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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:60"/><i><font color="Green">E49</font></i>: 
<span class="p1"><a href="complsp1.html#K16" target="_self">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="complsp1.html#K16" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:60_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complsp1.html#T21" target="_self">Th21</a></i>;<br/><a NAME="E3:60_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K15">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p3">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="complsp1.html#E1:60"><i><font color="Green">E49</font></i></a>, <a class="ref" href="rvsum_1.html#T116">RVSUM_1:116</a></i>;<br/><a NAME="E4:60_1"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K12">sqr</a> <span class="p2">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="square_1.html#T92">SQUARE_1:92</a></i>;<br/><a NAME="E5:60_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complsp1.html#K15" target="_self">abs</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="rvsum_1.html#T121">RVSUM_1:121</a></i>;<br/><a NAME="E6:60_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="complsp1.html#E1:60_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="complsp1.html#T59" target="_self">Th59</a></i>;<br/><a NAME="E7:60_1"/><b>then </b>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="complsp1.html#E1:60_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><a NAME="E8:60_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K5">0c</a> 
 <b>by </b><i><a class="ref" href="complex1.html#T131">COMPLEX1:131</a></i>;<br/><b>hence </b><a NAME="E9:60_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> <a href="complex1.html#K5">0c</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="complsp1.html#E1:60_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:60"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> <a href="complex1.html#K5">0c</a> 
 
<b>by </b><i><a class="ref" href="complsp1.html#T23" target="_self">Th23</a></i>;<br/>
<b>hence </b><a NAME="E4:60"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="complsp1.html#K11" target="_self">0c</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
