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

<b>set </b><font color="Maroon" title="c4">K</font> =  <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<b>set </b><font color="Maroon" title="c5">L</font> =  <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font>;<br/>
<b>set </b><font color="Maroon" title="c6">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="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">Cv</font> =  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</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="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">aCv</font> =  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</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="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">mCv</font> =  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</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="c3">f</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c10">R</font> =  <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c11">wq</font> =  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c12">Cw</font> =  <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c13">aCw</font> =  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">mCw</font> =  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>;<br/>
<a NAME="E1:112_1_1"/><i><font color="Green" title="E53">A1</font></i>: 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</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="c3">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c3">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>;<br/>
<a NAME="E2:112_1_1"/><i><font color="Green" title="E54">A2</font></i>: 
 <a href="vectsp10.html#K2" title="VECTSP10:func.2">CosetSet</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">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="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>;<br/>
<a NAME="E3:112_1_1"/><i><font color="Green" title="E55">A3</font></i>: 
 <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="hermitan.html#T58" target="_self" title="HERMITAN:th.58">Th58</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b2">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span> holds <br/>$3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>;<br/>
<div><i><font color="Green" title="E56">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="c3">f</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c16">B</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c17">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="E2:112_1_1_1"/><i><font color="Green" title="E57">A5</font></i>: 
<font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>consider </b><font color="Maroon" title="c18">b</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E4:112_1_1_1"/><i><font color="Green" title="E58">A6</font></i>: 
<font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">b</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>take </b><font color="Maroon" title="c19">y</font> = <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">a</font>,<font color="Maroon" title="c18">b</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">a1</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font>;<br/><b>let </b><font color="Maroon" title="c21">b1</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font>;<br/><b>assume </b><a NAME="E1:112_1_1_1_1"/>
<font color="Maroon" title="c15">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 ;<br/><a NAME="E2:112_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">a1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E2:112_1_1_1"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="vectsp_4.html#T59" title="VECTSP_4:th.59">VECTSP_4:59</a></i>;<br/><b>then consider </b><font color="Maroon" title="c22">vv</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="E4:112_1_1_1_1"/><i><font color="Green" title="E59">A7</font></i>: 
( <font color="Maroon" title="c22">vv</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c22">vv</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T57" title="VECTSP_4:th.57">VECTSP_4:57</a></i>;<br/><b>assume </b><a NAME="E5:112_1_1_1_1"/>
<font color="Maroon" title="c16">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 ;<br/><a NAME="E6:112_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c21">b1</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E4:112_1_1_1"><i><font color="Green" title="E58">A6</font></i></a>, <a class="ref" href="vectsp_4.html#T59" title="VECTSP_4:th.59">VECTSP_4:59</a></i>;<br/><b>then consider </b><font color="Maroon" title="c23">ww</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E8:112_1_1_1_1"/><i><font color="Green" title="E60">A8</font></i>: 
( <font color="Maroon" title="c23">ww</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span> &amp; <font color="Maroon" title="c18">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">b1</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">ww</font> )
 <b>by </b><i><a class="ref" href="vectsp_4.html#T57" title="VECTSP_4:th.57">VECTSP_4:57</a></i>;<br/><a NAME="E9:112_1_1_1_1"/>
<font color="Maroon" title="c22">vv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E4:112_1_1_1_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E10:112_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c22">vv</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="c3">f</font>
 <b>by </b><i><a class="ref" href="bilinear.html#D19" title="BILINEAR:def.19">BILINEAR:def 19</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">aa</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="E12:112_1_1_1_1"/><i><font color="Green" title="E61">A9</font></i>: 
( <font color="Maroon" title="c24">aa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">vv</font> &amp; ( for <font color="Olive" title="b1">w0</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font> holds  <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c24">aa</font>,<font color="Olive" title="b1">w0</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="E13:112_1_1_1_1"/>
<font color="Maroon" title="c23">ww</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E8:112_1_1_1_1"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="struct_0.html#D5" title="STRUCT_0:def.5">STRUCT_0:def 5</a></i>;<br/><a NAME="E14:112_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c23">ww</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="bilinear.html#K13" title="BILINEAR:func.13">rightker</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="bilinear.html#D20" title="BILINEAR:def.20">BILINEAR:def 20</a></i>;<br/><b>then consider </b><font color="Maroon" title="c25">bb</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E16:112_1_1_1_1"/><i><font color="Green" title="E62">A10</font></i>: 
