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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">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>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="E2:92"/><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/>
<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> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>assume </b><b>that </b><br/><a NAME="E4:92"/><i><font color="Green">E31</font></i>: 
 <a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font> <a href="hidden.html#R1">=</a> 0
 <b>and </b><br/><a NAME="E5:92"/><i><font color="Green">E32</font></i>: 
<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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div><i><font color="Green">E81</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1_1"/><i><font color="Green">E129</font></i>: 
<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> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:92_1_1"/><i><font color="Green">E130</font></i>: 
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#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</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"><a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E3:92_1_1"/>
<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> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</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"><a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><span class="p4">(<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></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E17"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="hermitan.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E4:92_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="hermitan.html#T16" target="_self">Th16</a>, <a class="ref" href="hermitan.html#T17" target="_self">Th17</a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_1_2"/><i><font color="Green">E131</font></i>: 
0 <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">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/><a NAME="E2:92_1_2"/><i><font color="Green">E132</font></i>: 
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#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> 1</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="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E3:92_1_2"/>
<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> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> 1</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="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">1 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 2</span>)</span> <a href="real_1.html#K4">*</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>
 <b>by </b><i><a class="txt" href="hermitan.html#E17"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="hermitan.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E4:92_1_2"/>
contradiction
 <b>by </b><i><a class="ref" href="hermitan.html#T16" target="_self">Th16</a>, <a class="ref" href="hermitan.html#T17" target="_self">Th17</a>, <a class="ref" href="xreal_1.html#T134">XREAL_1:134</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:92_1"/>
contradiction
 <b>by </b><i><a class="ref" href="hermitan.html#T14" target="_self">Th14</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_2"/><i><font color="Green">E133</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div><i><font color="Green">E134</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_2_1"/><i><font color="Green">E135</font></i>: 
<font color="Maroon">A</font> <a href="xxreal_0.html#R1">&lt;</a> 0
 ;<br/><a NAME="E2:92_2_1"/>
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#K5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> 0</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="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default">0 <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hermitan.html#T11" target="_self">Th11</a>, </i>;<br/><b>hence </b><a NAME="E3:92_2_1"/>
contradiction
 <b>by </b><i><a class="ref" href="hermitan.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_2_2"/><i><font color="Green">E136</font></i>: 
<font color="Maroon">A</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> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></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"><a href="hermitan.html#K4" target="_self">signnorm</a> <font color="Maroon">f</font>,<font color="Maroon">w</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K6">/</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> = 
<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">A</font> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default">2 <a href="real_1.html#K4">*</a> <span class="p4"><a href="complfld.html#K3">|.</a><span class="default"><span class="p5">(<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></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<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></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E17"><i><font color="Green">Lemma48</font></i></a>, <a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</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"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K4">*</a> 2</span>)</span> <a href="real_1.html#K4">*</a> <span class="p3"><a href="complfld.html#K3">|.</a><span class="default"><span class="p4">(<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></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><span class="p3">(<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></span>)</span>

<br/>.= 
<font color="Maroon">A</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="real_1.html#K3">+</a> <font color="Maroon">A</font></span>)</span>
<b>by </b><i><a class="ref" href="hermitan.html#T14" target="_self">Th14</a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
 <a href="real_1.html#K1">-</a> <font color="Maroon">A</font>

;<br/><a NAME="E3:92_2_2"/><b>then </b>
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="hermitan.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E4:92_2_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">A</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E5:92_2_2"/>
contradiction
 <b>by </b><i><a class="ref" href="hermitan.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:92_2"/>
contradiction
 <b>by </b><i><a class="ref" href="hermitan.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:92"/>
contradiction
 <b>by </b><i/>;<br/>


</div>
