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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="realset2.html#V3">trivial</a> <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being   <a href="clvect_1.html#NM1">VECTOR</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:38"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="anproj_1.html#D8">ANPROJ_1:def 8</a></i>;<br/>
<a NAME="E3:38"/><i><font color="Green">E23</font></i>: 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="clopban2.html#E2:38"><i><font color="Green">E22</font></i></a>, <a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/>
<a NAME="E4:38"/><b>then </b><i><font color="Green">E24</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
 
<b>by </b><i><a class="ref" href="clvect_1.html#T106">CLVECT_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> 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="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<a NAME="E6:38"/>
 <a href="complex1.html#K4">Im</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>;<br/>
<b>then </b><i><font color="Green">E25</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T136">COMPLEX1:136</a></i>
<br/>.= 
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="txt" href="clopban2.html#E4:38"><i><font color="Green">E24</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
;<br/>
<i><font color="Green">E26</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K12">"</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K12">"</a> </span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#D7">COMPLEX1:def 7</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K6">1r</a>  <a href="complex1.html#K13">/</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
1 <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T134">COMPLEX1:134</a>, <a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
1 <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9">XCMPLX_0:def 9</a></i>
<br/>.= 
<span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K2">"</a> 
<b>by </b><i><a class="txt" href="clopban2.html#E7:38"><i><font color="Green">E25</font></i></a></i>
;<br/>
<b>take </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K12">"</a> </span>)</span> <a href="clvect_1.html#K1">*</a> <font color="Maroon">c<sub>2</sub></font>;<br/>

<b>thus </b><span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="clvect_1.html#K5">.||</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="complex1.html#K12">"</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span>
<b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="clopban2.html#E3:38"><i><font color="Green">E23</font></i></a>, <a class="txt" href="clopban2.html#E8:38"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
;<br/>


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