#K17">.|</a></span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, </i>;<br/>
<a NAME="E9:64_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c12</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c1</font></span><a href="complex1.html#K17">.|</a></span> <a href="binop_2.html#K9">+</a> <font color="Maroon">abs2</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, <a class="txt" href="complsp1.html#E23"><i><font color="Green">Lemma43</font></i></a></i>;<br/>
<a NAME="E10:64_2"/><b>then </b><i><font color="Green">E109</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c12</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">abs1</font> <a href="binop_2.html#K9">+</a> <font color="Maroon">abs2</font>
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, <a class="txt" href="complsp1.html#E23"><i><font color="Green">Lemma43</font></i></a></i>;<br/>
<a NAME="E11:64_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E12:64_2"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:64_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c12</font></span><a href="complex1.html#K17">.|</a></span> &amp; <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c12</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">abs12</font> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, <a class="ref" href="complsp1.html#T23" target="_self">Th23</a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="rvsum_1.html#T26">RVSUM_1:26</a></i>;<br/>
<a NAME="E14:64_2"/><b>then </b>
( <font color="Maroon">abs'12</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">abs12</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">abs'12</font> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, <a class="txt" href="complsp1.html#E23"><i><font color="Green">Lemma43</font></i></a></i>;<br/>
<a NAME="E15:64_2"/><b>then </b>
<font color="Maroon">abs'12</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">abs12</font> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="square_1.html#T77">SQUARE_1:77</a></i>;<br/>
<a NAME="E16:64_2"/><b>then </b>
<font color="Maroon">abs'12</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>
<b>hence </b><a NAME="E17:64_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="rvsum_1.html#K4">+</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T79">RVSUM_1:79</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:64"/><b>then </b><i><font color="Green">E110</font></i>: 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="complsp1.html#E34"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="rvsum_1.html#T112">RVSUM_1:112</a></i>;<br/>
<a NAME="E6:64"/><i><font color="Green">E111</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">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="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/>0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:64_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>

<b>set </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>;<br/>
<b>reconsider </b><font color="Maroon">c1</font> = <font color="Maroon">z1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>, <font color="Maroon">c2</font> = <font color="Maroon">z2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complsp1.html#T24" target="_self">Th24</a>, </i>;<br/>
<a NAME="E3:64_3"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c1</font></span><a href="complex1.html#K17">.|</a></span> &amp; <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c2</font></span><a href="complex1.html#K17">.|</a></span> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T24" target="_self">Th24</a>, <a class="txt" href="complsp1.html#E23"><i><font color="Green">Lemma43</font></i></a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E4:64_3"/><b>then </b>
( <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c1</font></span><a href="complex1.html#K17">.|</a></span> <a href="binop_2.html#K11">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c2</font></span><a href="complex1.html#K17">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c1</font></span><a href="complex1.html#K17">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c2</font></span><a href="complex1.html#K17">.|</a></span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a>, <a class="ref" href="rvsum_1.html#T87">RVSUM_1:87</a></i>;<br/>
<a NAME="E5:64_3"/><b>then </b>
0 <a href="binop_2.html#K11">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c2</font></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<b>hence </b><a NAME="E6:64_3"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:64"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<a NAME="E8:64"/><b>then </b>
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="rvsum_1.html#T122">RVSUM_1:122</a>, <a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E9:64"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T23" target="_self">Th23</a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E10:64"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp;  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E11:64"/><b>then </b>
(  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span> &amp; 2 <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2 <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"><span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/>
<a NAME="E12:64"/><b>then </b>
(  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p0">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p0">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E13:64"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K14">mlt</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E14:64"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:64"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T97">SQUARE_1:97</a></i>;<br/>
<a NAME="E16:64"/><b>then </b>
(  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  &amp; <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span> &amp;  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="binop_2.html#K11">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p0">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="square_1.html#D4">SQUARE_1:def 4</a></i>;<br/>
<a NAME="E17:64"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p5">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a>  )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/>
<a NAME="E18:64"/><b>then </b>
( 0 <a href="binop_2.html#K23">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p0">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p0">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T18" target="_self">Th18</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="ref" href="square_1.html#T94">SQUARE_1:94</a></i>;<br/>
<a NAME="E19:64"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<a NAME="E20:64"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p4">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z2</font></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 
;<br/>
<a NAME="E21:64"/><b>then </b>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <span class="p4">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z1</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z2</font></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 
;<br/>
<b>hence </b><a NAME="E22:64"/>
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="complsp1.html#K9" target="_self">+</a> <font color="Maroon">z2</font></span>)</span></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z1</font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="binop_2.html#K9">+</a> <span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">z2</font></span><a href="complsp1.html#K16" target="_self">.|</a></span>
 ;<br/>


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