( <font color="Maroon" title="c25">bb</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">ww</font> &amp; ( for <font color="Olive" title="b1">v0</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="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">v0</font>,<font color="Maroon" title="c25">bb</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#E3:112_1_1"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/><b>thus </b><font color="Maroon" title="c19">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E4:112_1_1_1_1"><i><font color="Green" title="E59">A7</font></i></a>, <a class="txt" href="hermitan.html#E8:112_1_1_1_1"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="bilinear.html#T29" title="BILINEAR:th.29">BILINEAR:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><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> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E16:112_1_1_1_1"><i><font color="Green" title="E62">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><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> </span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E12:112_1_1_1_1"><i><font color="Green" title="E61">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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> </span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">vv</font>,<font color="Maroon" title="c23">ww</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><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> </span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E12:112_1_1_1_1"><i><font color="Green" title="E61">A9</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">a1</font>,<font color="Maroon" title="c21">b1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:112_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">A</font>,<font color="Maroon" title="c16">B</font>,<font color="Maroon" title="c19">y</font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">ff</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> <b> such that </b><br/><a NAME="E6:112_1_1"/><i><font color="Green" title="E57">A11</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="c3">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">B</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>,<font color="Maroon" title="c15">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font>]
 <b>from </b><i><a class="ref" href="binop_1.html#S3" title="BINOP_1:sch.3">BINOP_1:sch 3</a>(<a class="txt" href="hermitan.html#E4:112_1_1"><i><font color="Green" title="E56">A4</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c16">ff</font> = <font color="Maroon" title="c15">ff</font> as   <a href="bilinear.html#NM1" title="BILINEAR:NM.1">Form</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="c3">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> ;<br/>
<div><i><font color="Green" title="E58">A12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c18">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E2:112_1_1_2"/><i><font color="Green" title="E59">A13</font></i>: 
<font color="Maroon" title="c17">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>set </b><font color="Maroon" title="c19">ffw</font> =  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">A</font>, <font color="Maroon" title="c21">B</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="c3">f</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c22">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="E2:112_1_1_2_1"/><i><font color="Green" title="E60">A14</font></i>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">b</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="E4:112_1_1_2_1"/><i><font color="Green" title="E61">A15</font></i>: 
<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E5:112_1_1_2_1"/><i><font color="Green" title="E62">A16</font></i>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</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="c3">f</font></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</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="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>;<br/><a NAME="E6:112_1_1_2_1"/><i><font color="Green" title="E63">A17</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</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="c3">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E1:112_1_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_2_1"><i><font color="Green" title="E60">A14</font></i></a>, <a class="txt" href="hermitan.html#E4:112_1_1_2_1"><i><font color="Green" title="E61">A15</font></i></a>, <a class="ref" href="vectsp10.html#D3" title="VECTSP10:def.3">VECTSP10:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c21">B</font></span>)</span>,<font color="Maroon" title="c17">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span>,<font color="Maroon" title="c18">w</font>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_2"><i><font color="Green" title="E59">A13</font></i></a>, <a class="txt" href="hermitan.html#E5:112_1_1_2_1"><i><font color="Green" title="E62">A16</font></i></a>, <a class="txt" href="hermitan.html#E6:112_1_1_2_1"><i><font color="Green" title="E63">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c18">w</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c18">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T27" title="BILINEAR:th.27">BILINEAR:27</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c23">b</font>,<font color="Maroon" title="c18">w</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_2"><i><font color="Green" title="E59">A13</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_2_1"><i><font color="Green" title="E60">A14</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c21">B</font>,<font color="Maroon" title="c17">ww</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_2"><i><font color="Green" title="E59">A13</font></i></a>, <a class="txt" href="hermitan.html#E4:112_1_1_2_1"><i><font color="Green" title="E61">A15</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c21">B</font>,<font color="Maroon" title="c17">ww</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:112_1_1_2"/>
