<?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="Red">::  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="Red">::  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="E1:131"/><span class="lab"><font color="Green" title="E64">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#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></span>;<br/>
<a NAME="E2:131"/><span class="lab"><font color="Green" title="E65">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#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></span>;<br/>
<a NAME="E3:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">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#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E4:131"/><span class="lab"><font color="Green" title="E67">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="131_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="Red">::  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="E1:131_1"/><span class="lab"><font color="Green" title="E68">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="Red">::  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#T74" target="_self" title="SEQ_4:th.74">Th21</a></span>;<br/>
<a NAME="E3:131_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="#E1:131_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="ref" href="seq_4.html#T106" target="_self" title="SEQ_4:th.106">Th59</a></span>;<br/>
<a NAME="E4:131_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">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#T87" title="RVSUM_1:th.87">RVSUM_1:87</a></span>;<br/>
<a NAME="E5:131_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#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:131_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="#E4:131_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:131"/>
 <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#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/>
<a NAME="E6:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">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#T122" title="RVSUM_1:th.122">RVSUM_1:122</a>, <a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></span>;<br/>
<a NAME="E7:131"/>
 <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="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E8:131"/><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="E9:131"/><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="#E4:131"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="txt" href="#E6:131"><span class="lab"><font color="Green" title="E68">A7</font></span></a>, <a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>;<br/>
<a NAME="E10:131"/><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#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/>
<a NAME="E11:131"/><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#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E12:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">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#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E13:131"/><span class="lab"><font color="Green" title="E70">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="131_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="Red">::  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="E1:131_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="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E2:131_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">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="E3:131_2"/><span class="lab"><font color="Green" title="E72">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="Red">::  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#T74" target="_self" title="SEQ_4:th.74">Th21</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="txt" href="seq_4.html#E137"><span class="lab"><font color="Green" title="E62">Lm5</font></span></a></span>;<br/>
<a NAME="E6:131_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#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E7:131_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">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="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="seq_4.html#T106" target="_self" title="SEQ_4:th.106">Th59</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="txt" href="seq_4.html#E137"><span class="lab"><font color="Green" title="E62">Lm5</font></span></a></span>;<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="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="seq_4.html#T74" target="_self" title="SEQ_4:th.74">Th21</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="txt" href="seq_4.html#E137"><span class="lab"><font color="Green" title="E62">Lm5</font></span></a></span>;<br/>
<a NAME="E11:131_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#T142" title="COMPLEX1:th.142">COMPLEX1:142</a></span>;<br/>
<a NAME="E12:131_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="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="seq_4.html#T75" target="_self" title="SEQ_4:th.75">Th24</a></span>;<br/>
<a NAME="E13:131_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="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="seq_4.html#T106" target="_self" title="SEQ_4:th.106">Th59</a></span>;<br/>
<a NAME="E14:131_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="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="seq_4.html#T106" target="_self" title="SEQ_4:th.106">Th59</a></span>;<br/>
<a NAME="E15:131_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="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="txt" href="#E2:131_2"><span class="lab"><font color="Green" title="E71">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="E16:131_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="#E3:131_2"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="seq_4.html#T106" target="_self" title="SEQ_4:th.106">Th59</a></span>;<br/>
<a NAME="E17:131_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="#E7:131_2"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/>
<a NAME="E18:131_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="E19:131_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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E14:131"/><span class="lab"><font color="Green" title="E71">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#T116" title="RVSUM_1:th.116">RVSUM_1:116</a></span>;<br/>
<a NAME="E15:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">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#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E16:131"/><span class="lab"><font color="Green" title="E73">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="#E2:131"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</a></span>;<br/>
<a NAME="E17:131"/><span class="lab"><font color="Green" title="E74">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="#E14:131"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="ref" href="square_1.html#D4" title="SQUARE_1:def.4">SQUARE_1:def 4</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#T98" title="RVSUM_1:th.98">RVSUM_1:98</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#T119" title="RVSUM_1:th.119">RVSUM_1:119</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#T119" title="RVSUM_1:th.119">RVSUM_1:119</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#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></span>
;<br/>
<a NAME="E19:131"/><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="#E13:131"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="rvsum_1.html#T112" title="RVSUM_1:th.112">RVSUM_1:112</a></span>;<br/>
<a NAME="E20:131"/><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="#E12:131"><span class="lab"><font color="Green" title="E69">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="E21:131"/><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="#E2:131"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="txt" href="#E14:131"><span class="lab"><font color="Green" title="E71">A13</font></span></a>, <a class="ref" href="square_1.html#T97" title="SQUARE_1:th.97">SQUARE_1:97</a></span>;<br/>
<a NAME="E22:131"/><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="#E16:131"><span class="lab"><font color="Green" title="E73">A15</font></span></a>, <a class="txt" href="#E17:131"><span class="lab"><font color="Green" title="E74">A16</font></span></a>, <a class="txt" href="#E1:131"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:131"/>
<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="#E3:131"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E15:131"><span class="lab"><font color="Green" title="E72">A14</font></span></a>, <a class="ref" href="square_1.html#T89" title="SQUARE_1:th.89">SQUARE_1:89</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
