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<b>let </b><font color="Maroon">V</font> be   <a href="vectsp_1.html#NM6">VectSp</a> of  <a href="complfld.html#K1">F_Complex</a> ;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>, <font color="Maroon">w</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="hermitan.html#V5" target="_self">diagReR+0valued</a> <a href="hermitan.html#NM3">hermitan-Form</a> of <font color="Maroon">V</font>;<br/>





<b>set </b><font color="Maroon">v4</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font>;<br/>
<b>set </b><font color="Maroon">w1</font> = <font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">v1</font>;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font>;<br/>
<b>reconsider </b><font color="Maroon">A</font> =  <a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">v1</font>, <font color="Maroon">B</font> = <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>, <font color="Maroon">C</font> =  <a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> <b> such that </b><br/><a NAME="E3:93"/><i><font color="Green">E30</font></i>: 
( <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">a</font></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K3">Re</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font></span>)</span></span>)</span> <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">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> &amp;  <a href="complex1.html#K4">Im</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">w</font>,<font color="Maroon">v1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i/>;<br/>
<i><font color="Green">E31</font></i>: <font color="Maroon">B</font> = 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font></span>)</span> <a href="complfld.html#K2">*'</a> </span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T139">COMPLEX1:139</a></i>
;<br/>
<a NAME="E5:93"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">B</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a NAME="E6:93"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 implies <font color="Maroon">B</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:93_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">B</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">2 <a href="real_1.html#K4">*</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T77">XCMPLX_1:77</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T92">XCMPLX_1:92</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font>
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T128">XCMPLX_1:128</a></i>
;<br/>
<a NAME="E3:93_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="hermitan.html#E17"><i><font color="Green">Lemma48</font></i></a>, </i>;<br/>
<a NAME="E4:93_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T129">XREAL_1:129</a></i>;<br/>
<a NAME="E5:93_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/>
<a NAME="E6:93_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/>
<a NAME="E7:93_2"/><b>then </b>
0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<b>hence </b><a NAME="E8:93_2"/>
<font color="Maroon">B</font> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">A</font> <a href="real_1.html#K4">*</a> <font color="Maroon">C</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:93"/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="binop_1.html#K2">.</a> <font color="Maroon">v1</font>,<font color="Maroon">w</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">v1</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#T14" target="_self">Th14</a>, , <a class="ref" href="hermitan.html#T15" target="_self">Th15</a>, <a class="txt" href="hermitan.html#E4:93"><i><font color="Green">E31</font></i></a></i>;<br/>


</div>
