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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font> holds  <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Olive" title="b2">z2</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Olive" title="b1">z1</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Olive" title="b2">z2</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="seq_4.html#K14" title="SEQ_4:func.14">COMPLEX</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c2">z1</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z2</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span></span><br/>





<a NAME="E1:112"/><span class="lab"><font color="Green" title="E63">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T86" title="RVSUM_1:th.86">RVSUM_1:86</a></span>;<br/>
<a NAME="E2:112"/><span class="lab"><font color="Green" title="E64">A2</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T86" title="RVSUM_1:th.86">RVSUM_1:86</a></span>;<br/>
<a NAME="E3:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A3</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D2" title="SQUARE_1:def.2">SQUARE_1:def 2</a></span>;<br/>
<a NAME="E4:112"/><span class="lab"><font color="Green" title="E66">A4</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">r</font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>;<br/>
<span class="kw">assume </span><a NAME="E1:112_1"/><span class="lab"><font color="Green" title="E67">A5</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">c1</font> = <font color="Maroon" title="c2">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c7">c2</font> = <font color="Maroon" title="c3">z2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T57" target="_self" title="SEQ_4:th.57">Th74</a></span>;<br/>
<a NAME="E3:112_1"/>
( <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp; <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c7">c2</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112_1"><span class="lab"><font color="Green" title="E67">A5</font></span></a>, <a class="ref" href="seq_4.html#T89" target="_self" title="SEQ_4:th.89">Th106</a></span>;<br/>
<a NAME="E4:112_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A6</font></span>: 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c7">c2</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T60" title="RVSUM_1:th.60">RVSUM_1:60</a></span>;<br/>
<a NAME="E5:112_1"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c7">c2</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:112_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112_1"><span class="lab"><font color="Green" title="E68">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:112"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<a NAME="E6:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A7</font></span>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T92" title="RVSUM_1:th.92">RVSUM_1:92</a>, <a class="ref" href="square_1.html#T26" title="SQUARE_1:th.26">SQUARE_1:26</a></span>;<br/>
<a NAME="E7:112"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E8:112"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:112"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E66">A4</font></span></a>, <a class="txt" href="#E6:112"><span class="lab"><font color="Green" title="E67">A7</font></span></a>, <a class="ref" href="rvsum_1.html#T84" title="RVSUM_1:th.84">RVSUM_1:84</a>, <a class="ref" href="square_1.html#T22" title="SQUARE_1:th.22">SQUARE_1:22</a></span>;<br/>
<a NAME="E10:112"/><span class="kw">then </span>
2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T64" title="XREAL_1:th.64">XREAL_1:64</a></span>;<br/>
<a NAME="E11:112"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p0">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p0">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E12:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A8</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a></span>;<br/>
<a NAME="E13:112"/><span class="lab"><font color="Green" title="E69">A9</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">r2</font> = <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>;<br/>
<a NAME="E1:112_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E2:112_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:112_2"/><span class="lab"><font color="Green" title="E71">A11</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">c12</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T57" target="_self" title="SEQ_4:th.57">Th74</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">abs912</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E6:112_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c12</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T46" title="COMPLEX1:th.46">COMPLEX1:46</a></span>;<br/>
<a NAME="E7:112_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A12</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">abs912</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="seq_4.html#T89" target="_self" title="SEQ_4:th.89">Th106</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">abs1</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c9">abs2</font> = <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">c1</font> = <font color="Maroon" title="c2">z1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c11">c2</font> = <font color="Maroon" title="c3">z2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="seq_4.html#T57" target="_self" title="SEQ_4:th.57">Th74</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">abs12</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/>
<a NAME="E11:112_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">c1</font> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <font color="Maroon" title="c11">c2</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">c1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">c2</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T56" title="COMPLEX1:th.56">COMPLEX1:56</a></span>;<br/>
<a NAME="E12:112_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c12</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">c1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c11">c2</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="seq_4.html#T58" target="_self" title="SEQ_4:th.58">Th75</a></span>;<br/>
<a NAME="E13:112_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c12</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c10">c1</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c9">abs2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="seq_4.html#T89" target="_self" title="SEQ_4:th.89">Th106</a></span>;<br/>