 <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font> is <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">additive</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E59">A18</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="c3">f</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c18">v</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="E2:112_1_1_3"/><i><font color="Green" title="E60">A19</font></i>: 
<font color="Maroon" title="c17">vv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>set </b><font color="Maroon" title="c19">ffv</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">A</font>, <font color="Maroon" title="c21">B</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c22">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E2:112_1_1_3_1"/><i><font color="Green" title="E61">A20</font></i>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>consider </b><font color="Maroon" title="c23">b</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E4:112_1_1_3_1"/><i><font color="Green" title="E62">A21</font></i>: 
<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">b</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E5:112_1_1_3_1"/><i><font color="Green" title="E63">A22</font></i>: 
the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>;<br/><a NAME="E6:112_1_1_3_1"/><i><font color="Green" title="E64">A23</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K3" title="VECTSP10:func.3">addCoset</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c21">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E2:112_1_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_3_1"><i><font color="Green" title="E61">A20</font></i></a>, <a class="txt" href="hermitan.html#E4:112_1_1_3_1"><i><font color="Green" title="E62">A21</font></i></a>, <a class="ref" href="vectsp10.html#D3" title="VECTSP10:def.3">VECTSP10:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">A</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c21">B</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<span class="p1">(<span class="default">the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c21">B</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c23">b</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_3"><i><font color="Green" title="E60">A19</font></i></a>, <a class="txt" href="hermitan.html#E5:112_1_1_3_1"><i><font color="Green" title="E63">A22</font></i></a>, <a class="txt" href="hermitan.html#E6:112_1_1_3_1"><i><font color="Green" title="E64">A23</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c22">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c23">b</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T28" title="BILINEAR:th.28">BILINEAR:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c23">b</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_3"><i><font color="Green" title="E60">A19</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_3_1"><i><font color="Green" title="E61">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c21">B</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_3"><i><font color="Green" title="E60">A19</font></i></a>, <a class="txt" href="hermitan.html#E4:112_1_1_3_1"><i><font color="Green" title="E62">A21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c21">B</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">B</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:112_1_1_3"/>
 <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font> is <a href="hahnban1.html#V1" title="HAHNBAN1:attr.1">additive</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D11" title="HAHNBAN1:def.11">HAHNBAN1:def 11</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E60">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">ww</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c18">w</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E2:112_1_1_4"/><i><font color="Green" title="E61">A25</font></i>: 
<font color="Maroon" title="c17">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>set </b><font color="Maroon" title="c19">ffw</font> =  <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="c3">f</font></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c21">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>consider </b><font color="Maroon" title="c22">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="E2:112_1_1_4_1"/><i><font color="Green" title="E62">A26</font></i>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E3:112_1_1_4_1"/><i><font color="Green" title="E63">A27</font></i>: 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</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="c3">f</font></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</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="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>;<br/><a NAME="E4:112_1_1_4_1"/><i><font color="Green" title="E64">A28</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</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="c3">f</font></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E1:112_1_1"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_4_1"><i><font color="Green" title="E62">A26</font></i></a>, <a class="ref" href="vectsp10.html#D5" title="VECTSP10:def.5">VECTSP10:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c1">V</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c20">A</font></span>)</span>,<font color="Maroon" title="c17">ww</font>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span>,<font color="Maroon" title="c18">w</font>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_4"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="hermitan.html#E3:112_1_1_4_1"><i><font color="Green" title="E63">A27</font></i></a>, <a class="txt" href="hermitan.html#E4:112_1_1_4_1"><i><font color="Green" title="E64">A28</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c22">a</font>,<font color="Maroon" title="c18">w</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T32" title="BILINEAR:th.32">BILINEAR:32</a></i>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c20">A</font>,<font color="Maroon" title="c17">ww</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_4"><i><font color="Green" title="E61">A25</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_4_1"><i><font color="Green" title="E62">A26</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c21">r</font> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T10" title="BILINEAR:th.10">BILINEAR:10</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:112_1_1_4"/>
 <a href="bilinear.html#K10" title="BILINEAR:func.10">FunctionalSAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">ww</font> is <a href="hahnban1.html#V2" title="HAHNBAN1:attr.2">homogeneous</a>
