<?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>set </b><font color="Maroon">r</font> = <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span>;<br/>
<a NAME="E1:3"/><i><font color="Green">E30</font></i>: 
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0 implies  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_1"/><i><font color="Green">E31</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:3_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">z</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>take </b>
 <a href="complex1.html#K6">1r</a> 
;<br/>

<b>thus </b><a NAME="E3:3_1"/>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><a href="complex1.html#K6">1r</a> </span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="complex1.html#T12">COMPLEX1:12</a>, <a class="ref" href="complex1.html#T134">COMPLEX1:134</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:3"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> implies  ex <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">a</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:3_2"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span>
 ;<br/>

<b>take </b>
<span class="p1"><a href="arytm_0.html#K5">[*</a><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="real_1.html#K6">/</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span>,<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#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
;<br/>

<b>thus </b><a NAME="E2:3_2"/>
( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><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="real_1.html#K6">/</a> <span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><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="real_1.html#K6">/</a> <span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><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="real_1.html#K6">/</a> <span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">z</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, , <a class="ref" href="complex1.html#T130">COMPLEX1:130</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:3"/>
 ex <font color="Olive">b</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  st <br/>( <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">b</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complex1.html#K17">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="complex1.html#K9">*</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div>