<a NAME="E14:112_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c12</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">abs1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c9">abs2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="seq_4.html#T89" target="_self" title="SEQ_4:th.89">Th106</a></span>;<br/>
<a NAME="E15:112_2"/><span class="kw">then </span>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c6">c12</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">abs12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="txt" href="#E2:112_2"><span class="lab"><font color="Green" title="E70">A10</font></span></a>, <a class="ref" href="valued_1.html#D1" title="VALUED_1:def.1">VALUED_1:def 1</a></span>;<br/>
<a NAME="E16:112_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">abs912</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">abs12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112_2"><span class="lab"><font color="Green" title="E71">A11</font></span></a>, <a class="ref" href="seq_4.html#T89" target="_self" title="SEQ_4:th.89">Th106</a></span>;<br/>
<a NAME="E17:112_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">abs912</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">abs12</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:112_2"><span class="lab"><font color="Green" title="E72">A12</font></span></a>, <a class="ref" href="square_1.html#T15" title="SQUARE_1:th.15">SQUARE_1:15</a></span>;<br/>
<a NAME="E18:112_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">abs912</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T11" title="VALUED_1:th.11">VALUED_1:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:112_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T11" title="VALUED_1:th.11">VALUED_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:112"/><span class="lab"><font color="Green" title="E70">A13</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T86" title="RVSUM_1:th.86">RVSUM_1:86</a></span>;<br/>
<a NAME="E15:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A14</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="square_1.html#D2" title="SQUARE_1:def.2">SQUARE_1:def 2</a></span>;<br/>
<a NAME="E16:112"/><span class="lab"><font color="Green" title="E72">A15</font></span>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="ref" href="square_1.html#D2" title="SQUARE_1:def.2">SQUARE_1:def 2</a></span>;<br/>
<a NAME="E17:112"/><span class="lab"><font color="Green" title="E73">A16</font></span>: 
<span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:112"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="ref" href="square_1.html#D2" title="SQUARE_1:def.2">SQUARE_1:def 2</a></span>;<br/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">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#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T68" title="RVSUM_1:th.68">RVSUM_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span> <a href="rvsum_1.html#K5" title="RVSUM_1:func.5">+</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T89" title="RVSUM_1:th.89">RVSUM_1:89</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default">2 <a href="rvsum_1.html#K11" title="RVSUM_1:func.11">*</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T89" title="RVSUM_1:th.89">RVSUM_1:89</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>
;<br/>
<a NAME="E19:112"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K15" title="RVSUM_1:func.15">mlt</a> (<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span>,<span class="p5">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span>)</span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:112"><span class="lab"><font color="Green" title="E69">A9</font></span></a>, <a class="ref" href="rvsum_1.html#T82" title="RVSUM_1:th.82">RVSUM_1:82</a></span>;<br/>
<a NAME="E20:112"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:112"><span class="lab"><font color="Green" title="E68">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:112"/><span class="kw">then </span>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p2">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p4">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p2">(<span class="default">2 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p4">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p0">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p1">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E64">A2</font></span></a>, <a class="txt" href="#E14:112"><span class="lab"><font color="Green" title="E70">A13</font></span></a>, <a class="ref" href="square_1.html#T29" title="SQUARE_1:th.29">SQUARE_1:29</a></span>;<br/>
<a NAME="E22:112"/><span class="kw">then </span>
 <a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p3">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="square_1.html#K7" title="SQUARE_1:func.7">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" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p0">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c2">z1</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p4">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p5">(<span class="default"><a href="rvsum_1.html#K13" title="RVSUM_1:func.13">sqr</a> <span class="p0">(<span class="default"><a href="seq_4.html#K21" title="SEQ_4:func.21">abs</a> <font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:112"><span class="lab"><font color="Green" title="E72">A15</font></span></a>, <a class="txt" href="#E17:112"><span class="lab"><font color="Green" title="E73">A16</font></span></a>, <a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E63">A1</font></span></a>, <a class="ref" href="square_1.html#T26" title="SQUARE_1:th.26">SQUARE_1:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:112"/>
<span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="seq_4.html#K15" title="SEQ_4:func.15">+</a> <font color="Maroon" title="c3">z2</font></span>)</span></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c2">z1</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1"><a href="seq_4.html#K22" title="SEQ_4:func.22">|.</a><span class="default"><font color="Maroon" title="c3">z2</font></span><a href="seq_4.html#K22" title="SEQ_4:func.22">.|</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E65">A3</font></span></a>, <a class="txt" href="#E15:112"><span class="lab"><font color="Green" title="E71">A14</font></span></a>, <a class="ref" href="square_1.html#T22" title="SQUARE_1:th.22">SQUARE_1:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
