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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span>;<br/>
<a NAME="E1:9"/><i><font color="Green">E11</font></i>: 
( <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> 0 implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complfld.html#K3">.|</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<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>then </b><i><font color="Green">E13</font></i>: <font color="Maroon">c<sub>1</sub></font> = 
the <a href="struct_0.html#U2">Zero</a> of <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="ref" href="complfld.html#T94">COMPLFLD:94</a></i>
<br/>.= 
 <a href="complex1.html#K5">0c</a> 
<b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>
<br/>.= 
0

;<br/>
<b>take </b><font color="Maroon">c<sub>3</sub></font> = the <a href="vectsp_1.html#U1">unity</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>thus </b><a NAME="E3:9_1"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="complfld.html#T97">COMPLFLD:97</a></i>;<br/>

<a NAME="E4:9_1"/>
(  <a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="hermitan.html#E1"><i><font color="Green">Lemma1</font></i></a>, <a class="txt" href="hermitan.html#E2:9_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
(  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="hermitan.html#E12"><i><font color="Green">Lemma9</font></i></a>, <a class="txt" href="hermitan.html#E1:9_1"><i><font color="Green">E12</font></i></a>, <a class="ref" href="vectsp_1.html#D13">VECTSP_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:9"/>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complfld.html#K3">.|</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<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_2"/><i><font color="Green">E12</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span>
 ;<br/>

<b>take </b>
<span class="p1"><a href="hahnban1.html#K1">[**</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">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span>
;<br/>

<b>thus </b><a NAME="E2:9_2"/>
( <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</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">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span></span><a href="complfld.html#K3">.|</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="hahnban1.html#K1">[**</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">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1">[**</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">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</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">c<sub>1</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span></span>)</span></span><a href="hahnban1.html#K1">**]</a></span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="hermitan.html#E1:9_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="hermitan.html#T8" target="_self">Th8</a>, <a class="ref" href="complfld.html#T93">COMPLFLD:93</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  st <br/>( <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="complfld.html#K3">.|</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<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="complfld.html#K3">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="hermitan.html#E1:9"><i><font color="Green">E11</font></i></a>, <a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>


</div>