 <b>by </b><i><a class="ref" href="hahnban1.html#D12" title="HAHNBAN1:def.12">HAHNBAN1:def 12</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c17">vv</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="c3">f</font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon" title="c18">v</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="E2:112_1_1_5"/><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c17">vv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><b>set </b><font color="Maroon" title="c19">ffv</font> =  <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="112_1_1_5_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c20">A</font> be   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c21">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/><b>consider </b><font color="Maroon" title="c22">a</font> being   <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font><b> such that </b><br/><a NAME="E2:112_1_1_5_1"/><i><font color="Green" title="E62">A30</font></i>: 
<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">a</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#T23" title="VECTSP10:th.23">VECTSP10:23</a></i>;<br/><a NAME="E3:112_1_1_5_1"/><i><font color="Green" title="E63">A31</font></i>: 
the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="vectsp10.html#D6" title="VECTSP10:def.6">VECTSP10:def 6</a></i>;<br/><a NAME="E4:112_1_1_5_1"/><i><font color="Green" title="E64">A32</font></i>: 
<span class="p1">(<span class="default"><a href="vectsp10.html#K5" title="VECTSP10:func.5">lmultCoset</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c20">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="hermitan.html#E2:112_1_1"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_5_1"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="vectsp10.html#D5" title="VECTSP10:def.5">VECTSP10:def 5</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c20">A</font></span>)</span> = 
<font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<span class="p1">(<span class="default">the <a href="vectsp_1.html#U1" title="VECTSP_1:sel.1">lmult</a> of <span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</font>,<span class="p3">(<span class="default"><a href="bilinear.html#K16" title="BILINEAR:func.16">RKer</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c21">r</font>,<font color="Maroon" title="c20">A</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="vectsp_1.html#K4" title="VECTSP_1:func.4">*</a> <font color="Maroon" title="c22">a</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_5"><i><font color="Green" title="E61">A29</font></i></a>, <a class="txt" href="hermitan.html#E3:112_1_1_5_1"><i><font color="Green" title="E63">A31</font></i></a>, <a class="txt" href="hermitan.html#E4:112_1_1_5_1"><i><font color="Green" title="E64">A32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c18">v</font>,<font color="Maroon" title="c22">a</font></span>)</span>
<b>by </b><i><a class="ref" href="hermitan.html#T30" target="_self" title="HERMITAN:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c17">vv</font>,<font color="Maroon" title="c20">A</font></span>)</span>
<b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_5"><i><font color="Green" title="E61">A29</font></i></a>, <a class="txt" href="hermitan.html#E2:112_1_1_5_1"><i><font color="Green" title="E62">A30</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">r</font> <a href="complfld.html#K2" title="COMPLFLD:func.2">*'</a> </span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">A</font></span>)</span>
<b>by </b><i><a class="ref" href="bilinear.html#T9" title="BILINEAR:th.9">BILINEAR:9</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:112_1_1_5"/>
 <a href="bilinear.html#K9" title="BILINEAR:func.9">FunctionalFAF</a> <font color="Maroon" title="c16">ff</font>,<font color="Maroon" title="c17">vv</font> is <a href="hermitan.html#V1" title="HERMITAN:attr.1">cmplxhomogeneous</a>
 <b>by </b><i><a class="ref" href="hermitan.html#D1" target="_self" title="HERMITAN:def.1">Def1</a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c17">ff</font> = <font color="Maroon" title="c16">ff</font> as   <a href="hermitan.html#NM2" title="HERMITAN:NM.2">sesquilinear-Form</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="c3">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="hermitan.html#E8:112_1_1"><i><font color="Green" title="E58">A12</font></i></a>, <a class="txt" href="hermitan.html#E9:112_1_1"><i><font color="Green" title="E59">A18</font></i></a>, <a class="txt" href="hermitan.html#E10:112_1_1"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="hermitan.html#D4" target="_self" title="HERMITAN:def.4">Def4</a>, <a class="ref" href="bilinear.html#D12" title="BILINEAR:def.12">BILINEAR:def 12</a>, <a class="ref" href="bilinear.html#D13" title="BILINEAR:def.13">BILINEAR:def 13</a>, <a class="ref" href="bilinear.html#D15" title="BILINEAR:def.15">BILINEAR:def 15</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c17">ff</font>
;<br/>

<b>thus </b><a NAME="E13:112_1_1"/>
for <font color="Olive" title="b1">A</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">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="c3">f</font></span>)</span></span>)</span><br/> for <font color="Olive" title="b2">B</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> <font color="Maroon" title="c2">W</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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span></span>)</span><br/> for <font color="Olive" title="b3">v</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c1">V</font><br/> for <font color="Olive" title="b4">w</font> being  <a href="vectsp_1.html#NM6" title="VECTSP_1:NM.6">Vector</a> of <font color="Maroon" title="c2">W</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <span class="p1">(<span class="default"><a href="bilinear.html#K15" title="BILINEAR:func.15">LKer</a> <font color="Maroon" title="c3">f</font></span>)</span> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">w</font> <a href="vectsp_4.html#K3" title="VECTSP_4:func.3">+</a> <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="c3">f</font> <a href="hermitan.html#K3" title="HERMITAN:func.3">*'</a> </span>)</span></span>)</span> holds <br/><font color="Maroon" title="c17">ff</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b4">w</font>
 <b>by </b><i><a class="txt" href="hermitan.html#E6:112_1_1"><i><font color="Green" title="E57">A11</font></i></a></i>;<br/>


</div>
