<?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">p</font> be   <a href="hahnban1.html#NM4">Semi-Norm</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="vectsp_4.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">l</font> be   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">M</font>;<br/>







<b>assume </b><a NAME="E1:85"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">e</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">M</font><br/> for <font color="Olive">v</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>  st <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Olive">e</font> holds <br/><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Olive">v</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">RVSM</font> =  <a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font> as    <a href="rlsub_1.html#M1">Subspace</a> of  <a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#T5" target="_self">Th5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p1</font> = <font color="Maroon">p</font> as   <a href="hahnban.html#NM2">Banach-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> <b>by </b><i><a class="txt" href="hahnban1.html#E5:85"><i><font color="Green">E36</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">prRl</font> =  <a href="hahnban1.html#K11" target="_self">projRe</a> <font color="Maroon">l</font> as   <a href="hahnban.html#NM3">linear-Functional</a> of <font color="Maroon">RVSM</font> <b>by </b><i/>;<br/>
<a NAME="E5:85"/><i><font color="Green">E36</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E6:85"/><i><font color="Green">E37</font></i>: 
 <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">M</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">M</font> #) <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#G1">LoopStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>,the <a href="rlvect_1.html#U1">add</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span>,the <a href="struct_0.html#U2">Zero</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">M</font></span>)</span> #)
 
<b>by </b><i><a class="ref" href="hahnban1.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E7:85"/>
for <font color="Olive">x</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">RVSM</font><br/> for <font color="Olive">v</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">v</font> holds <br/><font color="Maroon">prRl</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">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">x</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">RVSM</font>;<br/><b>let </b><font color="Maroon">v</font> be   <a href="rlvect_1.html#NM1">VECTOR</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/>



<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">M</font> <b>by </b><i><a class="ref" href="hahnban1.html#T10" target="_self">Th10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">v1</font> = <font color="Maroon">v</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="hahnban1.html#T9" target="_self">Th9</a></i>;<br/>
<b>assume </b><a NAME="E3:85_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 ;<br/>

<a NAME="E4:85_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v1</font>
 
<b>by </b><i><a class="ref" href="hahnban1.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E5:85_1"/>
<font color="Maroon">prRl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hahnban1.html#E6:85"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E6:85_1"/><b>then </b>
<font color="Maroon">prRl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 
<b>by </b><i><a class="ref" href="complex1.html#T140">COMPLEX1:140</a></i>;<br/>
<b>hence </b><a NAME="E7:85_1"/>
<font color="Maroon">prRl</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="hahnban1.html#T11" target="_self">Th11</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


</div><b>end;</b></div>
<b>then consider </b><font color="Maroon">L1</font> being   <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><b> such that </b><br/><a NAME="E9:85"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">L1</font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">RVSM</font> <a href="hidden.html#R1">=</a> <font color="Maroon">prRl</font>
 <b>and </b><br/><a NAME="E10:85"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">e</font> being  <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> holds  <font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">e</font>
 <b>by </b><i><a class="ref" href="hahnban.html#T35">HAHNBAN:35</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L</font> =  <a href="hahnban1.html#K16" target="_self">prodReIm</a> <font color="Maroon">L1</font> as   <a href="hahnban1.html#NM2">linear-Functional</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">L</font>
;<br/>

<b>reconsider </b><font color="Maroon">tcM</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="vectsp_4.html#D2">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/><a NAME="E1:85_2"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#D2">VECTSP_4:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">V</font> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x2</font>, <font color="Maroon">ix1</font> = <a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x2</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <b>by </b><i><a class="ref" href="hahnban1.html#T9" target="_self">Th9</a></i>;<br/><i><font color="Green">E41</font></i>: <font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x1</font> = 
<span class="p1">(<span class="default"><a href="hahnban1.html#K11" target="_self">projRe</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T10" target="_self">Th10</a>, <a class="ref" href="hahnban1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="hahnban1.html#E6:85"><i><font color="Green">E37</font></i></a></i>
;<br/><b>reconsider </b><font color="Maroon">lx</font> = <font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</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/><a NAME="E6:85_2"/><i><font color="Green">E42</font></i>: 
( <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,1</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> 0 <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span> &amp; <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">lx</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</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">lx</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#T27">COMPLEX1:27</a></i>;<br/><font color="Maroon">lx</font> = 
<span class="p1">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</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">lx</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/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">lx</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><b>then </b><i><font color="Green">E84</font></i>: <a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> = 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,1</span><a href="arytm_0.html#K5">*]</a></span> <a href="complex1.html#K9">*</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</font></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">lx</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">lx</font></span>)</span></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">0 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">lx</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</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="hahnban1.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><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">lx</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><a NAME="E9:85_2"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><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">lx</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K4">Im</a> <font color="Maroon">lx</font></span>)</span></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#K3">Re</a> <font color="Maroon">lx</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#T27">COMPLEX1:27</a></i>;<br/><a NAME="E10:85_2"/><i><font color="Green">E88</font></i>: 
<a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <a href="hahnban1.html#K2" target="_self">i_FC</a>  <a href="vectsp_1.html#K6">*</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="vectsp_4.html#T22">VECTSP_4:22</a></i>;<br/><b>then </b><i><font color="Green">E89</font></i>: <font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">ix1</font> = 
<span class="p1">(<span class="default"><a href="hahnban1.html#K11" target="_self">projRe</a> <font color="Maroon">l</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">ix1</font>
<b>by </b><i><a class="ref" href="hahnban1.html#T10" target="_self">Th10</a>, <a class="ref" href="hahnban1.html#T11" target="_self">Th11</a>, <a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<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>
<b>by </b><i><a class="ref" href="hahnban1.html#T17" target="_self">Th17</a>, <a class="txt" href="hahnban1.html#E6:85"><i><font color="Green">E37</font></i></a></i>
<br/>.= 
 <a href="complex1.html#K3">Re</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><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">lx</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="complex1.html#K3">Re</a> <font color="Maroon">lx</font></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K4">Im</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T28">COMPLEX1:28</a>, </i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="nattra_1.html#K1">|</a> <font color="Maroon">tcM</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">L</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</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">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</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">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x2</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#E10:85"><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"><a href="complex1.html#K3">Re</a> <span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="funct_2.html#K3">.</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">L1</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">x2</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#E9:85"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hahnban1.html#T15" target="_self">Th15</a></i>
<br/>.= 
<font color="Maroon">l</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="complex1.html#T29">COMPLEX1:29</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:85"/>
<font color="Maroon">L</font> <a href="relset_1.html#K8">|</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>

