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

<b>let </b><font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">z</font> <a href="polyeq_3.html#K3" target="_self">^2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<br/>

<b>set </b><font color="Maroon">a</font> =  <a href="complex1.html#K3">Re</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="complex1.html#K4">Im</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">u</font> =  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font>;<br/>
<b>set </b><font color="Maroon">v</font> =  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font>;<br/>
<a NAME="E2:29"/><i><font color="Green">E47</font></i>: 
( <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/>
<a NAME="E3:29"/><b>then </b><i><font color="Green">E48</font></i>: 
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> &amp; <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> )
 
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<a NAME="E4:29"/>
<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/>
<a NAME="E5:29"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">z</font> <a href="polyeq_3.html#K3" target="_self">^2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 
;<br/>
<a NAME="E6:29"/><b>then </b><i><font color="Green">E50</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <font color="Maroon">s</font> &amp; <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K4">Im</a> <font color="Maroon">s</font> )
 
<b>by </b><i>, <a class="ref" href="complex1.html#T163">COMPLEX1:163</a>, , </i>;<br/>
<a NAME="E7:29"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E8:29"/>
( <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp; <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E9:29"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E10:29"/><b>then </b><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T89">SQUARE_1:89</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1"/>
(  <a href="complex1.html#K4">Im</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or  <a href="complex1.html#K4">Im</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>suppose </b></a><a NAME="E1:29_1_1"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E3:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/><a NAME="E4:29_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="real_2.html#T129">REAL_2:129</a></i>;<br/><a NAME="E5:29_1_1"/><b>then </b>
( (  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> 0 ) or (  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 <b>by </b><i><a class="ref" href="square_1.html#T25">SQUARE_1:25</a></i>;<br/><a NAME="E6:29_1_1"/><b>then </b>
( (  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> ) or (  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="square_1.html#T89">SQUARE_1:89</a>, <a class="ref" href="square_1.html#T90">SQUARE_1:90</a></i>;<br/><b>hence </b><a NAME="E7:29_1_1"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_2"><b>suppose </b></a><a NAME="E1:29_1_2"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">s</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:29_1_2"/><b>then </b>
2 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T163">COMPLEX1:163</a>, , , , </i>;<br/><a NAME="E3:29_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T135">XREAL_1:135</a></i>;<br/><a NAME="E4:29_1_2"/><b>then </b>
( (  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 ) or (  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> 0 ) )
 <b>by </b><i><a class="ref" href="xreal_1.html#T135">XREAL_1:135</a></i>;<br/><a NAME="E5:29_1_2"/><b>then </b>
( (  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> ) or (  <a href="complex1.html#K3">Re</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> &amp;  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span> ) )
 <b>by </b><i>, , <a class="ref" href="square_1.html#T89">SQUARE_1:89</a>, <a class="ref" href="square_1.html#T90">SQUARE_1:90</a></i>;<br/><b>hence </b><a NAME="E6:29_1_2"/>
( <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p0">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> or <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="polyeq_3.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="square_1.html#K7">sqrt</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">s</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span> <a href="polyeq_3.html#K1" target="_self">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
