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

<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="complfld.html#K1">F_Complex</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="bilinear.html#K16">RKer</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span>;<br/>
<a NAME="E1:117_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="bilinear.html#K13">rightker</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="hermitan.html#T58" target="_self">Th58</a></i>;<br/>
<b>thus </b><a NAME="E2:117_1_1"/>
 <a href="bilinear.html#K13">rightker</a> <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>,<a class="ref" href="bilinear.html#D25">BILINEAR:def 25</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:117_1_1_1"/>
<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>9</sub></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">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:117_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>9</sub></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>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E5:117_1_1_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23">VECTSP10:23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="117_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="hermitan.html#E5:117_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="hermitan.html#T62" target="_self">Th62</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
<b>by </b><i><a class="txt" href="hermitan.html#E3:117_1_1_1"><i><font color="Green">E61</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E7:117_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="bilinear.html#K13">rightker</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E8:117_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hermitan.html#E1:117_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="bilinear.html#D20">BILINEAR:def 20</a></i>;<br/>
<a NAME="E9:117_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="rlvect_1.html#R1">in</a>  <a href="bilinear.html#K16">RKer</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#D1">RLVECT_1:def 1</a></i>;<br/>
<b>then </b><font color="Maroon">c<sub>10</sub></font> <a href="vectsp_4.html#K3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span> = 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_4.html#T64">VECTSP_4:64</a></i>
<br/>.= 
 <a href="vectsp10.html#K4">zeroCoset</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span>

<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp10.html#T22">VECTSP10:22</a></i>
;<br/>
<b>hence </b><a NAME="E11:117_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="hermitan.html#E3:117_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="hermitan.html#E5:117_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></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="E3:117_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E4:117_1_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:117_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="vectsp_1.html#NM5">Vector</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><a href="hermitan.html#K7" target="_self">RQ*Form</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p2">(<span class="default"><a href="vectsp10.html#K6">VectQuot</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16">RKer</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="hermitan.html#K3" target="_self">*'</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 
<b>by </b><i><a class="ref" href="bilinear.html#T30">BILINEAR:30</a></i>;<br/>
<b>hence </b><a NAME="E6:117_1_1"/>
<font color="Maroon">c<sub>8</sub></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> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E4:117_1_1"><i><font color="Green">E61</font></i></a></i>;<br/>


</div>
