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<b>let </b><font color="Maroon" title="c1">V</font> be   <a href="vectsp_1.html#NM7" title="VECTSP_1:NM.7">VectSp</a> of  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="hermitan.html#V5" title="HERMITAN:attr.5">diagReR+0valued</a> <a href="hermitan.html#NM3" title="HERMITAN:NM.3">hermitan-Form</a> of <font color="Maroon" title="c1">V</font>;<br/>



<b>thus </b><a NAME="E1:102"/>
 <a href="bilinear.html#K14" title="BILINEAR:func.14">diagker</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <font color="Maroon" title="c2">f</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:102_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K14" title="BILINEAR:func.14">diagker</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E3:102_1"/><i><font color="Green" title="E47">A1</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">w</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font>;<br/><a NAME="E1:102_1_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="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K8" title="REAL_1:func.8">*</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="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">w</font>,<font color="Maroon" title="c5">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 <b>by </b><i><a class="txt" href="hermitan.html#E3:102_1"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="hermitan.html#T49" target="_self" title="HERMITAN:th.49">Th49</a>, <a class="ref" href="complfld.html#T93" title="COMPLFLD:th.93">COMPLFLD:93</a>, <a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/><a NAME="E2:102_1_1"/><b>then </b>
<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="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><a NAME="E3:102_1_1"/><b>then </b>
<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="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">w</font></span>)</span></span><a href="complfld.html#K3" title="COMPLFLD:func.3">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T6" title="XCMPLX_1:th.6">XCMPLX_1:6</a></i>;<br/><b>hence </b><a NAME="E4:102_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="complfld.html#T94" title="COMPLFLD:th.94">COMPLFLD:94</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:102_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="hermitan.html#E3:102_1"><i><font color="Green" title="E47">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:102"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <font color="Maroon" title="c2">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:102"/><i><font color="Green" title="E47">A2</font></i>: 
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; ( for <font color="Olive" title="b1">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font> holds  <font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  ) )
 ;<br/>
<a NAME="E5:102"/>
<font color="Maroon" title="c2">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 
<b>by </b><i><a class="txt" href="hermitan.html#E4:102"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:102"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K14" title="BILINEAR:func.14">diagker</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="hermitan.html#E4:102"><i><font color="Green" title="E47">A2</font></i></a></i>;<br/>


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