<b>let </b><font color="Maroon">e</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/>

<b>reconsider </b><font color="Maroon">Le</font> = <font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</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>reconsider </b><font color="Maroon">Ledz</font> = <span class="p1">(<span class="default"><font color="Maroon">Le</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K13">/</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a>  <b>by </b><i><a class="ref" href="complfld.html#D1">COMPLFLD:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e1</font> = <font color="Maroon">e</font>, <font color="Maroon">Ledze</font> = <font color="Maroon">Ledz</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">e</font> as   <a href="rlvect_1.html#NM1">VECTOR</a> of <span class="p1">(<span class="default"><a href="hahnban1.html#K10" target="_self">RealVS</a> <font color="Maroon">V</font></span>)</span> <b>by </b><i><a class="ref" href="hahnban1.html#T9" target="_self">Th9</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">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 or <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</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">E90</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><i><font color="Green">E91</font></i>: <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span></span><a href="complex1.html#K17">.|</a></span> = 
 <a href="complex1.html#K18">abs</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>

;<br/><i><font color="Green">E92</font></i>: <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">Ledz</font></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Le</font> <a href="complex1.html#K15">*'</a> </span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T153">COMPLEX1:153</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K6">/</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="hahnban1.html#T16" target="_self">Th16</a>, <a class="ref" href="complex1.html#T139">COMPLEX1:139</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="hahnban1.html#T15" target="_self">Th15</a>, <a class="ref" href="xcmplx_1.html#T60">XCMPLX_1:60</a></i>
;<br/><i><font color="Green">E93</font></i>: <font color="Maroon">p1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ledze</font> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">Ledz</font></span><a href="complfld.html#K3">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font>
<b>by </b><i/>
;<br/><a NAME="E5:85_3_1"/><i><font color="Green">E94</font></i>: 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span>
 <b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>;<br/><b>then </b><span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span>,0</span><a href="arytm_0.html#K5">*]</a></span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Le</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="xcmplx_0.html#K7">/</a> <span class="p2">(<span class="default"><span class="p3"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default">0 <a href="xcmplx_0.html#K3">*</a> <a href="complex1.html#K7">&lt;i&gt;</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">Le</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">Ledz</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span>
<b>by </b><i><a class="ref" href="complfld.html#T6">COMPLFLD:6</a>, <a class="ref" href="hahnban1.html#T17" target="_self">Th17</a></i>
<br/>.= 
<font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">Ledz</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">e</font></span>)</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">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">Ledz</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">e</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">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">Ledz</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">e</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#E10:85"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ledze</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">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">Ledz</font> <a href="vectsp_1.html#K6">*</a> <font color="Maroon">e</font></span>)</span></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><a NAME="E7:85_3_1"/><b>then </b>
<font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">Ledze</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
 <b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><b>hence </b><a NAME="E8:85_3_1"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="hahnban1.html#T14" target="_self">Th14</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">E95</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Maroon">Le</font></span><a href="complex1.html#K17">.|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</a><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="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span>,<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#K15" target="_self">i-shift</a> <span class="p0">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span></span>)</span></span><a href="hahnban1.html#K1" target="_self">**]</a></span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="txt" href="hahnban1.html#E10:85"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span>,<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#K15" target="_self">i-shift</a> <span class="p0">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span></span><a href="complex1.html#K17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T27">COMPLEX1:27</a></i>
;<br/><a NAME="E3:85_3_2"/><b>then </b>
<span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hahnban1.html#K13" target="_self">RtoC</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</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">L1</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span></span>)</span></span><a href="arytm_0.html#K5">*]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="arytm_0.html#K5">[*</a><span class="default">0,0</span><a href="arytm_0.html#K5">*]</a></span>
 <b>by </b><i><a class="ref" href="hahnban1.html#T15" target="_self">Th15</a>, , <a class="ref" href="complex1.html#T131">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" target="_self">RtoC</a> <font color="Maroon">L1</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="arytm_0.html#T12">ARYTM_0:12</a></i>;<br/><a NAME="E5:85_3_2"/><b>then </b>
<font color="Maroon">L1</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E6:85_3_2"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">L</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">e</font>
 <b>by </b><i><a class="ref" href="hahnban1.html#T14" target="_self">Th14</a>, <a class="ref" href="hahnban1.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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