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

<b>let </b><font color="Maroon" title="c1">V</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">VectSp</a> of  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="hahnban1.html#NM4" title="HAHNBAN1:NM.4">Semi-Norm</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c3">M</font> be    <a href="vectsp_4.html#M1" title="VECTSP_4:mode.1">Subspace</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c4">l</font> be   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c3">M</font>;<br/>







<b>assume </b><a NAME="E1:85"/><i><font color="Green" title="E43">A1</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">M</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e</font> holds <br/><span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">RVSM</font> =  <a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c3">M</font> as    <a href="rlsub_1.html#M1" title="RLSUB_1:mode.1">Subspace</a> of  <a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#T30" target="_self" title="HAHNBAN1:th.30">Th30</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p1</font> = <font color="Maroon" title="c2">p</font> as   <a href="hahnban.html#NM2" title="HAHNBAN:NM.2">Banach-Functional</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span> <b>by </b><i><a class="ref" href="hahnban1.html#T32" target="_self" title="HAHNBAN1:th.32">Th32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">prRl</font> =  <a href="hahnban1.html#K11" title="HAHNBAN1:func.11">projRe</a> <font color="Maroon" title="c4">l</font> as   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <font color="Maroon" title="c5">RVSM</font> <b>by </b><i><a class="ref" href="hahnban1.html#T33" target="_self" title="HAHNBAN1:th.33">Th33</a></i>;<br/>
<a NAME="E5:85"/><i><font color="Green" title="E44">A2</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c1">V</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c1">V</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#D22" target="_self" title="HAHNBAN1:def.22">Def22</a></i>;<br/>
<a NAME="E6:85"/><i><font color="Green" title="E45">A3</font></i>: 
 <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <font color="Maroon" title="c3">M</font>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <font color="Maroon" title="c3">M</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#G1" title="RLVECT_1:aggr.1">LoopStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c3">M</font></span>)</span>,the <a href="rlvect_1.html#U1" title="RLVECT_1:sel.1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c3">M</font></span>)</span>,the <a href="struct_0.html#U2" title="STRUCT_0:sel.2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c3">M</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#D22" target="_self" title="HAHNBAN1:def.22">Def22</a></i>;<br/>
<a NAME="E7:85"/>
for <font color="Olive" title="b1">x</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c5">RVSM</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v</font> holds <br/><font color="Maroon" title="c7">prRl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">v</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c5">RVSM</font>;<br/><b>let </b><font color="Maroon" title="c9">v</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c8">x</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c3">M</font> <b>by </b><i><a class="txt" href="hahnban1.html#E6:85"><i><font color="Green" title="E45">A3</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">v1</font> = <font color="Maroon" title="c9">v</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="txt" href="hahnban1.html#E5:85"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E3:85_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v</font>
 ;<br/>

<a NAME="E4:85_1"/><b>then </b><i><font color="Green" title="E46">A4</font></i>: 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x1</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">v1</font>
 
<b>by </b><i><a class="txt" href="hahnban1.html#E1:85"><i><font color="Green" title="E43">A1</font></i></a></i>;<br/>
<a NAME="E5:85_1"/>
<font color="Maroon" title="c7">prRl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hahnban1.html#D23" target="_self" title="HAHNBAN1:def.23">Def23</a></i>;<br/>
<a NAME="E6:85_1"/><b>then </b>
<font color="Maroon" title="c7">prRl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x1</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T140" title="COMPLEX1:th.140">COMPLEX1:140</a></i>;<br/>
<b>hence </b><a NAME="E7:85_1"/>
<font color="Maroon" title="c7">prRl</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">v</font>
 <b>by </b><i><a class="txt" href="hahnban1.html#E4:85_1"><i><font color="Green" title="E46">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon" title="c8">L1</font> being   <a href="hahnban.html#NM3" title="HAHNBAN:NM.3">linear-Functional</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span><b> such that </b><br/><a NAME="E9:85"/><i><font color="Green" title="E46">A5</font></i>: 
<font color="Maroon" title="c8">L1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">RVSM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">prRl</font>
 <b>and </b><br/><a NAME="E10:85"/><i><font color="Green" title="E47">A6</font></i>: 
for <font color="Olive" title="b1">e</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span> holds  <font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">e</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T35" title="HAHNBAN:th.35">HAHNBAN:35</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">L</font> =  <a href="hahnban1.html#K16" title="HAHNBAN1:func.16">prodReIm</a> <font color="Maroon" title="c8">L1</font> as   <a href="hahnban1.html#NM2" title="HAHNBAN1:NM.2">linear-Functional</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#T35" target="_self" title="HAHNBAN1:th.35">Th35</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">L</font>
;<br/>

