<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">K</font> =  <a href="complfld.html#K1">F_Complex</a> ;<br/>
<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> , <font color="Maroon">W</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">f</font> be   <a href="hermitan.html#NM2">sesquilinear-Form</a> of <font color="Maroon">V</font>,<font color="Maroon">W</font>;<br/>



<b>set </b><font color="Maroon">vq</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">wq</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">wqr</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">vql</font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><a href="bilinear.html#K15">LKer</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">rlf</font> =  <a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">qf</font> =  <a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font>;<br/>
<b>set </b><font color="Maroon">lrf</font> =  <a href="bilinear.html#K17">LQForm</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:120"/>
 <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:120_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:120_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:120"/>
 <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:120_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:120_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E3:120"/>
 <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:120_3"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:120_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E4:120"/>
 <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_4"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:120_4"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">w</font> = <font color="Maroon">ww</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">w</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <span class="p2">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:120_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E5:120"/>
 <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_5"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:120_5"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#D2" target="_self">Def2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:120_5"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E6:120"/>
 <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_6"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">vv</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:120_6"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">vv</font> &amp; ( for <font color="Olive">ww</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Olive">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#D2" target="_self">Def2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">ww</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:120_6"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K12">leftker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E7:120"/>
 <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:120_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:120_7"/><i><font color="Green">E129</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:120_7"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/>


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

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

<b>assume </b><a NAME="E8:120"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">ww</font> being   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">W</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E10:120"/><i><font color="Green">E130</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ww</font> &amp; ( for <font color="Olive">vv</font> being  <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> holds  <span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">vv</font>,<font color="Maroon">ww</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a>  ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="120_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">vv</font> be   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <font color="Maroon">f</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">vv</font> as   <a href="vectsp_1.html#NM5">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><a href="bilinear.html#K15">LKer</a> <span class="p3">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="hermitan.html#D2" target="_self">Def2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">vv</font>,<font color="Maroon">ww</font> = 
<span class="p1">(<span class="default"><a href="bilinear.html#K17">LQForm</a> <span class="p2">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">v</font>,<font color="Maroon">ww</font>
<b>by </b><i/>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:120"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K8" target="_self">Q*Form</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
