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

<b>let </b><font color="Maroon">V</font> be  non <a href="struct_0.html#V3">empty</a> <a href="vectsp_1.html#NM6">VectSp</a> of  <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">l</font> be   <a href="hahnban.html#NM3">linear-Functional</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>;<br/>



<a NAME="E1:84"/><i><font color="Green">E35</font></i>: 
 <a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font> is <a href="hahnban1.html#V1" target="_self">additive</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">y</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D11" target="_self">HAHNBAN1:def 11</a><br/>

<b>thus </b><span class="p1">(<span class="default"><a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span> = 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" target="_self">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <span class="p5">(<span class="default"><font color="Maroon">x</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</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"><span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>

<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="rlvect_1.html#K4">+</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" target="_self">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="rlvect_1.html#K4">+</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">y</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" target="_self">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="rlvect_1.html#K4">+</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" target="_self">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" target="_self">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E2:84"/>
 <a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font> is <a href="hahnban1.html#V2" target="_self">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="hahnban1.html#D12" target="_self">HAHNBAN1:def 12</a><br/><b>let </b><font color="Maroon">r</font> be   <a href="vectsp_1.html#NM4">Scalar</a> of ;<br/>



<b>reconsider </b><font color="Maroon">r1</font> = <font color="Maroon">r</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="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">a</font> =  <a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font>;<br/>
<b>set </b><font color="Maroon">b</font> =  <a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font>;<br/>
<a NAME="E2:84_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K3">*</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="E3:84_2"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
 
;<br/>
<i><font color="Green">E38</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></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">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></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">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></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">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="rlvect_1.html#K4">+</a> <span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="hahnban1.html#K1" target="_self">[**</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">r1</font></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p3">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T9" target="_self">Th9</a></i>
;<br/>
<i><font color="Green">E39</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="group_1.html#K10">*</a> <span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="group_1.html#D4">GROUP_1:def 4</a></i>
<br/>.= 
<a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <a href="hahnban1.html#K2" target="_self">i_FC</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/>
<a NAME="E6:84_2"/>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span>
 
<b>by </b><i><a class="ref" href="complfld.html#T4">COMPLFLD:4</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>;<br/>
<b>then </b><i><font color="Green">E40</font></i>: <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> = 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <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">r1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="real_1.html#K3">+</a> <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">r1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>

;<br/>
<i><font color="Green">E41</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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">r1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></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">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <a href="hahnban1.html#K2" target="_self">i_FC</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <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">r1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <a href="hahnban1.html#K2" target="_self">i_FC</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <a href="hahnban1.html#K2" target="_self">i_FC</a> </span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p5">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p0">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p0">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="group_1.html#K10">*</a> <span class="p5">(<span class="default"><a href="rlvect_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="group_1.html#K2">1.</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span></span>)</span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p5">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T15" target="_self">Th15</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p5">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p5">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span>,0</span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="rlvect_1.html#K4">+</a> <span class="p4"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default">0,<span class="p5">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span></span>)</span> <a href="vectsp_1.html#K6">*</a> <span class="p3">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3"><a href="hahnban1.html#K1" target="_self">[**</a><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">r1</font></span>)</span> <a href="real_1.html#K3">+</a> 0</span>)</span>,<span class="p4">(<span class="default">0 <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span> <a href="vectsp_1.html#K6">*</a> <span class="p3">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="group_1.html#K10">*</a> <font color="Maroon">r</font></span>)</span> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T10" target="_self">Th10</a>, <a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a></i>
;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" target="_self">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="txt" href="hahnban1.html#E7:84_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <span class="p5">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="txt" href="hahnban1.html#E5:84_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</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">r1</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p0">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D26">VECTSP_1:def 26</a>, <a class="ref" href="hahnban1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">r1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <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"><span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T9" target="_self">Th9</a>, <a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K10">*</a> <span class="p1"><a href="hahnban1.html#K1" target="_self">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" target="_self">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">l</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span>
<b>by </b><i><a class="txt" href="hahnban1.html#E5:84_2"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban1.html#E7:84_2"><i><font color="Green">E40</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:84"/>
 <a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">l</font> is   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="hahnban1.html#T8" target="_self">Th8</a></i>;<br/>


</div>