<b>reconsider </b><font color="Maroon" title="c10">tcM</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">M</font>;<br/><a NAME="E1:85_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2" title="VECTSP_4:def.2">VECTSP_4:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c12">x2</font> = <font color="Maroon" title="c11">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">x1</font> = <font color="Maroon" title="c12">x2</font>, <font color="Maroon" title="c14">ix1</font> = <a href="hahnban1.html#K2" title="HAHNBAN1:func.2">i_FC</a>  <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c12">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span> <b>by </b><i><a class="txt" href="hahnban1.html#E5:85"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/><i><font color="Green" title="E48">A7</font></i>: <font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">x1</font> = 
<span class="p1">(<span class="default"><a href="hahnban1.html#K11" title="HAHNBAN1:func.11">projRe</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E6:85"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="hahnban1.html#E9:85"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D23" target="_self" title="HAHNBAN1:def.23">Def23</a></i>
;<br/><b>reconsider </b><font color="Maroon" title="c15">lx</font> = <font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/><a NAME="E6:85_2"/><i><font color="Green" title="E49">A8</font></i>: 
( <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default">0,1</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> &amp; <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span> )
 <b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>;<br/><font color="Maroon" title="c15">lx</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>
;<br/><b>then </b><i><font color="Green" title="E50">A9</font></i>: <a href="hahnban1.html#K2" title="HAHNBAN1:func.2">i_FC</a>  <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span> = 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default">0,1</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6" title="COMPLFLD:th.6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
<b>by </b><i><a class="txt" href="hahnban1.html#E6:85_2"><i><font color="Green" title="E49">A8</font></i></a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>
;<br/><a NAME="E9:85_2"/><i><font color="Green" title="E51">A10</font></i>: 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>;<br/><a NAME="E10:85_2"/><i><font color="Green" title="E52">A11</font></i>: 
<a href="hahnban1.html#K2" title="HAHNBAN1:func.2">i_FC</a>  <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="hahnban1.html#K2" title="HAHNBAN1:func.2">i_FC</a>  <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c12">x2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T22" title="VECTSP_4:th.22">VECTSP_4:22</a></i>;<br/><b>then </b><i><font color="Green" title="E53">A12</font></i>: <font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">ix1</font> = 
<span class="p1">(<span class="default"><a href="hahnban1.html#K11" title="HAHNBAN1:func.11">projRe</a> <font color="Maroon" title="c4">l</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ix1</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E6:85"><i><font color="Green" title="E45">A3</font></i></a>, <a class="txt" href="hahnban1.html#E9:85"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="hahnban1.html#K2" title="HAHNBAN1:func.2">i_FC</a>  <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hahnban1.html#E10:85_2"><i><font color="Green" title="E52">A11</font></i></a>, <a class="ref" href="hahnban1.html#D23" target="_self" title="HAHNBAN1:def.23">Def23</a></i>
<br/>.= 
 <a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <font color="Maroon" title="c15">lx</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <font color="Maroon" title="c15">lx</font></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="txt" href="hahnban1.html#E8:85_2"><i><font color="Green" title="E50">A9</font></i></a>, <a class="ref" href="hahnban1.html#D12" target="_self" title="HAHNBAN1:def.12">Def12</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4" title="COMPLEX1:func.4">Im</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T28" title="COMPLEX1:th.28">COMPLEX1:28</a>, <a class="txt" href="hahnban1.html#E9:85_2"><i><font color="Green" title="E51">A10</font></i></a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="nattra_1.html#K1" title="NATTRA_1:func.1">|</a> <font color="Maroon" title="c10">tcM</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> = 
<font color="Maroon" title="c9">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x2</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" title="HAHNBAN1:func.15">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x2</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#D28" target="_self" title="HAHNBAN1:def.28">Def28</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3" title="COMPLEX1:func.3">Re</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><a href="hahnban1.html#K2" title="HAHNBAN1:func.2">i_FC</a>  <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c12">x2</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#D27" target="_self" title="HAHNBAN1:def.27">Def27</a>, <a class="txt" href="hahnban1.html#E4:85_2"><i><font color="Green" title="E48">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c4">l</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">x</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E11:85_2"><i><font color="Green" title="E53">A12</font></i></a>, <a class="ref" href="complex1.html#T29" title="COMPLEX1:th.29">COMPLEX1:29</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:85"/>
<font color="Maroon" title="c9">L</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">e</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <font color="Maroon" title="c1">V</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c12">Le</font> = <font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">Ledz</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">Le</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1" title="COMPLFLD:def.1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">e1</font> = <font color="Maroon" title="c11">e</font>, <font color="Maroon" title="c15">Ledze</font> = <font color="Maroon" title="c13">Ledz</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">e</font> as   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" title="HAHNBAN1:func.10">RealVS</a> <font color="Maroon" title="c1">V</font></span>)</span> <b>by </b><i><a class="txt" href="hahnban1.html#E5:85"><i><font color="Green" title="E44">A2</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:85_3"/>
( <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_1"><b>suppose </b></a><a NAME="E1:85_3_1"/><i><font color="Green" title="E48">A13</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><i><font color="Green" title="E49">A14</font></i>: <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self" title="HAHNBAN1:th.4">Th4</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>

