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<b>let </b><font color="Maroon" title="c1">V</font> be   <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">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>set </b><font color="Maroon" title="c3">vq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c4">vr</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">qf</font> =  <a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">qhf</font> =  <a href="hermitan.html#K9" title="HERMITAN:func.9">ScalarForm</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>set </b><font color="Maroon" title="c7">K</font> =  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<b>thus </b><a NAME="E1:128"/>
 <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K9" title="HERMITAN:func.9">ScalarForm</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span>
  <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="128_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">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="E1:128_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K9" title="HERMITAN:func.9">ScalarForm</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:128_1"/><i><font color="Green" title="E67">A1</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K9" title="HERMITAN:func.9">ScalarForm</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">A</font>,<font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">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="128_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">B</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c10">B</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K9" title="HERMITAN:func.9">ScalarForm</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c11">w</font>

<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<b>by </b><i><a class="txt" href="hermitan.html#E3:128_1"><i><font color="Green" title="E67">A1</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:128_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E3:128_1"><i><font color="Green" title="E67">A1</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c8">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="E2:128"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c9">A</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:128"/><i><font color="Green" title="E67">A2</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">A</font> &amp; ( for <font color="Olive" title="b1">B</font> being  <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">A</font>,<font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">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="128_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">B</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c10">B</font> as   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#T59" target="_self" title="HERMITAN:th.59">Th59</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K9" title="HERMITAN:func.9">ScalarForm</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c10">B</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" title="HERMITAN:func.8">Q*Form</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c9">A</font>,<font color="Maroon" title="c11">w</font>

<br/>.= 
 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
<b>by </b><i><a class="txt" href="hermitan.html#E4:128"><i><font color="Green" title="E67">A2</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:128"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K12" title="BILINEAR:func.12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K9" title="HERMITAN:func.9">ScalarForm</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E4:128"><i><font color="Green" title="E67">A2</font></i></a></i>;<br/>


</div>