;<br/><i><font color="Green" title="E50">A15</font></i>: <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c13">Ledz</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Le</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T153" title="COMPLEX1:th.153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="txt" href="hahnban1.html#E2:85_3_1"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="complex1.html#T139" title="COMPLEX1:th.139">COMPLEX1:139</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="hahnban1.html#E1:85_3_1"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
;<br/><i><font color="Green" title="E51">A16</font></i>: <font color="Maroon" title="c6">p1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">Ledze</font> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><font color="Maroon" title="c13">Ledz</font></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D19" target="_self" title="HAHNBAN1:def.19">Def19</a></i>
<br/>.= 
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font>
<b>by </b><i><a class="txt" href="hahnban1.html#E3:85_3_1"><i><font color="Green" title="E50">A15</font></i></a></i>
;<br/><a NAME="E5:85_3_1"/><i><font color="Green" title="E52">A17</font></i>: 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>;<br/><b>then </b><span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">Le</font> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="xcmplx_0.html#K7" title="XCMPLX_0:func.7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c12">Le</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self" title="HAHNBAN1:th.5">Th5</a></i>
<br/>.= 
<font color="Maroon" title="c13">Ledz</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#T6" title="COMPLFLD:th.6">COMPLFLD:6</a>, <a class="txt" href="hahnban1.html#E5:85_3_1"><i><font color="Green" title="E52">A17</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">Ledz</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">e</font></span>)</span>
<b>by </b><i><a class="ref" href="hahnban1.html#D12" target="_self" title="HAHNBAN1:def.12">Def12</a></i>
<br/>.= 
<span class="p1"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">Ledz</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">e</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" title="HAHNBAN1:func.15">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">Ledz</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">e</font></span>)</span></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#D28" target="_self" title="HAHNBAN1:def.28">Def28</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">Ledze</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" title="HAHNBAN1:func.15">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c13">Ledz</font> <a href="vectsp_1.html#K6" title="VECTSP_1:func.6">*</a> <font color="Maroon" title="c11">e</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>
;<br/><a NAME="E7:85_3_1"/><b>then </b>
<font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">Ledze</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/><b>hence </b><a NAME="E8:85_3_1"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font>
 <b>by </b><i><a class="txt" href="hahnban1.html#E10:85"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="hahnban1.html#E4:85_3_1"><i><font color="Green" title="E51">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="85_3_2"><b>suppose </b></a><a NAME="E1:85_3_2"/><i><font color="Green" title="E48">A18</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c12">Le</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2"><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">[**</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span>,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hahnban1.html#K15" title="HAHNBAN1:func.15">i-shift</a> <span class="p0">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" title="HAHNBAN1:func.1">**]</a></span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#D28" target="_self" title="HAHNBAN1:def.28">Def28</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span>,<span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="hahnban1.html#K15" title="HAHNBAN1:func.15">i-shift</a> <span class="p0">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27" title="COMPLEX1:th.27">COMPLEX1:27</a></i>
;<br/><a NAME="E3:85_3_2"/><b>then </b>
<span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span>,<span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K15" title="HAHNBAN1:func.15">i-shift</a> <span class="p5">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="arytm_0.html#K5" title="ARYTM_0:func.5">[*</a><span class="default">0,0</span><a href="arytm_0.html#K5" title="ARYTM_0:func.5">*]</a></span>
 <b>by </b><i><a class="txt" href="hahnban1.html#E1:85_3_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="hahnban1.html#E53"><i><font color="Green" title="E32">Lm4</font></i></a>, <a class="ref" href="complex1.html#T131" title="COMPLEX1:th.131">COMPLEX1:131</a></i>;<br/><a NAME="E4:85_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="hahnban1.html#K13" title="HAHNBAN1:func.13">RtoC</a> <font color="Maroon" title="c8">L1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T12" title="ARYTM_0:th.12">ARYTM_0:12</a></i>;<br/><a NAME="E5:85_3_2"/><b>then </b>
<font color="Maroon" title="c8">L1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><b>hence </b><a NAME="E6:85_3_2"/>
<span class="p1"><a href="complfld.html#K3" title="COMPLFLD:func.3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">e</font>
 <b>by </b><i><a class="txt" href="hahnban1.html#E10:85"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="hahnban1.html#E1:85_3_2"><i><font color="Green" title="E48">A18</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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